Research ArticleImage Encryption Using the Chaotic Josephus Matrix

Gelan Yang1 Huixia Jin1 and Na Bai2

1 Department of Computer Science Hunan City University Yiyang Hunan 413000 China2 School of Electronics and Information Engineering Anhui University Hefei Anhui 230039 China

Correspondence should be addressed to Na Bai realbaingmailcom

Received 3 October 2013 Accepted 14 January 2014 Published 6 March 2014

Academic Editor Yue Wu

Copyright copy 2014 Gelan Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a new image encryption solution using the chaotic Josephus matrix It extends the conventional Josephustraversing to a matrix form and proposes a treatment to improve the randomness of this matrix by mixing chaotic maps It alsoderives the corresponding encryption primitives controlled by the chaotic Josephus matrix In this way it builds up an imageencryption system with very high sensitivities in both encryption key and input image Our simulation results demonstrate thatan encrypted image of using this method is very random-like that is a uniform-like pixel histogram and very low correlations inadjacent pixels The design idea of this method is also applicable to data encryption of other types like audio and video

1 Introduction

Encryption is the process of transforming information(referred to as plaintext) using an algorithm (referred to asa cipher) to make the encrypted information (referred toas ciphertext) unreadable to anyone except those authorizedusers with special knowledge (referred to as a key) [1]Encryption has been long used by militaries and govern-ments to facilitate secret communications Since the digitalrevolution in the 1980s the demands of digital encryption invarious applications have quickly increased because digitalstorage and communication are widely used Encryption isnow commonly used in protecting information within manytypes of civilian systems like personal emails and patientdocuments

Digital image is a major data type of two dimensionsAlthough a digital image can be extracted in order andbecomes a one-dimensional data its distinctive character-istics make conventional ciphers developed for one dimen-sional data unsuitable [2] for example those based on DataEncryption Standard (DES) [3] and Advanced EncryptionStandard (AES) [4] As a result digital image encryption hasbecome an attractive research area in the past decade [5ndash11]

The chaotic map is considered a wise choice for dataencryption because of its ergodicity mixing property high

sensitivity to the initial conditions high deterministic prop-erties high unpredictable random behaviors and so forth[2 5 9 12ndash15] However the chaotic encryption methodis criticized for its vulnerability to attacks via certain basinstructures [16] its low efficiency for encrypting the wholeimage [2] its deteriorated randomness property from its useof the finite precision with fixed-point arithmetic [17] andnonuniform distribution of the chaos sequence [18]

In computer science [19] and mathematics [20] theJosephus problem is a theoretical problem related to a certaincounting-out game If the counting-out order is recordedas a sequence a Josephus traversing is obtained because allelements in the game are traversed without repetition TheJosephus traversing has already been used in data encryptionfield for years The Josephus traversing is simple to realizeand fast to compute but previous attempts focused more orless on the scrambling purposes [21 22]However scramblingbased encryption is vulnerable to statistical attack ciphertext-only attack and known plaintext attack [23] because it neverchanges pixel values

In order to achieve higher security level many recentefforts adopt the hybrid idea to use one encryption systemto suppress disadvantages of another system while keepingadvantages unchanged For example [24] incorporates thechaotic map to DES [25 26] combine the chaotic map into

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 632060 13 pageshttpdxdoiorg1011552014632060

2 Mathematical Problems in Engineering

Input 119905 the initial total number of persons in a circle119904 the starting position in the circle119899 the counting period

Output 119902120587 the Josephus permutation sequence according to parameter set (119905 119904 119899)

count = 0 done = 0 pos = 119904 label = zeros(1 119905) 119902120587= [] initial settings

while (simdone) main looptodo = label(pos)if (todo == 0) if this person has not been taken out

count = count + 1if (count == 119899) if this is the 119899th person

119902120587(end + 1) = pos count = 0 label(pos) = 1

if (length(119902120587) == 119905)

done = 1end

endendpos = pos + 1if (pos gt 119905)

pos = 1end

end

Algorithm 1 The generation of a Josephus permutation sequence

conventional transform domain encryption [27 28] addchaotic map to Sudoku puzzles for encryption In this paperwe develop a new image encryption method by combin-ing the ideas of the chaotic map the Josephus traversingsequence and conventional substitution and transpositionciphersThe remainder of the paper is organized as follows inSection 2 the Josephus permutation and the Logistic chaoticmap are briefly reviewed in Section 3 the CJPM and itsgenerator are given in Section 4 the proposed image encryp-tion method based on CJPM is fully discussed including itsflowchart and functions for each part in Section 5 simulationresults are shown and various security analyses are applied inSection 6 the paper is concluded

2 Preliminary

21 Josephus Permutation The Josephus permutation orJosephus problem is well known in computer science andmathematics It is named after Flavius Josephus a Jewishhistorian lived in the 1st century It is a theoretical problemrelated to a certain counting-out game that works by having 119905people standing in a circle with consecutive tags from 1 to119905 Starting at predetermined person you count around thecircle Once you reach the 119899th person take them out of thecircle and have the members to close the circle Then repeatthe process until only one person is left That person winsthe game If we record the tags of people who have beentaken out at each round as a sequence then this sequenceis a permutation of a natural number sequence and is calledJosephus permutation sequence

It is clear that three parameters are involved in the Jose-phus problem namely the initial total number of persons ina circle 119905 the starting position in the circle 119904 and the countingperiod 119899 Therefore a Josephus permutation sequence 119902

120587

can be denoted as follows where 119869 denotes the Josephuspermutation according to the set of parameters 119905 119904 and 119899A Josephus permutation sequence can be easily implementedby linked lists and dynamic arrays Algorithm 1 describes a

119902120587= 119869 (119905 119904 119899) (1)

For example

119902120587= 119869 (18 1 4)

= [4 8 12 16 2 7 13 18 6 14 3 11 5 17 15 1 10 9]

(2)

119902120587= 119869 (18 1 7)

= [7 14 3 11 1 10 2 13 6 18 16 15 17 5 12 4 8 9]

(3)

119902120587= 119869 (18 4 7)

= [10 17 6 14 4 13 5 16 9 3 1 18 2 8 15 7 11 12]

(4)

It is clear that (1) compared with the length 119905 natural numbersequence a Josephus permutation sequence 119902

120587changes a lot

especially considering that none of the twoneighbor numbersis consecutive (2) for a fixed length 119905 different pairs of (119904 119899)give distinct Josephus permutation sequences However it isweak in that (1) 119902

120587rsquos very first several elements divulge the

parameter of counting period 119899 (2) the difference betweentwo 119902

120587s may disclose the difference between their starting

positions for example the difference between the two firstelements of 119902

120587s in (3) and (4) is 3 which is the difference

of their parameters of starting positions Therefore it is notcompletely random for a Josephus permutation sequence

Mathematical Problems in Engineering 3

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1Logistic map (r = 36)

x[i]

x[i + 1]

x[i + 2]

x[i + 3]

y = x

(a)

0

01

02

03

04

05

06

07

08

09

1

2 25 3 35 4

Bifurcation diagram of the logistic map

(b)

Figure 1 The Logistic map

22 Chaotic Logistic Map The Logistic map is a polynomialmapping of degree two It was introduced by the biologistRobet May in 1976 [31] The Logistic map is written as (5)where symbol L is used to denote the Logistic map 119883

119894isin

[0 1] and represents the population at year 119894 and hence 1198830

represents the initial population at year 0 119903 is a positivenumber and represents the combined rate for reproductionand starvation [32] This map is often cited as an example ofhow complex chaotic behaviors can arise from a very simplenonlinear dynamic equation Consider

119883119894+1=L (119883

119894) = 119903119883

119894(1 minus 119883

119894) (5)

The Logistic map has been well studied The plots of thefirst few iterations of the Logistic map and its bifurcationdiagram are shown in Figure 1 It is well known that when119903 isin [357 4] (approximately) the Logistic map has chaoticbehaviors for most values but there are still certain isolatedranges of 119903 that show nonchaotic behavior for example119903 asymp 383 which corresponds to a big gap in its bifurcationdiagram

In reality the Logistic sequence is controlled by a set ofparameters of (119883

0 119903119873119898) where (119883

0 119903) are parameters in

the Logistic map119873 is the length of sequence and119898 denotesthe number of thrown-away samples Therefore a Logisticsequence 119883 can be denoted as (6) Based on this logisticsequence119883 of length119873 sorted sequence1198831015840 can be obtainedby sorting 119883 in the ascending order It is certain that 1198831015840 is apermutation of the original119883Therefore119883 and1198831015840 satisfy (7)

where 119901120587is a permutation mapping sequence and 119894 denotes

sequence element index Consider

119883 =L (1198830 119903119873119898) (6)

1198831015840

119894= 119883119901120587(119894)

(7)

119883 = [04000 09120 03050 08055 05954 09154

02943 07892 06322 08836 03908

09047 03277 08372 05179 09488

01846 05721]

(8)

1198831015840

= [01846 02943 03050 03277 03908 04000

05179 05721 05954 06322 07892

08055 08372 08836 09047 09120

09154 09488]

(9)

119901120587= 119871 (04 38 18 0)

= [17 7 3 13 11 1 15 18 5 9 8 4 14 10 12 2 6 16]

(10)

For example if (1198830 119903 119873119898) = (04 38 18 0) then 119883

and 1198831015840 are shown in (8) and (9) respectively Correspond-ingly the permutation sequence 119901

120587is determined as (10)

4 Mathematical Problems in Engineering

12

3

6

9

1

2

4

57

8

10

11

(a) Conventional Josephusproblem

P120587(3)

P120587(6)

P120587(9)

P120587(1)

P120587(2)

P120587(4)

P120587(5)P120587(7)

P120587(8)

P120587(10)

P120587(11)P120587(12)

(b) Chaotic Josephus problem

Figure 2 The chaotic Josephus problem

CJPS at length RCCJPM at size of R-by-C

Rearrangement

Figure 3 Rearranging a CJPS to a CJPM

3 Chaotic Josephus PermutationMatrix (CJPM)

31 Chaotic Josephus Permutation Sequence (CJPS) Fromprevious sections it is clear that the Josephus permutationsequence 119902

120587and the chaotic permutation sequence 119901

120587are

obtained from different mechanismsThe Josephus permuta-tion sequence 119902

120587is easy to obtain but not completely random-

like while the chaotic permutation sequence 119901120587is random-

like but requires a large amount of computations for main-taining accuracy It is desirable that a permutation sequencebe random-like and only costs moderate computations

In order to achieve the above objective the new Jose-phus permutation sequence based on a chaotic permutationsequence 119901

120587is defined whose initial positions on the circle

are not consecutive numbers like those in the conventionalJosephus problem This new problem can be restated as fol-lows (1) 119905 people with numbered tags stand in a circle (thesenumbers together form a permutation sequence accordingto sorting a chaotic sequence) (2) starting at predeterminedperson you count around the circle until you reach the 119899thperson take him out of the circle and have the members toclose the circle and record his tag number (3) repeat theprocess until only one person is left If we record the tags ofpeople who have been taken out of the circle as a sequencethen a chaotic Josephus permutation sequence is obtained

It is noticeable that the chaotic Josephus problem hasparameters for both the chaotic Logistic map and for theconventional Josephus problem In other words a chaoticJosephus permutation sequence 119888119902

120587is determined by (11)

where parameters 119905 119904 119899 have the same meanings as in (1)1198830

and 119903 are parameters in the Logistic map and 119898 determines

the number of thrown-away samples in the chaotic Logisticsequence Consider

119888119902120587= 119888119869 (119905 119904 119899 119883

0 119903 119898) (11)

In the conventional Josephus problem (controlled byparameters (119905 119904 119899) see (1)) a natural number sequence 1 to119905 is used to denote the tags for people standing in the circlewhile a permutated sequence 119901

120587(controlled by parameters

(1198830 119903119873119898) see (6)) is used in the chaotic Josephus problem

as the tags In order to match the sequence lengths of 119901120587

and 119902120587 119905 = 119873 is the condition that has to be satisfied

The conventional Josephus problem and the chaotic Josephusproblem for 119905 = 119873 = 12 are illustrated in Figure 2

Therefore the new Josephus permutation sequence 119888119902120587

can be denoted as a composed function of a conventionalJosephus permutation sequence 119902

120587and a chaotic permutation

sequence 119901120587

119888119902120587= 119901120587∘ 119902120587= 119901120587(119902120587) (12)

For example if 119902120587

= 119869(18 1 4) in (2) and119901120587

= 119871(04 38 18 0) in (10) are preknown 119888119902120587

=

119888119869(18 1 4 04 38 0) is obtained as (13) shows by using(12) Similarly (14)ndash(16) can be also obtained Consider

119888119902120587= 119888119869 (18 1 4 04 38 0)

= [13 18 4 2 7 15 14 16 1 10 3 8 11 6 12 17 9 5]

(13)

119888119902120587= 119888119869 (18 1 7 04 38 0)

= [15 10 3 8 17 9 7 14 1 16 2 12 6 11 4 13 18 5]

(14)

119888119902120587= 119888119869 (18 4 7 04 38 0)

= [9 6 1 10 13 14 11 2 5 3 17 16 7 18 12 15 8 4]

(15)

119888119902120587= 119888119869 (18 4 7 040001 38 0)

= [8 6 1 10 11 12 13 2 9 15 7 18 3 5 14 17 4 16]

(16)

Compared to the previous conventional Josephus permu-tation sequence (see (2)ndash(4)) the chaotic Josephus permuta-tion sequences (see (12)ndash(15)) are more random-like because(1) the difference between its very first elements is not relatedto its counting period anymore (2) the difference betweentwo 119888119902

120587s which are only different in their starting positions

is no longer equal to the difference of their starting positions(3) slight perturbations in chaotic map parameters lead tobig changes in resulting 119888119902

120587s (4) two neighbor elements in

119888119902120587may or may not be consecutive Therefore the chaotic

Josephus permutation sequence ismore random-like than theconventional Josephus permutation sequence

32 Chaotic Josephus Permutation Matrix (CJPM) Based onchaotic Josephus permutation sequence(s) a chaotic Jose-phus permutation matrix can be generated via various ways

Mathematical Problems in Engineering 5

1

2

3

4

5

6

7

8

9

10

20

30

40

50

60

2 4 6 8 0

(a) CJPM(119877 119862 1 4 1198830 119903 0)

2

3

4

5

6

7

8

9

2 4 6 81 0

10

20

30

40

50

60

(b) CJPM(119877 119862 1 7 1198830 119903 0)

2

3

4

5

6

7

8

9

2 4 6 81 0

10

20

30

40

50

60

(c) CJPM(119877 119862 4 7 1198830 119903 0)

1

2

3

4

5

6

7

8

9

2 4 6 80

10

20

30

40

50

60

(d) CJPM(119877 119862 1 4 1198830+ 119890 119903 0)

1

2

3

4

5

6

7

8

9

2 4 6 80

10

20

30

40

50

60

(e) CJPM(119877 119862 1 4 1198830+ 119890 119903 10)

2 4 6 81

2

3

4

5

6

7

8

9

0

10

20

30

40

50

60

(f) CJPM(119877 119862 1 4 1198830+ 119890 119903 + 119890 10)

Figure 4 Parametric CJPMs (Note 119877 = 8 119862 = 81198830= 04 119903 = 38 and 119890 = 00001)

PlaintextP

Initial key

CiphertextC

CJPMgenerator

Encryption key

M

One blockPixel

permutationPixel

substitutionS

Figure 5 Image encryption method based on CJPM

Among these methods Algorithm 2 illustrates a straightfor-ward method to obtain a CJPM via a CJPS by rearrangingCJPS elements to a matrix

It is noticeable that Algorithm 2 is equivalent to rearrang-ing a sequence of elements into a matrix following the orderillustrated in Figure 3

Therefore a CJPM is determined by the same set ofparameters controlling a CJPS In order to emphasize thematrix property the parameter 119905 in CJPS is replaced by twoparameters of height 119877 and width 119862 where 119905 = 119877119862 Similarlya CJPM is uniquely determined by a set of parameters(119877 119862 119904 119899 119883

0 119903 119898) as for a CJPS Mathematically this claim

can be denoted as

119872 = CJPM (119877 119862 119904 119899 1198830 119903 119898) (17)

tN

xr yr w

p = L(X0 s n)

cq = cJ(t s n X0 r m)

Initial key

q = J(t s n)

R C s nr m

X0

Figure 6 Key functions in CJPM

Figure 4 illustrates various CJPMs via Algorithm 2according to different parameter sets It is clear that CJPMis very sensitive to its parameters and small changes in theparameter set lead to distinct CJPMs

4 Image Encryption AlgorithmBased on CJPM

In 1949 Claude Shannon the father of ldquoInformationTheoryrdquoproposed that confusion and diffusion are two properties ofthe operation of a secure cipher where the term confusionrefers to making the relationship between the encryptionkey and the ciphertext a very complex and developed one

6 Mathematical Problems in Engineering

Center of gravity

PlaintextP

Reference

CJPM M

Initial key Encryption key

point (xr yr)R C s t r m

X0 = sin(arctan(ygxg))(xg yg)

Figure 7 The internal structure of CJPM generator

PlaintextP

M +M998400

P998400

SMod(P998400 F)Mod(MF)

Figure 8 The internal structure of CJPM generator

[33] and the term diffusion refers to the property that theredundancy in the statistics of the plaintext is ldquodissipatedrdquo inthe statistics of the ciphertext [33] In otherwords for a securecipher it has to have good confusion and diffusion properties(1) different ciphertexts are desired to have similar statistics(2) any slight change in a plaintext is desired to lead to bigdifference in its ciphertext The image encryption algorithmbased on CJPM is proposed in this section to meet these twocriteria

41 Flowchart of Image Encryption Algorithm Based on CJPMSince the CJPM is parametric and random-like it can beused for image encryption directly However considering therequirements from confusion and diffusion properties theencryption procedure can be described as Figure 5 showsThe plaintext image is first sent to the CJPM generator whichis a preparation stage for generating a CJPM119872 for future useLater this CJPM119872 is used as a reference matrix to permuteand substitute image pixels for each image block in the stagesof pixel permutation and pixel substitution respectively Thedecryption procedure is simply to reverse the encryptionprocedure

42 Key Schedule It is clear that the CJPM is the coreof the cipher and thus key is related to the used CJPMreference matrix 119872 Initial key is composed of parameters(119909119903 119910119903 119908 119903 119898 119877 119862 119904 119899) where (119909

119903 119910119903 119908) is used in CJPM

generator for obtaining plaintext-dependent parameter 1198830

used in the Logistic map (119877 119862) are used as the parameter 119905in (1) and the parameter119873 in (6) The functions of each partof the initial key are shown in Figure 6

The output encryption key is composed of (119877 119862

119904 119899 1198830 119903 119898) all of which are directly required for deter-

mining a CJPM according to (17) Among these parameters119877 119862 119904 119899 and 119898 are restricted to integers 119909

119903 119910119903 119903 and 119883

0

are decimals More specifically 119877 and 119862 should be positive

integers smaller than the plaintext image size 119904 and 119899 shouldbe positive integers below the product of 119877119862 119903 should be anumber in between [36 4] (119909

119903 119910119903) is an arbitrary point on

119909119910 plain with weight 119908 and119898 is a nonnegative integer

43 CJPM Generator In order to enhance the resistance todifferential attacks the CJPM generator used in Figure 5 isdesigned to be plaintext dependent Recall that a CJPM isdetermined by a set of parameters (119877 119862 119904 119899 119883

0 119903 119898) shown

in (17) In the CJPM generator for image encryption only theparameter119883

0is not directly given by the initial key but by the

plaintext and a reference point (119909119903 119910119903) controlling the weight

in calculating the center of gravity Once 1198830is generated

it is stored in the encryption key The whole procedure oftranslating the initial key to a plaintext-dependent CJPMmatrix119872 and encryption key is shown in Figure 7

A plaintext is considered as an object of pixels where itsupper-left corner pixel is the reference point located at (1 1)Correspondingly pixels next to it along 119909 and 119910 directionsare (2 1) and (1 2) respectively The center of gravity ofthis plaintext is calculated via (18) where 119875

119894denotes the

119894th pixel intensity value and 119909119894and 119910

119894denote the location

of the 119894th pixel in the image with respect to the upper-leftcorner Once the center of gravity (119909

119892 119910119892) is obtained the

initial value of Logistic map 1198830is also determined via (19)

where arctan(sdot) is the arc tangent function and sin(sdot) is thesine function It is easy to verify that the range of (19) is[0 1] which satisfies restrictions for the initial value119883

0in the

Logisticmap Finally all required parameters for aCJPM thatis (119877 119862 119904 119899 119883

0 119903 119898) are obtained and thus a CJPM 119872 is

generated Meanwhile the used parameters are stored as theencryption key which can be used in the decryption processConsider

119909119892=119909119903sdot 119908 + sum119909

119894119875119894

119908 + sum119875119894

119910119892=119910119903sdot 119908 + sum119910

119894119875119894

119908 + sum119875119894

(18)

1198830= 05 [sin(arctan(

119910119892

119909119892

)) + 1] (19)

It is worth noting that the plaintext-dependent CJPMgenerator guarantees that the proposed cipher has gooddiffusion property any slight changes in plaintext lead tobig difference in ciphertext This is because the resulting

Mathematical Problems in Engineering 7

(a) Original image (b) 32-by-32 (c) 64-by-64 (d) 128-by-128

0100200300400500600700800900

1000

0 50 100 150 200 250

(e) Histogram of (a)

0

100

200

300

400

500

600

0 50 100 150 200 250

(f) Histogram of (b)

0

100

200

300

400

500

600

0 50 100 150 200 250

(g) Histogram of (c)

0

100

200

300

400

500

600

0 50 100 150 200 250

(h) Histogram of (d)

Figure 9 Pixel substitution results for CJPM at various sizes

(a) Original image (b) 32-by-32 (c) 64-by-32 (d) 64-by-64

(e) 64-by-128 (f) 128-by-128 (g) 256-by-128 (h) 256-by-256

Figure 10 Pixel permutation results for CJPM at various sizes

CJPM matrix119872 is dependent on the parameter 1198830and the

parameter 1198830is dependent on the center of gravity for the

plaintext while the center of the plaintext gravity alters forany slight change in plaintext Furthermore this 119883

0is the

parameter in the chaotic map and thus any slight change

in initial value leads to a completely different trajectory asthe bifurcation diagram in Figure 1 shows Consequentlya completely different CJPM is obtained as the referencematrix Eventually this new reference matrix leads to adistinct ciphertext It can be demonstrated that without

8 Mathematical Problems in Engineering

Input 119888119902120587 a CJPS at length 119905 = 119877 times 119862

Output119872 a CJPM of size 119877-by-119862for 119894 = 1 119877

for 119895 = 1 119862119872(119894 119895) = 119888119902

120587((119894 minus 1) 119877 + 119895)

endend

Algorithm 2 Reshape a CJPS into a CJPM

preknowing the reference pointrsquos coordinate (119909119903 119910119903) and its

weight 119908 there is no way to slightly change plaintext imagepixels so that its gravity center is unchanged

44 Pixel Substitution Pixel substitution refers to the pro-cess of changing pixel values From the point of viewof statistics this process is to change the statistics of aplaintext image so that the statistics of resulting ciphertextimage is completely different Moreover it is desired thatdifferent ciphertext images have similar statistics whichimplies that ciphertext images tell little information aboutkeys and plaintext images As a result the confusion propertyis achieved

The proposed pixel substitution block is shown inFigure 8 where symbol 119865 = 2

119887 and 119887 denotes the numberof bits supported by the format of the plaintext image Forexample if plaintext 119875 is an 8-bit gray image then 119865 = 28 =256 if plaintext119875 is a binary image then 119865 = 21 = 2 It can benoticed that the reference CJPM119872 is first to convert to1198721015840whose format is compatible with the format of the plaintextimage later1198721015840 and 119875 are added over the space of 119865 finallythe encrypted image 119878 is obtained

Because elements in a CJPM matrix 119872 are uniformlydistributed on integer set [1 119905] after ldquoModrdquo operation 1198721015840still has a uniform-like distribution for its elements on[0 119865] As a result when this 1198721015840 is used to randomly shiftthe pixel value in plaintext the resulting pixel value inciphertext has an equal opportunity to be any value on [0 119865]As a result a uniform-like histogram is achieved in theciphertext

Pixel substitution results based on CJPMs are shown inFigure 9 It is clear that histograms before and after pixelsubstitution are very different and ciphertext histograms arevery flat compared to plaintext ones It is also noticeable thatas the size of the referenceCJPM increases the ciphertext has abetter encryption quality from the point view of human visualinspection

45 Pixel Permutation Pixel permutation refers to the pro-cess scrambling the positions of pixels in plaintext to disguiseinformation contained in an image Denote an image beforeand after pixel permutation as 119861 and 119860 respectively Assumethe way of indexing image pixels is the same as the order torearrange elements in a CJPM as Figure 3 shows Then thepixel permutation process can be mathematically defined asa permutation 119891

120587between domain 119861 and range 119860 forall119894 119895 isin

1 2 119905 exist119860119894= 119861119895= 119861119891120587(119894) where 119905 is the total

number of pixels in the image As we mentioned in previoussections a CJPM is generated from a CJPS which is apermutation sequence Therefore a CJPM can be directlyused for pixel permutation that is given a CJPM 119872 and itspixel permutation can be defined as 119891

120587(119894) = 119872(119894)

For example Figure 10 illustrates pixel permutationresults of the ldquoTuftsrdquo logo image for different CJPMsIt is clear that pixels are well shuffled within the imageblock As long as the size of CJPMprocessing imageblock increases the resulting shuffled image looksbetter and better When the block size reaches to orover 128-by-128 pixels in the plaintext are almost evenlyshuffled

It is clear that images after pixel permutation look verydifferent from the plaintext image It is also worth notingthat a CJPM also depends on a set of parameters besides thesize and that any change in other parameters will lead to acompletely different permutated image

5 Simulation Results and Security Analysis

An excellent encryption method should be both robustand effective Robustness means that the cipher should beapplicable to any plaintext image written in a supportedformat Effectiveness implies that the cipher is able to generateeligible ciphertext images which hide information frompossible intruders

In this section we focus on discussing the performanceof the CJPM based image cipher described in Section 4 It isworth noting that all following computer simulations are rununder MATLAB 2010a and Windows XP environment withCore 2 Quad 26GHz processors

51 Histogram Analysis Histogram analysis is one of themost straightforward evaluations for ciphertext quality for itdirectly analyzes the pixel distribution of a ciphertext image

Figure 11 illustrates image encryption results for variousplaintext images image ldquo113rdquo is a binary handwriting scannedimage selected from ICDAR 2009 database image ldquoLenardquois a commonly used image of gray type image ldquo5113rdquoand ldquotestpat1krdquo are used to mimic the possible complexpatterns in plaintext and they are both selected from USC-SIPI database It is worth noting that the used CJPM is at sizeof 256-by-256

Mathematical Problems in Engineering 9

0 10

051

152

253

354

0100200300400500600700800

0 50 100 150 200 2500

10002000300040005000600070008000

0 50 100 150 200 2500

051

152

253

354

455

0 50 100 150 200 250

0 10

05

1

15

2

25

0100200300400500600

0 50 100 150 200 2500

100200300400500600

0 50 100 150 200 2500

100020003000400050006000700080009000

10000

0 50 100 150 200 250

113 Lenna Testpat1k

times104

times106 times104

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Plai

ntex

tP

Plai

ntex

tP

Plai

ntex

tP

Plai

ntex

tP

Ciph

erte

xtC

Ciph

erte

xtC

Ciph

erte

xtC

Ciph

erte

xtC

times106

Hist

ogra

m o

fC

Hist

ogra

m o

fC

Hist

ogra

m o

fC

Hist

ogra

m o

fC

5113

Figure 11 Histogram analysis for image encryption based on CJPM

It is clear that no matter what histogram a plaintextimage has the histogram of its ciphertext image is flatwhich implies that the pixel distribution is almost uniformComplex patterns and large homogenous regions in plaintextimages are completely unintelligible and become random-like in ciphertext images These results imply that the pro-posed image encryption method based on CJPM is robustand effective for various image formats and contents

52 Adjacent Pixel Autocorrelation (APAC) Analysis Highcorrelations of adjacent pixels can be utilized to carryout cryptanalysis Therefore a secure encryption algorithmshould break the high correlation relationship between adja-cent pixels

In statistics the autocorrelation 119877119886of a random process

119883 describes the correlation between values of the process atdifferent points in time as a function of the two times or of thetime difference The autocorrelation coefficient 119877

119886is defined

in (20) where 119889 is the time difference 120583 is the mean valuedefined by (21) and 120590 is the standard deviation defined by(22) the definition of mathematical expectation is given in(23)

119877119886(119898) =

119864 [(119883119905minus 120583) (119883

119905+119889minus 120583)]

1205902

(20)

120583 = 119864 [119883] (21)

10 Mathematical Problems in Engineering

Plai

ntex

t

Plai

ntex

t

Ciph

erte

xt

Ciph

erte

xt

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 250

Plai

ntex

tCi

pher

text

0

50

100

150

200

250

0

50

100

150

200

250

DiagonalVerticalHorizontal

DiagonalVerticalHorizontal

Figure 12 APAC analysis for image encryption based on CJPM

Table 1 Adjacent pixel autocorrelation analysis

Correlation coefficients (10minus3) Horizontal Vertical DiagonalPlaintext Lena 9399652 9709000 9709894

Ciphertext

CJPM minus22281 minus03709 3265[6] 20970 161870 17805[8] minus25000 minus10000 minus93000[5] 57765 28434 20662[29] minus127212 minus602579 624427[9] minus134000 12000 398000[30] minus158900 minus653800 minus323100[16] 815860 minus400530 minus47150[10] 1257000 581000 504000

120590 = radic119864 [(119883 minus 120583)2

] (22)

119864 [119909] =

119873

sum

119894=1

119909119894

119873 (23)

The closer to zero this coefficient is the weaker the rela-tionship two different time functions have Specifically inadjacent pixel correlation test we let 119883 be the image pixelsequence and let 119889 be 1 that is compare to the adjacent pixelsequence

Based on the reference direction there are three waysof extracting a two-dimensional image to a one-dimensionalsequence and they are the horizontal adjacent correlation

coefficient the vertical adjacent correlation coefficient andthe diagonal adjacent correlation coefficient

ldquoLenardquo image in the 2nd column of Figure 11 is used asthe test plaintext image because its APAC is widely reportedby other encryption methods Peer comparison results ofthe proposed CJPM cipher and cited encryption methods onAPAC are shown in Table 1 (best results are bolded)

In addition Figure 12 shows the result of randomlyselected 1024 pairs of two adjacent pixels from the plaintextand the ciphertext along horizontal vertical and diagonaldirections where 119909- and 119910-axes denote the intensity values ofa randomly selected pixel and its adjacent pixel respectivelyIt is clear that after applying the CJPM based image cipherthe high correlations between adjacent pixels in plaintext arebroken

53 Plaintext Sensitivity Analysis In order to test the resis-tance of the cipher to differential attacks plaintext sensitivityanalysis is required for a secure cipher In differential attacksan adversary attempts to extract meaningful relationshipbetween a plaintext image and its ciphertext image bymakinga slight change usually only one pixel in the plaintext imagewhile encrypting the plaintext image with the same encryp-tion key By comparing the change in ciphertext imagesthe encryption key might be cracked and furthermore theinformation contained in ciphertext might be leaked

Although there are othermeasures [34 35] to evaluate theresistance of plaintext attacks two classic measures are theNumber of Pixel Change Rate (NPCR) and Unified Average

Mathematical Problems in Engineering 11

Before changing one pixel After changing one pixel DifferencePl

aint

ext i

mag

eCi

pher

text

imag

e

Figure 13 Plaintext sensitivity analysis (differential attacks) for image encryption based on CJPM

(a) Encrypted image (1 1 128128 1 1 4 0)

(b) Decrypted image (1 1 128128 1 1 4 0)

(c) Decrypted image (1 1 128128 1 2 4 0)

(d) Decrypted image (1 1 128128 1 1 39999 0)

Figure 14 Key sensitivity analysis

Changing Intensity (UACI) [5]TheNPCR is used tomeasurethe percentage of the number of pixels changed in ciphertextafter making a slight change in plaintext Therefore thetheoretical greatest upper-bound of the NPCR is 100 TheUACI is used to measure the averaged intensity change forpixels in ciphertext images after making a slight change ina plaintext image It is demonstrable that the UACI of an idealcipher for 8-bit gray images is about 03346 [36]

Suppose ciphertext images before and after one pixelchange in a plaintext image are 1198621 and 1198622 respectively thepixel values at grid (119894 119895) in 1198621 and 1198622 are denoted as 1198621(119894 119895)and 1198622(119894 119895) a bipolar array 119863 is defined as (24) then the

NPCR and UACI can be mathematically defined as (25) and(26) respectively where symbol119873 denotes the total numberof pixels in the ciphertext symbol 119865 denotes the largestsupported pixel value compatible with the ciphertext imageformat and | sdot | is the absolute value function Consider

119863(119894 119895) =

1 if 1198621 (119894 119895) = 1198622 (119894 119895)

0 if 1198621 (119894 119895) = 1198622

(119894 119895)

(24)

12 Mathematical Problems in Engineering

Table 2 NPCR and UACI analyses on ldquoLenardquo image

CPJMsize 16-by-16 32-by-32 64-by-64 128-by-128 256-by-256

NPCR 996170 996246 995895 995850 996338UACI 338205 338379 334048 334076 334040

NPCR = sum

119894119895

119863(119894 119895)

119873times 100 (25)

UACI = sum

119894119895

100381610038161003816100381610038161198621

(119894 119895) minus 1198622

(119894 119895)10038161003816100381610038161003816

(119865 sdot 119873)times 100 (26)

Table 2 shows the NPCR and UACI results for one pixelchange in the ldquoLenardquo image It is clear that for CJPMs atvarious sizes the proposed cipher has good performances inboth the NPCR and UACI analyses Simulation results fit theexpectations of the ideal cipher very well

Figure 13 shows a differential attack on ldquoLenardquo imagewhile keeping the encryption key unchanged It is noticeablethat the difference between two plaintext images is thepixel on Lenarsquos nose However the ciphertext images are sodifferent that the image of their difference is still random-like As a result the one pixel change in the plaintext image isldquodissipatedrdquo in ciphertext From this point view this plaintextsensitivity is closely related to the diffusion property of acipher In other words it is reasonable to claim that a cipherhas good NPCR and UACI results if it has good diffusionproperties

54 Key Space Analysis The encryption key in the proposedimage encryption method using CPJM is composed of a setof parameters (119909

119903 119910119903 119908 119877 119862 119904 119899 119903 119898) where (119909

119903 119910119903) is an

arbitrary point on 119909119910 plain119908 is a nonnegative decimal119877 and119862 should be positive integers smaller than the plaintext imagesize 119904 and 119899 should be positive integers below the productof 119877119862 119903 should be a number in between [36 4] and 119898 is anonnegative integer Therefore theoretically the key space ofthe proposed cipher is infinitely large

Because the chaotic Logistic map is used as the triggerfor pseudorandom sequences the proposed cipher has highkey sensitivities as well The results of key sensitivity analysisare shown in Figure 14 where the set of parameters writtenin parenthesis is the used key It is clear that unless thecorrect decryption key is applied a ciphertext image cannotbe restored

6 Conclusion

In this paper we discussed the generation of a chaotic Jose-phus permutation matrix by using the conventional Josephuspermutation sequences and the logistic chaotic map Theproposed CJPM is parametric and is uniquely dependent onthe set of parameters which is sufficiently large to providea secure size of key space As another heritage from thechaotic Logistic map the CJPM is highly sensitive to itsinitial values (parameters) Any slight change in parameters

leads to significant differences in resulting CJPM Simulationresults show that (1) the ciphertext image is random-likefrom the perspective of human visual inspection (2) theencryption quality is almost independent of the plaintextimage (3) the proposed encryptionmethod is able to encryptplaintext images with large homogeneous regions to secureciphertext images (4) histogram analysis also shows thatdifferent ciphertext images tend to have the uniform distri-bution on [0 119865] (5) adjacent pixel correlation analysis showsthat neighbor pixels in ciphertext have lower correlationsthan many existing encryption methods (6) the proposedcipher is highly sensitive to encryption key and plaintext (7)experimental UACI results of the proposed cipher are veryclose to those of the ideal one

The proposed cipher can be used for various image typesfor example binary images 8-bit gray images 16-bit grayimages RBG images and so forth The same encryption ideamay also be applied to audio video or other types of digitalformats

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Scientific Research Fundof Hunan Provincial Education Department under Grant(no 12B023) It is also supported by the National NaturalScience Foundation of China (nos 61204039 and 61106029)and Scientific Research Foundation for Returned ScholarsMinistry of Human Resources and Social Security of thePeoplersquos Republic of China

References

[1] D StinsonCryptographyTheory andPractice CRCPress 2006[2] M Yang N Bourbakis and S Li ldquoData-image-video encryp-

tionrdquo IEEE Potentials vol 23 no 3 pp 28ndash34 2004[3] FIPS PUB 46 Data Encryption Standard National Bureau of

Standards 1977[4] FIPS PUB 197 Avanced Encryption Standard Announcing the

Advanced Encryption Standard (AES) 2001[5] Y Mao and G Chen Chaos-Based Image Encryption Springer

Berlin Germany 2005[6] H Yang K-W Wong X Liao W Zhang and P Wei ldquoA fast

image encryption and authentication scheme based on chaoticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3507ndash3517 2010

[7] J Hu and F Han ldquoA pixel-based scrambling scheme for digitalmedical images protectionrdquo Journal of Network and ComputerApplications vol 32 no 4 pp 788ndash794 2009

[8] Z Shuo C Ruhua J Yingchun and G Shiping ldquoAn imageencryption algorithm based on multiple chaos and wavelettransformrdquo in Proceedings of the 2nd International CongressImage and Signal Processing (CISP rsquo09) pp 1ndash5 2009

Mathematical Problems in Engineering 13

[9] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010

[10] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009

[11] X Liao S Lai and Q Zhou ldquoA novel image encryptionalgorithm based on self-adaptive wave transmissionrdquo SignalProcessing vol 90 no 9 pp 2714ndash2722 2010

[12] S Fu-Yan L Shu-Tang and L Zong-Wang ldquoImage encryptionusing high-dimension chaotic systemrdquo Chinese Physics vol 16no 12 pp 3616ndash3623 2007

[13] V Patidar N K Pareek and K K Sud ldquoA new substitution-diffusion based image cipher using chaotic standard and logisticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 7 pp 3056ndash3075 2009

[14] Y Wu J P Noonan G Yang and H Jin ldquoImage encryptionusing the two-dimensional logistic chaotic maprdquo Journal ofElectronic Imaging vol 21 no 1 Article ID 013014 2012

[15] Y Wu J P Noonan and S Agaian ldquoA wheel-switch chaoticsystem for image encryptionrdquo in Proceedings of the InternationalConference on System Science and Engineering (ICSSE rsquo11) pp23ndash27 June 2011

[16] L Zhang X Liao andXWang ldquoAn image encryption approachbased on chaoticmapsrdquoChaos Solitons and Fractals vol 24 no3 pp 759ndash765 2005

[17] H S Kwok and W K S Tang ldquoA fast image encryption systembased on chaotic maps with finite precision representationrdquoChaos Solitons and Fractals vol 32 no 4 pp 1518ndash1529 2007

[18] J Jian Y Shi C Hu Q Ma and J Li ldquoEncryption of digitalimage based on chaos systemrdquo in Computer and ComputingTechnologies in Agriculture II vol 2 pp 1145ndash1151 2009

[19] P Van-Roy and S Haridi Concepts Techniques and Models ofComputer Programming MIT Press 2004

[20] P SchumerMathematical Journeys Wiley-Interscience 2004[21] G Ye X Huang and C Zhu ldquoImage encryption algorithm of

double scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007

[22] XDesheng andXYueshan ldquoDigital image scrambling based onjosephus traversingrdquo Computer Engineering and Applicationsno 10 pp 44ndash46 2005

[23] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2005

[24] Y-P Zhang Z-J Zhai W Liu X Nie S-P Cao and W-D Dai ldquoDigital image encryption algorithm based on chaosand improved DESrdquo in Proceedings of the IEEE InternationalConference on Systems Man and Cybernetics (SMC rsquo09) pp474ndash479 October 2009

[25] J Lang R Tao and Y Wang ldquoImage encryption based on themultiple-parameter discrete fractional Fourier transform andchaos functionrdquo Optics Communications vol 283 no 10 pp2092ndash2096 2010

[26] S Yang and S Sun ldquoVideo encryptionmethod based on chaoticmaps in DCT domainrdquo Progress in Natural Science vol 18 no10 pp 1299ndash1304 2008

[27] V Wu J P Noonan and S Agaian ldquoBinary data encryptionusing the Sudoku block cipherrdquo in Proceedings of the IEEEInternational Conference on SystemsMan andCybernetics (SMCrsquo10) pp 3915ndash3921 October 2010

[28] Y Wu Y Zhou J P Noonan K Panetta and S Agaian ldquoImageencryption using the Sudoku matrixrdquo in Proceedings of theMobileMultimediaImage Processing Security and ApplicationsApril 2010

[29] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004

[30] H Gao Y Zhang S Liang and D Li ldquoA new chaotic algorithmfor image encryptionrdquo Chaos Solitons and Fractals vol 29 no2 pp 393ndash399 2006

[31] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[32] J Thompson and H Stewart Nonlinear Dynamics and ChaosJohn Wiley amp Sons 2002

[33] C E Shannon ldquoCommunication Theory of Secrecy SystemsrdquoBell Systems Technical Journal vol 28 pp 656ndash715 1949

[34] Y Wu Y Zhou G Saveriades S Agaian J P Noonan andP Natarajan ldquoLocal Shannon entropy measure with statisticaltests for image randomnessrdquo Information Sciences vol 222 no323 342 pages 2013

[35] Y Wu S Agaian and J P Noona ldquoA novel method of testingimage randomness with applications to image shuffling andencryptionrdquo in Defense Security and Sensing Proceedings ofthe SPIE pp 875507ndash875507 2013

[36] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI random-ness tests for image encryptionrdquo Cyber Journals pp 31ndash38 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Input 119905 the initial total number of persons in a circle119904 the starting position in the circle119899 the counting period

Output 119902120587 the Josephus permutation sequence according to parameter set (119905 119904 119899)

count = 0 done = 0 pos = 119904 label = zeros(1 119905) 119902120587= [] initial settings

while (simdone) main looptodo = label(pos)if (todo == 0) if this person has not been taken out

count = count + 1if (count == 119899) if this is the 119899th person

119902120587(end + 1) = pos count = 0 label(pos) = 1

if (length(119902120587) == 119905)

done = 1end

endendpos = pos + 1if (pos gt 119905)

pos = 1end

end

Algorithm 1 The generation of a Josephus permutation sequence

conventional transform domain encryption [27 28] addchaotic map to Sudoku puzzles for encryption In this paperwe develop a new image encryption method by combin-ing the ideas of the chaotic map the Josephus traversingsequence and conventional substitution and transpositionciphersThe remainder of the paper is organized as follows inSection 2 the Josephus permutation and the Logistic chaoticmap are briefly reviewed in Section 3 the CJPM and itsgenerator are given in Section 4 the proposed image encryp-tion method based on CJPM is fully discussed including itsflowchart and functions for each part in Section 5 simulationresults are shown and various security analyses are applied inSection 6 the paper is concluded

2 Preliminary

21 Josephus Permutation The Josephus permutation orJosephus problem is well known in computer science andmathematics It is named after Flavius Josephus a Jewishhistorian lived in the 1st century It is a theoretical problemrelated to a certain counting-out game that works by having 119905people standing in a circle with consecutive tags from 1 to119905 Starting at predetermined person you count around thecircle Once you reach the 119899th person take them out of thecircle and have the members to close the circle Then repeatthe process until only one person is left That person winsthe game If we record the tags of people who have beentaken out at each round as a sequence then this sequenceis a permutation of a natural number sequence and is calledJosephus permutation sequence

It is clear that three parameters are involved in the Jose-phus problem namely the initial total number of persons ina circle 119905 the starting position in the circle 119904 and the countingperiod 119899 Therefore a Josephus permutation sequence 119902

120587

can be denoted as follows where 119869 denotes the Josephuspermutation according to the set of parameters 119905 119904 and 119899A Josephus permutation sequence can be easily implementedby linked lists and dynamic arrays Algorithm 1 describes a

119902120587= 119869 (119905 119904 119899) (1)

For example

119902120587= 119869 (18 1 4)

= [4 8 12 16 2 7 13 18 6 14 3 11 5 17 15 1 10 9]

(2)

119902120587= 119869 (18 1 7)

= [7 14 3 11 1 10 2 13 6 18 16 15 17 5 12 4 8 9]

(3)

119902120587= 119869 (18 4 7)

= [10 17 6 14 4 13 5 16 9 3 1 18 2 8 15 7 11 12]

(4)

It is clear that (1) compared with the length 119905 natural numbersequence a Josephus permutation sequence 119902

120587changes a lot

especially considering that none of the twoneighbor numbersis consecutive (2) for a fixed length 119905 different pairs of (119904 119899)give distinct Josephus permutation sequences However it isweak in that (1) 119902

120587rsquos very first several elements divulge the

parameter of counting period 119899 (2) the difference betweentwo 119902

120587s may disclose the difference between their starting

positions for example the difference between the two firstelements of 119902

120587s in (3) and (4) is 3 which is the difference

of their parameters of starting positions Therefore it is notcompletely random for a Josephus permutation sequence

Mathematical Problems in Engineering 3

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1Logistic map (r = 36)

x[i]

x[i + 1]

x[i + 2]

x[i + 3]

y = x

(a)

0

01

02

03

04

05

06

07

08

09

1

2 25 3 35 4

Bifurcation diagram of the logistic map

(b)

Figure 1 The Logistic map

22 Chaotic Logistic Map The Logistic map is a polynomialmapping of degree two It was introduced by the biologistRobet May in 1976 [31] The Logistic map is written as (5)where symbol L is used to denote the Logistic map 119883

119894isin

[0 1] and represents the population at year 119894 and hence 1198830

represents the initial population at year 0 119903 is a positivenumber and represents the combined rate for reproductionand starvation [32] This map is often cited as an example ofhow complex chaotic behaviors can arise from a very simplenonlinear dynamic equation Consider

119883119894+1=L (119883

119894) = 119903119883

119894(1 minus 119883

119894) (5)

The Logistic map has been well studied The plots of thefirst few iterations of the Logistic map and its bifurcationdiagram are shown in Figure 1 It is well known that when119903 isin [357 4] (approximately) the Logistic map has chaoticbehaviors for most values but there are still certain isolatedranges of 119903 that show nonchaotic behavior for example119903 asymp 383 which corresponds to a big gap in its bifurcationdiagram

In reality the Logistic sequence is controlled by a set ofparameters of (119883

0 119903119873119898) where (119883

0 119903) are parameters in

the Logistic map119873 is the length of sequence and119898 denotesthe number of thrown-away samples Therefore a Logisticsequence 119883 can be denoted as (6) Based on this logisticsequence119883 of length119873 sorted sequence1198831015840 can be obtainedby sorting 119883 in the ascending order It is certain that 1198831015840 is apermutation of the original119883Therefore119883 and1198831015840 satisfy (7)

where 119901120587is a permutation mapping sequence and 119894 denotes

sequence element index Consider

119883 =L (1198830 119903119873119898) (6)

1198831015840

119894= 119883119901120587(119894)

(7)

119883 = [04000 09120 03050 08055 05954 09154

02943 07892 06322 08836 03908

09047 03277 08372 05179 09488

01846 05721]

(8)

1198831015840

= [01846 02943 03050 03277 03908 04000

05179 05721 05954 06322 07892

08055 08372 08836 09047 09120

09154 09488]

(9)

119901120587= 119871 (04 38 18 0)

= [17 7 3 13 11 1 15 18 5 9 8 4 14 10 12 2 6 16]

(10)

For example if (1198830 119903 119873119898) = (04 38 18 0) then 119883

and 1198831015840 are shown in (8) and (9) respectively Correspond-ingly the permutation sequence 119901

120587is determined as (10)

4 Mathematical Problems in Engineering

12

3

6

9

1

2

4

57

8

10

11

(a) Conventional Josephusproblem

P120587(3)

P120587(6)

P120587(9)

P120587(1)

P120587(2)

P120587(4)

P120587(5)P120587(7)

P120587(8)

P120587(10)

P120587(11)P120587(12)

(b) Chaotic Josephus problem

Figure 2 The chaotic Josephus problem

CJPS at length RCCJPM at size of R-by-C

Rearrangement

Figure 3 Rearranging a CJPS to a CJPM

3 Chaotic Josephus PermutationMatrix (CJPM)

31 Chaotic Josephus Permutation Sequence (CJPS) Fromprevious sections it is clear that the Josephus permutationsequence 119902

120587and the chaotic permutation sequence 119901

120587are

obtained from different mechanismsThe Josephus permuta-tion sequence 119902

120587is easy to obtain but not completely random-

like while the chaotic permutation sequence 119901120587is random-

like but requires a large amount of computations for main-taining accuracy It is desirable that a permutation sequencebe random-like and only costs moderate computations

In order to achieve the above objective the new Jose-phus permutation sequence based on a chaotic permutationsequence 119901

120587is defined whose initial positions on the circle

are not consecutive numbers like those in the conventionalJosephus problem This new problem can be restated as fol-lows (1) 119905 people with numbered tags stand in a circle (thesenumbers together form a permutation sequence accordingto sorting a chaotic sequence) (2) starting at predeterminedperson you count around the circle until you reach the 119899thperson take him out of the circle and have the members toclose the circle and record his tag number (3) repeat theprocess until only one person is left If we record the tags ofpeople who have been taken out of the circle as a sequencethen a chaotic Josephus permutation sequence is obtained

It is noticeable that the chaotic Josephus problem hasparameters for both the chaotic Logistic map and for theconventional Josephus problem In other words a chaoticJosephus permutation sequence 119888119902

120587is determined by (11)

where parameters 119905 119904 119899 have the same meanings as in (1)1198830

and 119903 are parameters in the Logistic map and 119898 determines

the number of thrown-away samples in the chaotic Logisticsequence Consider

119888119902120587= 119888119869 (119905 119904 119899 119883

0 119903 119898) (11)

In the conventional Josephus problem (controlled byparameters (119905 119904 119899) see (1)) a natural number sequence 1 to119905 is used to denote the tags for people standing in the circlewhile a permutated sequence 119901

120587(controlled by parameters

(1198830 119903119873119898) see (6)) is used in the chaotic Josephus problem

as the tags In order to match the sequence lengths of 119901120587

and 119902120587 119905 = 119873 is the condition that has to be satisfied

The conventional Josephus problem and the chaotic Josephusproblem for 119905 = 119873 = 12 are illustrated in Figure 2

Therefore the new Josephus permutation sequence 119888119902120587

can be denoted as a composed function of a conventionalJosephus permutation sequence 119902

120587and a chaotic permutation

sequence 119901120587

119888119902120587= 119901120587∘ 119902120587= 119901120587(119902120587) (12)

For example if 119902120587

= 119869(18 1 4) in (2) and119901120587

= 119871(04 38 18 0) in (10) are preknown 119888119902120587

=

119888119869(18 1 4 04 38 0) is obtained as (13) shows by using(12) Similarly (14)ndash(16) can be also obtained Consider

119888119902120587= 119888119869 (18 1 4 04 38 0)

= [13 18 4 2 7 15 14 16 1 10 3 8 11 6 12 17 9 5]

(13)

119888119902120587= 119888119869 (18 1 7 04 38 0)

= [15 10 3 8 17 9 7 14 1 16 2 12 6 11 4 13 18 5]

(14)

119888119902120587= 119888119869 (18 4 7 04 38 0)

= [9 6 1 10 13 14 11 2 5 3 17 16 7 18 12 15 8 4]

(15)

119888119902120587= 119888119869 (18 4 7 040001 38 0)

= [8 6 1 10 11 12 13 2 9 15 7 18 3 5 14 17 4 16]

(16)

Compared to the previous conventional Josephus permu-tation sequence (see (2)ndash(4)) the chaotic Josephus permuta-tion sequences (see (12)ndash(15)) are more random-like because(1) the difference between its very first elements is not relatedto its counting period anymore (2) the difference betweentwo 119888119902

120587s which are only different in their starting positions

is no longer equal to the difference of their starting positions(3) slight perturbations in chaotic map parameters lead tobig changes in resulting 119888119902

120587s (4) two neighbor elements in

119888119902120587may or may not be consecutive Therefore the chaotic

Josephus permutation sequence ismore random-like than theconventional Josephus permutation sequence

32 Chaotic Josephus Permutation Matrix (CJPM) Based onchaotic Josephus permutation sequence(s) a chaotic Jose-phus permutation matrix can be generated via various ways

Mathematical Problems in Engineering 5

1

2

3

4

5

6

7

8

9

10

20

30

40

50

60

2 4 6 8 0

(a) CJPM(119877 119862 1 4 1198830 119903 0)

2

3

4

5

6

7

8

9

2 4 6 81 0

10

20

30

40

50

60

(b) CJPM(119877 119862 1 7 1198830 119903 0)

2

3

4

5

6

7

8

9

2 4 6 81 0

10

20

30

40

50

60

(c) CJPM(119877 119862 4 7 1198830 119903 0)

1

2

3

4

5

6

7

8

9

2 4 6 80

10

20

30

40

50

60

(d) CJPM(119877 119862 1 4 1198830+ 119890 119903 0)

1

2

3

4

5

6

7

8

9

2 4 6 80

10

20

30

40

50

60

(e) CJPM(119877 119862 1 4 1198830+ 119890 119903 10)

2 4 6 81

2

3

4

5

6

7

8

9

0

10

20

30

40

50

60

(f) CJPM(119877 119862 1 4 1198830+ 119890 119903 + 119890 10)

Figure 4 Parametric CJPMs (Note 119877 = 8 119862 = 81198830= 04 119903 = 38 and 119890 = 00001)

PlaintextP

Initial key

CiphertextC

CJPMgenerator

Encryption key

M

One blockPixel

permutationPixel

substitutionS

Figure 5 Image encryption method based on CJPM

Among these methods Algorithm 2 illustrates a straightfor-ward method to obtain a CJPM via a CJPS by rearrangingCJPS elements to a matrix

It is noticeable that Algorithm 2 is equivalent to rearrang-ing a sequence of elements into a matrix following the orderillustrated in Figure 3

Therefore a CJPM is determined by the same set ofparameters controlling a CJPS In order to emphasize thematrix property the parameter 119905 in CJPS is replaced by twoparameters of height 119877 and width 119862 where 119905 = 119877119862 Similarlya CJPM is uniquely determined by a set of parameters(119877 119862 119904 119899 119883

0 119903 119898) as for a CJPS Mathematically this claim

can be denoted as

119872 = CJPM (119877 119862 119904 119899 1198830 119903 119898) (17)

tN

xr yr w

p = L(X0 s n)

cq = cJ(t s n X0 r m)

Initial key

q = J(t s n)

R C s nr m

X0

Figure 6 Key functions in CJPM

Figure 4 illustrates various CJPMs via Algorithm 2according to different parameter sets It is clear that CJPMis very sensitive to its parameters and small changes in theparameter set lead to distinct CJPMs

4 Image Encryption AlgorithmBased on CJPM

In 1949 Claude Shannon the father of ldquoInformationTheoryrdquoproposed that confusion and diffusion are two properties ofthe operation of a secure cipher where the term confusionrefers to making the relationship between the encryptionkey and the ciphertext a very complex and developed one

6 Mathematical Problems in Engineering

Center of gravity

PlaintextP

Reference

CJPM M

Initial key Encryption key

point (xr yr)R C s t r m

X0 = sin(arctan(ygxg))(xg yg)

Figure 7 The internal structure of CJPM generator

PlaintextP

M +M998400

P998400

SMod(P998400 F)Mod(MF)

Figure 8 The internal structure of CJPM generator

[33] and the term diffusion refers to the property that theredundancy in the statistics of the plaintext is ldquodissipatedrdquo inthe statistics of the ciphertext [33] In otherwords for a securecipher it has to have good confusion and diffusion properties(1) different ciphertexts are desired to have similar statistics(2) any slight change in a plaintext is desired to lead to bigdifference in its ciphertext The image encryption algorithmbased on CJPM is proposed in this section to meet these twocriteria

41 Flowchart of Image Encryption Algorithm Based on CJPMSince the CJPM is parametric and random-like it can beused for image encryption directly However considering therequirements from confusion and diffusion properties theencryption procedure can be described as Figure 5 showsThe plaintext image is first sent to the CJPM generator whichis a preparation stage for generating a CJPM119872 for future useLater this CJPM119872 is used as a reference matrix to permuteand substitute image pixels for each image block in the stagesof pixel permutation and pixel substitution respectively Thedecryption procedure is simply to reverse the encryptionprocedure

42 Key Schedule It is clear that the CJPM is the coreof the cipher and thus key is related to the used CJPMreference matrix 119872 Initial key is composed of parameters(119909119903 119910119903 119908 119903 119898 119877 119862 119904 119899) where (119909

119903 119910119903 119908) is used in CJPM

generator for obtaining plaintext-dependent parameter 1198830

used in the Logistic map (119877 119862) are used as the parameter 119905in (1) and the parameter119873 in (6) The functions of each partof the initial key are shown in Figure 6

The output encryption key is composed of (119877 119862

119904 119899 1198830 119903 119898) all of which are directly required for deter-

mining a CJPM according to (17) Among these parameters119877 119862 119904 119899 and 119898 are restricted to integers 119909

119903 119910119903 119903 and 119883

0

are decimals More specifically 119877 and 119862 should be positive

integers smaller than the plaintext image size 119904 and 119899 shouldbe positive integers below the product of 119877119862 119903 should be anumber in between [36 4] (119909

119903 119910119903) is an arbitrary point on

119909119910 plain with weight 119908 and119898 is a nonnegative integer

43 CJPM Generator In order to enhance the resistance todifferential attacks the CJPM generator used in Figure 5 isdesigned to be plaintext dependent Recall that a CJPM isdetermined by a set of parameters (119877 119862 119904 119899 119883

0 119903 119898) shown

in (17) In the CJPM generator for image encryption only theparameter119883

0is not directly given by the initial key but by the

plaintext and a reference point (119909119903 119910119903) controlling the weight

in calculating the center of gravity Once 1198830is generated

it is stored in the encryption key The whole procedure oftranslating the initial key to a plaintext-dependent CJPMmatrix119872 and encryption key is shown in Figure 7

A plaintext is considered as an object of pixels where itsupper-left corner pixel is the reference point located at (1 1)Correspondingly pixels next to it along 119909 and 119910 directionsare (2 1) and (1 2) respectively The center of gravity ofthis plaintext is calculated via (18) where 119875

119894denotes the

119894th pixel intensity value and 119909119894and 119910

119894denote the location

of the 119894th pixel in the image with respect to the upper-leftcorner Once the center of gravity (119909

119892 119910119892) is obtained the

initial value of Logistic map 1198830is also determined via (19)

where arctan(sdot) is the arc tangent function and sin(sdot) is thesine function It is easy to verify that the range of (19) is[0 1] which satisfies restrictions for the initial value119883

0in the

Logisticmap Finally all required parameters for aCJPM thatis (119877 119862 119904 119899 119883

0 119903 119898) are obtained and thus a CJPM 119872 is

generated Meanwhile the used parameters are stored as theencryption key which can be used in the decryption processConsider

119909119892=119909119903sdot 119908 + sum119909

119894119875119894

119908 + sum119875119894

119910119892=119910119903sdot 119908 + sum119910

119894119875119894

119908 + sum119875119894

(18)

1198830= 05 [sin(arctan(

119910119892

119909119892

)) + 1] (19)

It is worth noting that the plaintext-dependent CJPMgenerator guarantees that the proposed cipher has gooddiffusion property any slight changes in plaintext lead tobig difference in ciphertext This is because the resulting

Mathematical Problems in Engineering 7

(a) Original image (b) 32-by-32 (c) 64-by-64 (d) 128-by-128

0100200300400500600700800900

1000

0 50 100 150 200 250

(e) Histogram of (a)

0

100

200

300

400

500

600

0 50 100 150 200 250

(f) Histogram of (b)

0

100

200

300

400

500

600

0 50 100 150 200 250

(g) Histogram of (c)

0

100

200

300

400

500

600

0 50 100 150 200 250

(h) Histogram of (d)

Figure 9 Pixel substitution results for CJPM at various sizes

(a) Original image (b) 32-by-32 (c) 64-by-32 (d) 64-by-64

(e) 64-by-128 (f) 128-by-128 (g) 256-by-128 (h) 256-by-256

Figure 10 Pixel permutation results for CJPM at various sizes

CJPM matrix119872 is dependent on the parameter 1198830and the

parameter 1198830is dependent on the center of gravity for the

plaintext while the center of the plaintext gravity alters forany slight change in plaintext Furthermore this 119883

0is the

parameter in the chaotic map and thus any slight change

in initial value leads to a completely different trajectory asthe bifurcation diagram in Figure 1 shows Consequentlya completely different CJPM is obtained as the referencematrix Eventually this new reference matrix leads to adistinct ciphertext It can be demonstrated that without

8 Mathematical Problems in Engineering

Input 119888119902120587 a CJPS at length 119905 = 119877 times 119862

Output119872 a CJPM of size 119877-by-119862for 119894 = 1 119877

for 119895 = 1 119862119872(119894 119895) = 119888119902

120587((119894 minus 1) 119877 + 119895)

endend

Algorithm 2 Reshape a CJPS into a CJPM

preknowing the reference pointrsquos coordinate (119909119903 119910119903) and its

weight 119908 there is no way to slightly change plaintext imagepixels so that its gravity center is unchanged

44 Pixel Substitution Pixel substitution refers to the pro-cess of changing pixel values From the point of viewof statistics this process is to change the statistics of aplaintext image so that the statistics of resulting ciphertextimage is completely different Moreover it is desired thatdifferent ciphertext images have similar statistics whichimplies that ciphertext images tell little information aboutkeys and plaintext images As a result the confusion propertyis achieved

The proposed pixel substitution block is shown inFigure 8 where symbol 119865 = 2

119887 and 119887 denotes the numberof bits supported by the format of the plaintext image Forexample if plaintext 119875 is an 8-bit gray image then 119865 = 28 =256 if plaintext119875 is a binary image then 119865 = 21 = 2 It can benoticed that the reference CJPM119872 is first to convert to1198721015840whose format is compatible with the format of the plaintextimage later1198721015840 and 119875 are added over the space of 119865 finallythe encrypted image 119878 is obtained

Because elements in a CJPM matrix 119872 are uniformlydistributed on integer set [1 119905] after ldquoModrdquo operation 1198721015840still has a uniform-like distribution for its elements on[0 119865] As a result when this 1198721015840 is used to randomly shiftthe pixel value in plaintext the resulting pixel value inciphertext has an equal opportunity to be any value on [0 119865]As a result a uniform-like histogram is achieved in theciphertext

Pixel substitution results based on CJPMs are shown inFigure 9 It is clear that histograms before and after pixelsubstitution are very different and ciphertext histograms arevery flat compared to plaintext ones It is also noticeable thatas the size of the referenceCJPM increases the ciphertext has abetter encryption quality from the point view of human visualinspection

45 Pixel Permutation Pixel permutation refers to the pro-cess scrambling the positions of pixels in plaintext to disguiseinformation contained in an image Denote an image beforeand after pixel permutation as 119861 and 119860 respectively Assumethe way of indexing image pixels is the same as the order torearrange elements in a CJPM as Figure 3 shows Then thepixel permutation process can be mathematically defined asa permutation 119891

120587between domain 119861 and range 119860 forall119894 119895 isin

1 2 119905 exist119860119894= 119861119895= 119861119891120587(119894) where 119905 is the total

number of pixels in the image As we mentioned in previoussections a CJPM is generated from a CJPS which is apermutation sequence Therefore a CJPM can be directlyused for pixel permutation that is given a CJPM 119872 and itspixel permutation can be defined as 119891

120587(119894) = 119872(119894)

For example Figure 10 illustrates pixel permutationresults of the ldquoTuftsrdquo logo image for different CJPMsIt is clear that pixels are well shuffled within the imageblock As long as the size of CJPMprocessing imageblock increases the resulting shuffled image looksbetter and better When the block size reaches to orover 128-by-128 pixels in the plaintext are almost evenlyshuffled

It is clear that images after pixel permutation look verydifferent from the plaintext image It is also worth notingthat a CJPM also depends on a set of parameters besides thesize and that any change in other parameters will lead to acompletely different permutated image

5 Simulation Results and Security Analysis

An excellent encryption method should be both robustand effective Robustness means that the cipher should beapplicable to any plaintext image written in a supportedformat Effectiveness implies that the cipher is able to generateeligible ciphertext images which hide information frompossible intruders

In this section we focus on discussing the performanceof the CJPM based image cipher described in Section 4 It isworth noting that all following computer simulations are rununder MATLAB 2010a and Windows XP environment withCore 2 Quad 26GHz processors

51 Histogram Analysis Histogram analysis is one of themost straightforward evaluations for ciphertext quality for itdirectly analyzes the pixel distribution of a ciphertext image

Figure 11 illustrates image encryption results for variousplaintext images image ldquo113rdquo is a binary handwriting scannedimage selected from ICDAR 2009 database image ldquoLenardquois a commonly used image of gray type image ldquo5113rdquoand ldquotestpat1krdquo are used to mimic the possible complexpatterns in plaintext and they are both selected from USC-SIPI database It is worth noting that the used CJPM is at sizeof 256-by-256

Mathematical Problems in Engineering 9

0 10

051

152

253

354

0100200300400500600700800

0 50 100 150 200 2500

10002000300040005000600070008000

0 50 100 150 200 2500

051

152

253

354

455

0 50 100 150 200 250

0 10

05

1

15

2

25

0100200300400500600

0 50 100 150 200 2500

100200300400500600

0 50 100 150 200 2500

100020003000400050006000700080009000

10000

0 50 100 150 200 250

113 Lenna Testpat1k

times104

times106 times104

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Plai

ntex

tP

Plai

ntex

tP

Plai

ntex

tP

Plai

ntex

tP

Ciph

erte

xtC

Ciph

erte

xtC

Ciph

erte

xtC

Ciph

erte

xtC

times106

Hist

ogra

m o

fC

Hist

ogra

m o

fC

Hist

ogra

m o

fC

Hist

ogra

m o

fC

5113

Figure 11 Histogram analysis for image encryption based on CJPM

It is clear that no matter what histogram a plaintextimage has the histogram of its ciphertext image is flatwhich implies that the pixel distribution is almost uniformComplex patterns and large homogenous regions in plaintextimages are completely unintelligible and become random-like in ciphertext images These results imply that the pro-posed image encryption method based on CJPM is robustand effective for various image formats and contents

52 Adjacent Pixel Autocorrelation (APAC) Analysis Highcorrelations of adjacent pixels can be utilized to carryout cryptanalysis Therefore a secure encryption algorithmshould break the high correlation relationship between adja-cent pixels

In statistics the autocorrelation 119877119886of a random process

119883 describes the correlation between values of the process atdifferent points in time as a function of the two times or of thetime difference The autocorrelation coefficient 119877

119886is defined

in (20) where 119889 is the time difference 120583 is the mean valuedefined by (21) and 120590 is the standard deviation defined by(22) the definition of mathematical expectation is given in(23)

119877119886(119898) =

119864 [(119883119905minus 120583) (119883

119905+119889minus 120583)]

1205902

(20)

120583 = 119864 [119883] (21)

10 Mathematical Problems in Engineering

Plai

ntex

t

Plai

ntex

t

Ciph

erte

xt

Ciph

erte

xt

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 250

Plai

ntex

tCi

pher

text

0

50

100

150

200

250

0

50

100

150

200

250

DiagonalVerticalHorizontal

DiagonalVerticalHorizontal

Figure 12 APAC analysis for image encryption based on CJPM

Table 1 Adjacent pixel autocorrelation analysis

Correlation coefficients (10minus3) Horizontal Vertical DiagonalPlaintext Lena 9399652 9709000 9709894

Ciphertext

CJPM minus22281 minus03709 3265[6] 20970 161870 17805[8] minus25000 minus10000 minus93000[5] 57765 28434 20662[29] minus127212 minus602579 624427[9] minus134000 12000 398000[30] minus158900 minus653800 minus323100[16] 815860 minus400530 minus47150[10] 1257000 581000 504000

120590 = radic119864 [(119883 minus 120583)2

] (22)

119864 [119909] =

119873

sum

119894=1

119909119894

119873 (23)

The closer to zero this coefficient is the weaker the rela-tionship two different time functions have Specifically inadjacent pixel correlation test we let 119883 be the image pixelsequence and let 119889 be 1 that is compare to the adjacent pixelsequence

Based on the reference direction there are three waysof extracting a two-dimensional image to a one-dimensionalsequence and they are the horizontal adjacent correlation

coefficient the vertical adjacent correlation coefficient andthe diagonal adjacent correlation coefficient

ldquoLenardquo image in the 2nd column of Figure 11 is used asthe test plaintext image because its APAC is widely reportedby other encryption methods Peer comparison results ofthe proposed CJPM cipher and cited encryption methods onAPAC are shown in Table 1 (best results are bolded)

In addition Figure 12 shows the result of randomlyselected 1024 pairs of two adjacent pixels from the plaintextand the ciphertext along horizontal vertical and diagonaldirections where 119909- and 119910-axes denote the intensity values ofa randomly selected pixel and its adjacent pixel respectivelyIt is clear that after applying the CJPM based image cipherthe high correlations between adjacent pixels in plaintext arebroken

53 Plaintext Sensitivity Analysis In order to test the resis-tance of the cipher to differential attacks plaintext sensitivityanalysis is required for a secure cipher In differential attacksan adversary attempts to extract meaningful relationshipbetween a plaintext image and its ciphertext image bymakinga slight change usually only one pixel in the plaintext imagewhile encrypting the plaintext image with the same encryp-tion key By comparing the change in ciphertext imagesthe encryption key might be cracked and furthermore theinformation contained in ciphertext might be leaked

Although there are othermeasures [34 35] to evaluate theresistance of plaintext attacks two classic measures are theNumber of Pixel Change Rate (NPCR) and Unified Average

Mathematical Problems in Engineering 11

Before changing one pixel After changing one pixel DifferencePl

aint

ext i

mag

eCi

pher

text

imag

e

Figure 13 Plaintext sensitivity analysis (differential attacks) for image encryption based on CJPM

(a) Encrypted image (1 1 128128 1 1 4 0)

(b) Decrypted image (1 1 128128 1 1 4 0)

(c) Decrypted image (1 1 128128 1 2 4 0)

(d) Decrypted image (1 1 128128 1 1 39999 0)

Figure 14 Key sensitivity analysis

Changing Intensity (UACI) [5]TheNPCR is used tomeasurethe percentage of the number of pixels changed in ciphertextafter making a slight change in plaintext Therefore thetheoretical greatest upper-bound of the NPCR is 100 TheUACI is used to measure the averaged intensity change forpixels in ciphertext images after making a slight change ina plaintext image It is demonstrable that the UACI of an idealcipher for 8-bit gray images is about 03346 [36]

Suppose ciphertext images before and after one pixelchange in a plaintext image are 1198621 and 1198622 respectively thepixel values at grid (119894 119895) in 1198621 and 1198622 are denoted as 1198621(119894 119895)and 1198622(119894 119895) a bipolar array 119863 is defined as (24) then the

NPCR and UACI can be mathematically defined as (25) and(26) respectively where symbol119873 denotes the total numberof pixels in the ciphertext symbol 119865 denotes the largestsupported pixel value compatible with the ciphertext imageformat and | sdot | is the absolute value function Consider

119863(119894 119895) =

1 if 1198621 (119894 119895) = 1198622 (119894 119895)

0 if 1198621 (119894 119895) = 1198622

(119894 119895)

(24)

12 Mathematical Problems in Engineering

Table 2 NPCR and UACI analyses on ldquoLenardquo image

CPJMsize 16-by-16 32-by-32 64-by-64 128-by-128 256-by-256

NPCR 996170 996246 995895 995850 996338UACI 338205 338379 334048 334076 334040

NPCR = sum

119894119895

119863(119894 119895)

119873times 100 (25)

UACI = sum

119894119895

100381610038161003816100381610038161198621

(119894 119895) minus 1198622

(119894 119895)10038161003816100381610038161003816

(119865 sdot 119873)times 100 (26)

Table 2 shows the NPCR and UACI results for one pixelchange in the ldquoLenardquo image It is clear that for CJPMs atvarious sizes the proposed cipher has good performances inboth the NPCR and UACI analyses Simulation results fit theexpectations of the ideal cipher very well

Figure 13 shows a differential attack on ldquoLenardquo imagewhile keeping the encryption key unchanged It is noticeablethat the difference between two plaintext images is thepixel on Lenarsquos nose However the ciphertext images are sodifferent that the image of their difference is still random-like As a result the one pixel change in the plaintext image isldquodissipatedrdquo in ciphertext From this point view this plaintextsensitivity is closely related to the diffusion property of acipher In other words it is reasonable to claim that a cipherhas good NPCR and UACI results if it has good diffusionproperties

54 Key Space Analysis The encryption key in the proposedimage encryption method using CPJM is composed of a setof parameters (119909

119903 119910119903 119908 119877 119862 119904 119899 119903 119898) where (119909

119903 119910119903) is an

arbitrary point on 119909119910 plain119908 is a nonnegative decimal119877 and119862 should be positive integers smaller than the plaintext imagesize 119904 and 119899 should be positive integers below the productof 119877119862 119903 should be a number in between [36 4] and 119898 is anonnegative integer Therefore theoretically the key space ofthe proposed cipher is infinitely large

Because the chaotic Logistic map is used as the triggerfor pseudorandom sequences the proposed cipher has highkey sensitivities as well The results of key sensitivity analysisare shown in Figure 14 where the set of parameters writtenin parenthesis is the used key It is clear that unless thecorrect decryption key is applied a ciphertext image cannotbe restored

6 Conclusion

In this paper we discussed the generation of a chaotic Jose-phus permutation matrix by using the conventional Josephuspermutation sequences and the logistic chaotic map Theproposed CJPM is parametric and is uniquely dependent onthe set of parameters which is sufficiently large to providea secure size of key space As another heritage from thechaotic Logistic map the CJPM is highly sensitive to itsinitial values (parameters) Any slight change in parameters

leads to significant differences in resulting CJPM Simulationresults show that (1) the ciphertext image is random-likefrom the perspective of human visual inspection (2) theencryption quality is almost independent of the plaintextimage (3) the proposed encryptionmethod is able to encryptplaintext images with large homogeneous regions to secureciphertext images (4) histogram analysis also shows thatdifferent ciphertext images tend to have the uniform distri-bution on [0 119865] (5) adjacent pixel correlation analysis showsthat neighbor pixels in ciphertext have lower correlationsthan many existing encryption methods (6) the proposedcipher is highly sensitive to encryption key and plaintext (7)experimental UACI results of the proposed cipher are veryclose to those of the ideal one

The proposed cipher can be used for various image typesfor example binary images 8-bit gray images 16-bit grayimages RBG images and so forth The same encryption ideamay also be applied to audio video or other types of digitalformats

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Scientific Research Fundof Hunan Provincial Education Department under Grant(no 12B023) It is also supported by the National NaturalScience Foundation of China (nos 61204039 and 61106029)and Scientific Research Foundation for Returned ScholarsMinistry of Human Resources and Social Security of thePeoplersquos Republic of China

References

[1] D StinsonCryptographyTheory andPractice CRCPress 2006[2] M Yang N Bourbakis and S Li ldquoData-image-video encryp-

tionrdquo IEEE Potentials vol 23 no 3 pp 28ndash34 2004[3] FIPS PUB 46 Data Encryption Standard National Bureau of

Standards 1977[4] FIPS PUB 197 Avanced Encryption Standard Announcing the

Advanced Encryption Standard (AES) 2001[5] Y Mao and G Chen Chaos-Based Image Encryption Springer

Berlin Germany 2005[6] H Yang K-W Wong X Liao W Zhang and P Wei ldquoA fast

image encryption and authentication scheme based on chaoticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3507ndash3517 2010

[7] J Hu and F Han ldquoA pixel-based scrambling scheme for digitalmedical images protectionrdquo Journal of Network and ComputerApplications vol 32 no 4 pp 788ndash794 2009

[8] Z Shuo C Ruhua J Yingchun and G Shiping ldquoAn imageencryption algorithm based on multiple chaos and wavelettransformrdquo in Proceedings of the 2nd International CongressImage and Signal Processing (CISP rsquo09) pp 1ndash5 2009

Mathematical Problems in Engineering 13

[9] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010

[10] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009

[11] X Liao S Lai and Q Zhou ldquoA novel image encryptionalgorithm based on self-adaptive wave transmissionrdquo SignalProcessing vol 90 no 9 pp 2714ndash2722 2010

[12] S Fu-Yan L Shu-Tang and L Zong-Wang ldquoImage encryptionusing high-dimension chaotic systemrdquo Chinese Physics vol 16no 12 pp 3616ndash3623 2007

[13] V Patidar N K Pareek and K K Sud ldquoA new substitution-diffusion based image cipher using chaotic standard and logisticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 7 pp 3056ndash3075 2009

[14] Y Wu J P Noonan G Yang and H Jin ldquoImage encryptionusing the two-dimensional logistic chaotic maprdquo Journal ofElectronic Imaging vol 21 no 1 Article ID 013014 2012

[15] Y Wu J P Noonan and S Agaian ldquoA wheel-switch chaoticsystem for image encryptionrdquo in Proceedings of the InternationalConference on System Science and Engineering (ICSSE rsquo11) pp23ndash27 June 2011

[16] L Zhang X Liao andXWang ldquoAn image encryption approachbased on chaoticmapsrdquoChaos Solitons and Fractals vol 24 no3 pp 759ndash765 2005

[17] H S Kwok and W K S Tang ldquoA fast image encryption systembased on chaotic maps with finite precision representationrdquoChaos Solitons and Fractals vol 32 no 4 pp 1518ndash1529 2007

[18] J Jian Y Shi C Hu Q Ma and J Li ldquoEncryption of digitalimage based on chaos systemrdquo in Computer and ComputingTechnologies in Agriculture II vol 2 pp 1145ndash1151 2009

[19] P Van-Roy and S Haridi Concepts Techniques and Models ofComputer Programming MIT Press 2004

[20] P SchumerMathematical Journeys Wiley-Interscience 2004[21] G Ye X Huang and C Zhu ldquoImage encryption algorithm of

double scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007

[22] XDesheng andXYueshan ldquoDigital image scrambling based onjosephus traversingrdquo Computer Engineering and Applicationsno 10 pp 44ndash46 2005

[23] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2005

[24] Y-P Zhang Z-J Zhai W Liu X Nie S-P Cao and W-D Dai ldquoDigital image encryption algorithm based on chaosand improved DESrdquo in Proceedings of the IEEE InternationalConference on Systems Man and Cybernetics (SMC rsquo09) pp474ndash479 October 2009

[25] J Lang R Tao and Y Wang ldquoImage encryption based on themultiple-parameter discrete fractional Fourier transform andchaos functionrdquo Optics Communications vol 283 no 10 pp2092ndash2096 2010

[26] S Yang and S Sun ldquoVideo encryptionmethod based on chaoticmaps in DCT domainrdquo Progress in Natural Science vol 18 no10 pp 1299ndash1304 2008

[27] V Wu J P Noonan and S Agaian ldquoBinary data encryptionusing the Sudoku block cipherrdquo in Proceedings of the IEEEInternational Conference on SystemsMan andCybernetics (SMCrsquo10) pp 3915ndash3921 October 2010

[28] Y Wu Y Zhou J P Noonan K Panetta and S Agaian ldquoImageencryption using the Sudoku matrixrdquo in Proceedings of theMobileMultimediaImage Processing Security and ApplicationsApril 2010

[29] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004

[30] H Gao Y Zhang S Liang and D Li ldquoA new chaotic algorithmfor image encryptionrdquo Chaos Solitons and Fractals vol 29 no2 pp 393ndash399 2006

[31] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[32] J Thompson and H Stewart Nonlinear Dynamics and ChaosJohn Wiley amp Sons 2002

[33] C E Shannon ldquoCommunication Theory of Secrecy SystemsrdquoBell Systems Technical Journal vol 28 pp 656ndash715 1949

[34] Y Wu Y Zhou G Saveriades S Agaian J P Noonan andP Natarajan ldquoLocal Shannon entropy measure with statisticaltests for image randomnessrdquo Information Sciences vol 222 no323 342 pages 2013

[35] Y Wu S Agaian and J P Noona ldquoA novel method of testingimage randomness with applications to image shuffling andencryptionrdquo in Defense Security and Sensing Proceedings ofthe SPIE pp 875507ndash875507 2013

[36] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI random-ness tests for image encryptionrdquo Cyber Journals pp 31ndash38 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

0 02 04 06 08 10

01

02

03

04

05

06

07

08

09

1Logistic map (r = 36)

x[i]

x[i + 1]

x[i + 2]

x[i + 3]

y = x

(a)

0

01

02

03

04

05

06

07

08

09

1

2 25 3 35 4

Bifurcation diagram of the logistic map

(b)

Figure 1 The Logistic map

22 Chaotic Logistic Map The Logistic map is a polynomialmapping of degree two It was introduced by the biologistRobet May in 1976 [31] The Logistic map is written as (5)where symbol L is used to denote the Logistic map 119883

119894isin

[0 1] and represents the population at year 119894 and hence 1198830

represents the initial population at year 0 119903 is a positivenumber and represents the combined rate for reproductionand starvation [32] This map is often cited as an example ofhow complex chaotic behaviors can arise from a very simplenonlinear dynamic equation Consider

119883119894+1=L (119883

119894) = 119903119883

119894(1 minus 119883

119894) (5)

The Logistic map has been well studied The plots of thefirst few iterations of the Logistic map and its bifurcationdiagram are shown in Figure 1 It is well known that when119903 isin [357 4] (approximately) the Logistic map has chaoticbehaviors for most values but there are still certain isolatedranges of 119903 that show nonchaotic behavior for example119903 asymp 383 which corresponds to a big gap in its bifurcationdiagram

In reality the Logistic sequence is controlled by a set ofparameters of (119883

0 119903119873119898) where (119883

0 119903) are parameters in

the Logistic map119873 is the length of sequence and119898 denotesthe number of thrown-away samples Therefore a Logisticsequence 119883 can be denoted as (6) Based on this logisticsequence119883 of length119873 sorted sequence1198831015840 can be obtainedby sorting 119883 in the ascending order It is certain that 1198831015840 is apermutation of the original119883Therefore119883 and1198831015840 satisfy (7)

where 119901120587is a permutation mapping sequence and 119894 denotes

sequence element index Consider

119883 =L (1198830 119903119873119898) (6)

1198831015840

119894= 119883119901120587(119894)

(7)

119883 = [04000 09120 03050 08055 05954 09154

02943 07892 06322 08836 03908

09047 03277 08372 05179 09488

01846 05721]

(8)

1198831015840

= [01846 02943 03050 03277 03908 04000

05179 05721 05954 06322 07892

08055 08372 08836 09047 09120

09154 09488]

(9)

119901120587= 119871 (04 38 18 0)

= [17 7 3 13 11 1 15 18 5 9 8 4 14 10 12 2 6 16]

(10)

For example if (1198830 119903 119873119898) = (04 38 18 0) then 119883

and 1198831015840 are shown in (8) and (9) respectively Correspond-ingly the permutation sequence 119901

120587is determined as (10)

4 Mathematical Problems in Engineering

12

3

6

9

1

2

4

57

8

10

11

(a) Conventional Josephusproblem

P120587(3)

P120587(6)

P120587(9)

P120587(1)

P120587(2)

P120587(4)

P120587(5)P120587(7)

P120587(8)

P120587(10)

P120587(11)P120587(12)

(b) Chaotic Josephus problem

Figure 2 The chaotic Josephus problem

CJPS at length RCCJPM at size of R-by-C

Rearrangement

Figure 3 Rearranging a CJPS to a CJPM

3 Chaotic Josephus PermutationMatrix (CJPM)

31 Chaotic Josephus Permutation Sequence (CJPS) Fromprevious sections it is clear that the Josephus permutationsequence 119902

120587and the chaotic permutation sequence 119901

120587are

obtained from different mechanismsThe Josephus permuta-tion sequence 119902

120587is easy to obtain but not completely random-

like while the chaotic permutation sequence 119901120587is random-

like but requires a large amount of computations for main-taining accuracy It is desirable that a permutation sequencebe random-like and only costs moderate computations

In order to achieve the above objective the new Jose-phus permutation sequence based on a chaotic permutationsequence 119901

120587is defined whose initial positions on the circle

are not consecutive numbers like those in the conventionalJosephus problem This new problem can be restated as fol-lows (1) 119905 people with numbered tags stand in a circle (thesenumbers together form a permutation sequence accordingto sorting a chaotic sequence) (2) starting at predeterminedperson you count around the circle until you reach the 119899thperson take him out of the circle and have the members toclose the circle and record his tag number (3) repeat theprocess until only one person is left If we record the tags ofpeople who have been taken out of the circle as a sequencethen a chaotic Josephus permutation sequence is obtained

It is noticeable that the chaotic Josephus problem hasparameters for both the chaotic Logistic map and for theconventional Josephus problem In other words a chaoticJosephus permutation sequence 119888119902

120587is determined by (11)

where parameters 119905 119904 119899 have the same meanings as in (1)1198830

and 119903 are parameters in the Logistic map and 119898 determines

the number of thrown-away samples in the chaotic Logisticsequence Consider

119888119902120587= 119888119869 (119905 119904 119899 119883

0 119903 119898) (11)

In the conventional Josephus problem (controlled byparameters (119905 119904 119899) see (1)) a natural number sequence 1 to119905 is used to denote the tags for people standing in the circlewhile a permutated sequence 119901

120587(controlled by parameters

(1198830 119903119873119898) see (6)) is used in the chaotic Josephus problem

as the tags In order to match the sequence lengths of 119901120587

and 119902120587 119905 = 119873 is the condition that has to be satisfied

The conventional Josephus problem and the chaotic Josephusproblem for 119905 = 119873 = 12 are illustrated in Figure 2

Therefore the new Josephus permutation sequence 119888119902120587

can be denoted as a composed function of a conventionalJosephus permutation sequence 119902

120587and a chaotic permutation

sequence 119901120587

119888119902120587= 119901120587∘ 119902120587= 119901120587(119902120587) (12)

For example if 119902120587

= 119869(18 1 4) in (2) and119901120587

= 119871(04 38 18 0) in (10) are preknown 119888119902120587

=

119888119869(18 1 4 04 38 0) is obtained as (13) shows by using(12) Similarly (14)ndash(16) can be also obtained Consider

119888119902120587= 119888119869 (18 1 4 04 38 0)

= [13 18 4 2 7 15 14 16 1 10 3 8 11 6 12 17 9 5]

(13)

119888119902120587= 119888119869 (18 1 7 04 38 0)

= [15 10 3 8 17 9 7 14 1 16 2 12 6 11 4 13 18 5]

(14)

119888119902120587= 119888119869 (18 4 7 04 38 0)

= [9 6 1 10 13 14 11 2 5 3 17 16 7 18 12 15 8 4]

(15)

119888119902120587= 119888119869 (18 4 7 040001 38 0)

= [8 6 1 10 11 12 13 2 9 15 7 18 3 5 14 17 4 16]

(16)

Compared to the previous conventional Josephus permu-tation sequence (see (2)ndash(4)) the chaotic Josephus permuta-tion sequences (see (12)ndash(15)) are more random-like because(1) the difference between its very first elements is not relatedto its counting period anymore (2) the difference betweentwo 119888119902

120587s which are only different in their starting positions

is no longer equal to the difference of their starting positions(3) slight perturbations in chaotic map parameters lead tobig changes in resulting 119888119902

120587s (4) two neighbor elements in

119888119902120587may or may not be consecutive Therefore the chaotic

Josephus permutation sequence ismore random-like than theconventional Josephus permutation sequence

32 Chaotic Josephus Permutation Matrix (CJPM) Based onchaotic Josephus permutation sequence(s) a chaotic Jose-phus permutation matrix can be generated via various ways

Mathematical Problems in Engineering 5

1

2

3

4

5

6

7

8

9

10

20

30

40

50

60

2 4 6 8 0

(a) CJPM(119877 119862 1 4 1198830 119903 0)

2

3

4

5

6

7

8

9

2 4 6 81 0

10

20

30

40

50

60

(b) CJPM(119877 119862 1 7 1198830 119903 0)

2

3

4

5

6

7

8

9

2 4 6 81 0

10

20

30

40

50

60

(c) CJPM(119877 119862 4 7 1198830 119903 0)

1

2

3

4

5

6

7

8

9

2 4 6 80

10

20

30

40

50

60

(d) CJPM(119877 119862 1 4 1198830+ 119890 119903 0)

1

2

3

4

5

6

7

8

9

2 4 6 80

10

20

30

40

50

60

(e) CJPM(119877 119862 1 4 1198830+ 119890 119903 10)

2 4 6 81

2

3

4

5

6

7

8

9

0

10

20

30

40

50

60

(f) CJPM(119877 119862 1 4 1198830+ 119890 119903 + 119890 10)

Figure 4 Parametric CJPMs (Note 119877 = 8 119862 = 81198830= 04 119903 = 38 and 119890 = 00001)

PlaintextP

Initial key

CiphertextC

CJPMgenerator

Encryption key

M

One blockPixel

permutationPixel

substitutionS

Figure 5 Image encryption method based on CJPM

Among these methods Algorithm 2 illustrates a straightfor-ward method to obtain a CJPM via a CJPS by rearrangingCJPS elements to a matrix

It is noticeable that Algorithm 2 is equivalent to rearrang-ing a sequence of elements into a matrix following the orderillustrated in Figure 3

Therefore a CJPM is determined by the same set ofparameters controlling a CJPS In order to emphasize thematrix property the parameter 119905 in CJPS is replaced by twoparameters of height 119877 and width 119862 where 119905 = 119877119862 Similarlya CJPM is uniquely determined by a set of parameters(119877 119862 119904 119899 119883

0 119903 119898) as for a CJPS Mathematically this claim

can be denoted as

119872 = CJPM (119877 119862 119904 119899 1198830 119903 119898) (17)

tN

xr yr w

p = L(X0 s n)

cq = cJ(t s n X0 r m)

Initial key

q = J(t s n)

R C s nr m

X0

Figure 6 Key functions in CJPM

Figure 4 illustrates various CJPMs via Algorithm 2according to different parameter sets It is clear that CJPMis very sensitive to its parameters and small changes in theparameter set lead to distinct CJPMs

4 Image Encryption AlgorithmBased on CJPM

In 1949 Claude Shannon the father of ldquoInformationTheoryrdquoproposed that confusion and diffusion are two properties ofthe operation of a secure cipher where the term confusionrefers to making the relationship between the encryptionkey and the ciphertext a very complex and developed one

6 Mathematical Problems in Engineering

Center of gravity

PlaintextP

Reference

CJPM M

Initial key Encryption key

point (xr yr)R C s t r m

X0 = sin(arctan(ygxg))(xg yg)

Figure 7 The internal structure of CJPM generator

PlaintextP

M +M998400

P998400

SMod(P998400 F)Mod(MF)

Figure 8 The internal structure of CJPM generator

[33] and the term diffusion refers to the property that theredundancy in the statistics of the plaintext is ldquodissipatedrdquo inthe statistics of the ciphertext [33] In otherwords for a securecipher it has to have good confusion and diffusion properties(1) different ciphertexts are desired to have similar statistics(2) any slight change in a plaintext is desired to lead to bigdifference in its ciphertext The image encryption algorithmbased on CJPM is proposed in this section to meet these twocriteria

41 Flowchart of Image Encryption Algorithm Based on CJPMSince the CJPM is parametric and random-like it can beused for image encryption directly However considering therequirements from confusion and diffusion properties theencryption procedure can be described as Figure 5 showsThe plaintext image is first sent to the CJPM generator whichis a preparation stage for generating a CJPM119872 for future useLater this CJPM119872 is used as a reference matrix to permuteand substitute image pixels for each image block in the stagesof pixel permutation and pixel substitution respectively Thedecryption procedure is simply to reverse the encryptionprocedure

42 Key Schedule It is clear that the CJPM is the coreof the cipher and thus key is related to the used CJPMreference matrix 119872 Initial key is composed of parameters(119909119903 119910119903 119908 119903 119898 119877 119862 119904 119899) where (119909

119903 119910119903 119908) is used in CJPM

generator for obtaining plaintext-dependent parameter 1198830

used in the Logistic map (119877 119862) are used as the parameter 119905in (1) and the parameter119873 in (6) The functions of each partof the initial key are shown in Figure 6

The output encryption key is composed of (119877 119862

119904 119899 1198830 119903 119898) all of which are directly required for deter-

mining a CJPM according to (17) Among these parameters119877 119862 119904 119899 and 119898 are restricted to integers 119909

119903 119910119903 119903 and 119883

0

are decimals More specifically 119877 and 119862 should be positive

integers smaller than the plaintext image size 119904 and 119899 shouldbe positive integers below the product of 119877119862 119903 should be anumber in between [36 4] (119909

119903 119910119903) is an arbitrary point on

119909119910 plain with weight 119908 and119898 is a nonnegative integer

43 CJPM Generator In order to enhance the resistance todifferential attacks the CJPM generator used in Figure 5 isdesigned to be plaintext dependent Recall that a CJPM isdetermined by a set of parameters (119877 119862 119904 119899 119883

0 119903 119898) shown

in (17) In the CJPM generator for image encryption only theparameter119883

0is not directly given by the initial key but by the

plaintext and a reference point (119909119903 119910119903) controlling the weight

in calculating the center of gravity Once 1198830is generated

it is stored in the encryption key The whole procedure oftranslating the initial key to a plaintext-dependent CJPMmatrix119872 and encryption key is shown in Figure 7

A plaintext is considered as an object of pixels where itsupper-left corner pixel is the reference point located at (1 1)Correspondingly pixels next to it along 119909 and 119910 directionsare (2 1) and (1 2) respectively The center of gravity ofthis plaintext is calculated via (18) where 119875

119894denotes the

119894th pixel intensity value and 119909119894and 119910

119894denote the location

of the 119894th pixel in the image with respect to the upper-leftcorner Once the center of gravity (119909

119892 119910119892) is obtained the

initial value of Logistic map 1198830is also determined via (19)

where arctan(sdot) is the arc tangent function and sin(sdot) is thesine function It is easy to verify that the range of (19) is[0 1] which satisfies restrictions for the initial value119883

0in the

Logisticmap Finally all required parameters for aCJPM thatis (119877 119862 119904 119899 119883

0 119903 119898) are obtained and thus a CJPM 119872 is

generated Meanwhile the used parameters are stored as theencryption key which can be used in the decryption processConsider

119909119892=119909119903sdot 119908 + sum119909

119894119875119894

119908 + sum119875119894

119910119892=119910119903sdot 119908 + sum119910

119894119875119894

119908 + sum119875119894

(18)

1198830= 05 [sin(arctan(

119910119892

119909119892

)) + 1] (19)

It is worth noting that the plaintext-dependent CJPMgenerator guarantees that the proposed cipher has gooddiffusion property any slight changes in plaintext lead tobig difference in ciphertext This is because the resulting

Mathematical Problems in Engineering 7

(a) Original image (b) 32-by-32 (c) 64-by-64 (d) 128-by-128

0100200300400500600700800900

1000

0 50 100 150 200 250

(e) Histogram of (a)

0

100

200

300

400

500

600

0 50 100 150 200 250

(f) Histogram of (b)

0

100

200

300

400

500

600

0 50 100 150 200 250

(g) Histogram of (c)

0

100

200

300

400

500

600

0 50 100 150 200 250

(h) Histogram of (d)

Figure 9 Pixel substitution results for CJPM at various sizes

(a) Original image (b) 32-by-32 (c) 64-by-32 (d) 64-by-64

(e) 64-by-128 (f) 128-by-128 (g) 256-by-128 (h) 256-by-256

Figure 10 Pixel permutation results for CJPM at various sizes

CJPM matrix119872 is dependent on the parameter 1198830and the

parameter 1198830is dependent on the center of gravity for the

plaintext while the center of the plaintext gravity alters forany slight change in plaintext Furthermore this 119883

0is the

parameter in the chaotic map and thus any slight change

in initial value leads to a completely different trajectory asthe bifurcation diagram in Figure 1 shows Consequentlya completely different CJPM is obtained as the referencematrix Eventually this new reference matrix leads to adistinct ciphertext It can be demonstrated that without

8 Mathematical Problems in Engineering

Input 119888119902120587 a CJPS at length 119905 = 119877 times 119862

Output119872 a CJPM of size 119877-by-119862for 119894 = 1 119877

for 119895 = 1 119862119872(119894 119895) = 119888119902

120587((119894 minus 1) 119877 + 119895)

endend

Algorithm 2 Reshape a CJPS into a CJPM

preknowing the reference pointrsquos coordinate (119909119903 119910119903) and its

weight 119908 there is no way to slightly change plaintext imagepixels so that its gravity center is unchanged

44 Pixel Substitution Pixel substitution refers to the pro-cess of changing pixel values From the point of viewof statistics this process is to change the statistics of aplaintext image so that the statistics of resulting ciphertextimage is completely different Moreover it is desired thatdifferent ciphertext images have similar statistics whichimplies that ciphertext images tell little information aboutkeys and plaintext images As a result the confusion propertyis achieved

The proposed pixel substitution block is shown inFigure 8 where symbol 119865 = 2

119887 and 119887 denotes the numberof bits supported by the format of the plaintext image Forexample if plaintext 119875 is an 8-bit gray image then 119865 = 28 =256 if plaintext119875 is a binary image then 119865 = 21 = 2 It can benoticed that the reference CJPM119872 is first to convert to1198721015840whose format is compatible with the format of the plaintextimage later1198721015840 and 119875 are added over the space of 119865 finallythe encrypted image 119878 is obtained

Because elements in a CJPM matrix 119872 are uniformlydistributed on integer set [1 119905] after ldquoModrdquo operation 1198721015840still has a uniform-like distribution for its elements on[0 119865] As a result when this 1198721015840 is used to randomly shiftthe pixel value in plaintext the resulting pixel value inciphertext has an equal opportunity to be any value on [0 119865]As a result a uniform-like histogram is achieved in theciphertext

Pixel substitution results based on CJPMs are shown inFigure 9 It is clear that histograms before and after pixelsubstitution are very different and ciphertext histograms arevery flat compared to plaintext ones It is also noticeable thatas the size of the referenceCJPM increases the ciphertext has abetter encryption quality from the point view of human visualinspection

45 Pixel Permutation Pixel permutation refers to the pro-cess scrambling the positions of pixels in plaintext to disguiseinformation contained in an image Denote an image beforeand after pixel permutation as 119861 and 119860 respectively Assumethe way of indexing image pixels is the same as the order torearrange elements in a CJPM as Figure 3 shows Then thepixel permutation process can be mathematically defined asa permutation 119891

120587between domain 119861 and range 119860 forall119894 119895 isin

1 2 119905 exist119860119894= 119861119895= 119861119891120587(119894) where 119905 is the total

number of pixels in the image As we mentioned in previoussections a CJPM is generated from a CJPS which is apermutation sequence Therefore a CJPM can be directlyused for pixel permutation that is given a CJPM 119872 and itspixel permutation can be defined as 119891

120587(119894) = 119872(119894)

For example Figure 10 illustrates pixel permutationresults of the ldquoTuftsrdquo logo image for different CJPMsIt is clear that pixels are well shuffled within the imageblock As long as the size of CJPMprocessing imageblock increases the resulting shuffled image looksbetter and better When the block size reaches to orover 128-by-128 pixels in the plaintext are almost evenlyshuffled

It is clear that images after pixel permutation look verydifferent from the plaintext image It is also worth notingthat a CJPM also depends on a set of parameters besides thesize and that any change in other parameters will lead to acompletely different permutated image

5 Simulation Results and Security Analysis

An excellent encryption method should be both robustand effective Robustness means that the cipher should beapplicable to any plaintext image written in a supportedformat Effectiveness implies that the cipher is able to generateeligible ciphertext images which hide information frompossible intruders

In this section we focus on discussing the performanceof the CJPM based image cipher described in Section 4 It isworth noting that all following computer simulations are rununder MATLAB 2010a and Windows XP environment withCore 2 Quad 26GHz processors

51 Histogram Analysis Histogram analysis is one of themost straightforward evaluations for ciphertext quality for itdirectly analyzes the pixel distribution of a ciphertext image

Figure 11 illustrates image encryption results for variousplaintext images image ldquo113rdquo is a binary handwriting scannedimage selected from ICDAR 2009 database image ldquoLenardquois a commonly used image of gray type image ldquo5113rdquoand ldquotestpat1krdquo are used to mimic the possible complexpatterns in plaintext and they are both selected from USC-SIPI database It is worth noting that the used CJPM is at sizeof 256-by-256

Mathematical Problems in Engineering 9

0 10

051

152

253

354

0100200300400500600700800

0 50 100 150 200 2500

10002000300040005000600070008000

0 50 100 150 200 2500

051

152

253

354

455

0 50 100 150 200 250

0 10

05

1

15

2

25

0100200300400500600

0 50 100 150 200 2500

100200300400500600

0 50 100 150 200 2500

100020003000400050006000700080009000

10000

0 50 100 150 200 250

113 Lenna Testpat1k

times104

times106 times104

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Plai

ntex

tP

Plai

ntex

tP

Plai

ntex

tP

Plai

ntex

tP

Ciph

erte

xtC

Ciph

erte

xtC

Ciph

erte

xtC

Ciph

erte

xtC

times106

Hist

ogra

m o

fC

Hist

ogra

m o

fC

Hist

ogra

m o

fC

Hist

ogra

m o

fC

5113

Figure 11 Histogram analysis for image encryption based on CJPM

It is clear that no matter what histogram a plaintextimage has the histogram of its ciphertext image is flatwhich implies that the pixel distribution is almost uniformComplex patterns and large homogenous regions in plaintextimages are completely unintelligible and become random-like in ciphertext images These results imply that the pro-posed image encryption method based on CJPM is robustand effective for various image formats and contents

52 Adjacent Pixel Autocorrelation (APAC) Analysis Highcorrelations of adjacent pixels can be utilized to carryout cryptanalysis Therefore a secure encryption algorithmshould break the high correlation relationship between adja-cent pixels

In statistics the autocorrelation 119877119886of a random process

119883 describes the correlation between values of the process atdifferent points in time as a function of the two times or of thetime difference The autocorrelation coefficient 119877

119886is defined

in (20) where 119889 is the time difference 120583 is the mean valuedefined by (21) and 120590 is the standard deviation defined by(22) the definition of mathematical expectation is given in(23)

119877119886(119898) =

119864 [(119883119905minus 120583) (119883

119905+119889minus 120583)]

1205902

(20)

120583 = 119864 [119883] (21)

10 Mathematical Problems in Engineering

Plai

ntex

t

Plai

ntex

t

Ciph

erte

xt

Ciph

erte

xt

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 250

Plai

ntex

tCi

pher

text

0

50

100

150

200

250

0

50

100

150

200

250

DiagonalVerticalHorizontal

DiagonalVerticalHorizontal

Figure 12 APAC analysis for image encryption based on CJPM

Table 1 Adjacent pixel autocorrelation analysis

Correlation coefficients (10minus3) Horizontal Vertical DiagonalPlaintext Lena 9399652 9709000 9709894

Ciphertext

CJPM minus22281 minus03709 3265[6] 20970 161870 17805[8] minus25000 minus10000 minus93000[5] 57765 28434 20662[29] minus127212 minus602579 624427[9] minus134000 12000 398000[30] minus158900 minus653800 minus323100[16] 815860 minus400530 minus47150[10] 1257000 581000 504000

120590 = radic119864 [(119883 minus 120583)2

] (22)

119864 [119909] =

119873

sum

119894=1

119909119894

119873 (23)

The closer to zero this coefficient is the weaker the rela-tionship two different time functions have Specifically inadjacent pixel correlation test we let 119883 be the image pixelsequence and let 119889 be 1 that is compare to the adjacent pixelsequence

Based on the reference direction there are three waysof extracting a two-dimensional image to a one-dimensionalsequence and they are the horizontal adjacent correlation

coefficient the vertical adjacent correlation coefficient andthe diagonal adjacent correlation coefficient

ldquoLenardquo image in the 2nd column of Figure 11 is used asthe test plaintext image because its APAC is widely reportedby other encryption methods Peer comparison results ofthe proposed CJPM cipher and cited encryption methods onAPAC are shown in Table 1 (best results are bolded)

In addition Figure 12 shows the result of randomlyselected 1024 pairs of two adjacent pixels from the plaintextand the ciphertext along horizontal vertical and diagonaldirections where 119909- and 119910-axes denote the intensity values ofa randomly selected pixel and its adjacent pixel respectivelyIt is clear that after applying the CJPM based image cipherthe high correlations between adjacent pixels in plaintext arebroken

53 Plaintext Sensitivity Analysis In order to test the resis-tance of the cipher to differential attacks plaintext sensitivityanalysis is required for a secure cipher In differential attacksan adversary attempts to extract meaningful relationshipbetween a plaintext image and its ciphertext image bymakinga slight change usually only one pixel in the plaintext imagewhile encrypting the plaintext image with the same encryp-tion key By comparing the change in ciphertext imagesthe encryption key might be cracked and furthermore theinformation contained in ciphertext might be leaked

Although there are othermeasures [34 35] to evaluate theresistance of plaintext attacks two classic measures are theNumber of Pixel Change Rate (NPCR) and Unified Average

Mathematical Problems in Engineering 11

Before changing one pixel After changing one pixel DifferencePl

aint

ext i

mag

eCi

pher

text

imag

e

Figure 13 Plaintext sensitivity analysis (differential attacks) for image encryption based on CJPM

(a) Encrypted image (1 1 128128 1 1 4 0)

(b) Decrypted image (1 1 128128 1 1 4 0)

(c) Decrypted image (1 1 128128 1 2 4 0)

(d) Decrypted image (1 1 128128 1 1 39999 0)

Figure 14 Key sensitivity analysis

Changing Intensity (UACI) [5]TheNPCR is used tomeasurethe percentage of the number of pixels changed in ciphertextafter making a slight change in plaintext Therefore thetheoretical greatest upper-bound of the NPCR is 100 TheUACI is used to measure the averaged intensity change forpixels in ciphertext images after making a slight change ina plaintext image It is demonstrable that the UACI of an idealcipher for 8-bit gray images is about 03346 [36]

Suppose ciphertext images before and after one pixelchange in a plaintext image are 1198621 and 1198622 respectively thepixel values at grid (119894 119895) in 1198621 and 1198622 are denoted as 1198621(119894 119895)and 1198622(119894 119895) a bipolar array 119863 is defined as (24) then the

NPCR and UACI can be mathematically defined as (25) and(26) respectively where symbol119873 denotes the total numberof pixels in the ciphertext symbol 119865 denotes the largestsupported pixel value compatible with the ciphertext imageformat and | sdot | is the absolute value function Consider

119863(119894 119895) =

1 if 1198621 (119894 119895) = 1198622 (119894 119895)

0 if 1198621 (119894 119895) = 1198622

(119894 119895)

(24)

12 Mathematical Problems in Engineering

Table 2 NPCR and UACI analyses on ldquoLenardquo image

CPJMsize 16-by-16 32-by-32 64-by-64 128-by-128 256-by-256

NPCR 996170 996246 995895 995850 996338UACI 338205 338379 334048 334076 334040

NPCR = sum

119894119895

119863(119894 119895)

119873times 100 (25)

UACI = sum

119894119895

100381610038161003816100381610038161198621

(119894 119895) minus 1198622

(119894 119895)10038161003816100381610038161003816

(119865 sdot 119873)times 100 (26)

Table 2 shows the NPCR and UACI results for one pixelchange in the ldquoLenardquo image It is clear that for CJPMs atvarious sizes the proposed cipher has good performances inboth the NPCR and UACI analyses Simulation results fit theexpectations of the ideal cipher very well

Figure 13 shows a differential attack on ldquoLenardquo imagewhile keeping the encryption key unchanged It is noticeablethat the difference between two plaintext images is thepixel on Lenarsquos nose However the ciphertext images are sodifferent that the image of their difference is still random-like As a result the one pixel change in the plaintext image isldquodissipatedrdquo in ciphertext From this point view this plaintextsensitivity is closely related to the diffusion property of acipher In other words it is reasonable to claim that a cipherhas good NPCR and UACI results if it has good diffusionproperties

54 Key Space Analysis The encryption key in the proposedimage encryption method using CPJM is composed of a setof parameters (119909

119903 119910119903 119908 119877 119862 119904 119899 119903 119898) where (119909

119903 119910119903) is an

arbitrary point on 119909119910 plain119908 is a nonnegative decimal119877 and119862 should be positive integers smaller than the plaintext imagesize 119904 and 119899 should be positive integers below the productof 119877119862 119903 should be a number in between [36 4] and 119898 is anonnegative integer Therefore theoretically the key space ofthe proposed cipher is infinitely large

Because the chaotic Logistic map is used as the triggerfor pseudorandom sequences the proposed cipher has highkey sensitivities as well The results of key sensitivity analysisare shown in Figure 14 where the set of parameters writtenin parenthesis is the used key It is clear that unless thecorrect decryption key is applied a ciphertext image cannotbe restored

6 Conclusion

In this paper we discussed the generation of a chaotic Jose-phus permutation matrix by using the conventional Josephuspermutation sequences and the logistic chaotic map Theproposed CJPM is parametric and is uniquely dependent onthe set of parameters which is sufficiently large to providea secure size of key space As another heritage from thechaotic Logistic map the CJPM is highly sensitive to itsinitial values (parameters) Any slight change in parameters

leads to significant differences in resulting CJPM Simulationresults show that (1) the ciphertext image is random-likefrom the perspective of human visual inspection (2) theencryption quality is almost independent of the plaintextimage (3) the proposed encryptionmethod is able to encryptplaintext images with large homogeneous regions to secureciphertext images (4) histogram analysis also shows thatdifferent ciphertext images tend to have the uniform distri-bution on [0 119865] (5) adjacent pixel correlation analysis showsthat neighbor pixels in ciphertext have lower correlationsthan many existing encryption methods (6) the proposedcipher is highly sensitive to encryption key and plaintext (7)experimental UACI results of the proposed cipher are veryclose to those of the ideal one

The proposed cipher can be used for various image typesfor example binary images 8-bit gray images 16-bit grayimages RBG images and so forth The same encryption ideamay also be applied to audio video or other types of digitalformats

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Scientific Research Fundof Hunan Provincial Education Department under Grant(no 12B023) It is also supported by the National NaturalScience Foundation of China (nos 61204039 and 61106029)and Scientific Research Foundation for Returned ScholarsMinistry of Human Resources and Social Security of thePeoplersquos Republic of China

References

[1] D StinsonCryptographyTheory andPractice CRCPress 2006[2] M Yang N Bourbakis and S Li ldquoData-image-video encryp-

tionrdquo IEEE Potentials vol 23 no 3 pp 28ndash34 2004[3] FIPS PUB 46 Data Encryption Standard National Bureau of

Standards 1977[4] FIPS PUB 197 Avanced Encryption Standard Announcing the

Advanced Encryption Standard (AES) 2001[5] Y Mao and G Chen Chaos-Based Image Encryption Springer

Berlin Germany 2005[6] H Yang K-W Wong X Liao W Zhang and P Wei ldquoA fast

image encryption and authentication scheme based on chaoticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3507ndash3517 2010

[7] J Hu and F Han ldquoA pixel-based scrambling scheme for digitalmedical images protectionrdquo Journal of Network and ComputerApplications vol 32 no 4 pp 788ndash794 2009

[8] Z Shuo C Ruhua J Yingchun and G Shiping ldquoAn imageencryption algorithm based on multiple chaos and wavelettransformrdquo in Proceedings of the 2nd International CongressImage and Signal Processing (CISP rsquo09) pp 1ndash5 2009

Mathematical Problems in Engineering 13

[9] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010

[10] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009

[11] X Liao S Lai and Q Zhou ldquoA novel image encryptionalgorithm based on self-adaptive wave transmissionrdquo SignalProcessing vol 90 no 9 pp 2714ndash2722 2010

[12] S Fu-Yan L Shu-Tang and L Zong-Wang ldquoImage encryptionusing high-dimension chaotic systemrdquo Chinese Physics vol 16no 12 pp 3616ndash3623 2007

[13] V Patidar N K Pareek and K K Sud ldquoA new substitution-diffusion based image cipher using chaotic standard and logisticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 7 pp 3056ndash3075 2009

[14] Y Wu J P Noonan G Yang and H Jin ldquoImage encryptionusing the two-dimensional logistic chaotic maprdquo Journal ofElectronic Imaging vol 21 no 1 Article ID 013014 2012

[15] Y Wu J P Noonan and S Agaian ldquoA wheel-switch chaoticsystem for image encryptionrdquo in Proceedings of the InternationalConference on System Science and Engineering (ICSSE rsquo11) pp23ndash27 June 2011

[16] L Zhang X Liao andXWang ldquoAn image encryption approachbased on chaoticmapsrdquoChaos Solitons and Fractals vol 24 no3 pp 759ndash765 2005

[17] H S Kwok and W K S Tang ldquoA fast image encryption systembased on chaotic maps with finite precision representationrdquoChaos Solitons and Fractals vol 32 no 4 pp 1518ndash1529 2007

[18] J Jian Y Shi C Hu Q Ma and J Li ldquoEncryption of digitalimage based on chaos systemrdquo in Computer and ComputingTechnologies in Agriculture II vol 2 pp 1145ndash1151 2009

[19] P Van-Roy and S Haridi Concepts Techniques and Models ofComputer Programming MIT Press 2004

[20] P SchumerMathematical Journeys Wiley-Interscience 2004[21] G Ye X Huang and C Zhu ldquoImage encryption algorithm of

double scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007

[22] XDesheng andXYueshan ldquoDigital image scrambling based onjosephus traversingrdquo Computer Engineering and Applicationsno 10 pp 44ndash46 2005

[23] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2005

[24] Y-P Zhang Z-J Zhai W Liu X Nie S-P Cao and W-D Dai ldquoDigital image encryption algorithm based on chaosand improved DESrdquo in Proceedings of the IEEE InternationalConference on Systems Man and Cybernetics (SMC rsquo09) pp474ndash479 October 2009

[25] J Lang R Tao and Y Wang ldquoImage encryption based on themultiple-parameter discrete fractional Fourier transform andchaos functionrdquo Optics Communications vol 283 no 10 pp2092ndash2096 2010

[26] S Yang and S Sun ldquoVideo encryptionmethod based on chaoticmaps in DCT domainrdquo Progress in Natural Science vol 18 no10 pp 1299ndash1304 2008

[27] V Wu J P Noonan and S Agaian ldquoBinary data encryptionusing the Sudoku block cipherrdquo in Proceedings of the IEEEInternational Conference on SystemsMan andCybernetics (SMCrsquo10) pp 3915ndash3921 October 2010

[28] Y Wu Y Zhou J P Noonan K Panetta and S Agaian ldquoImageencryption using the Sudoku matrixrdquo in Proceedings of theMobileMultimediaImage Processing Security and ApplicationsApril 2010

[29] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004

[30] H Gao Y Zhang S Liang and D Li ldquoA new chaotic algorithmfor image encryptionrdquo Chaos Solitons and Fractals vol 29 no2 pp 393ndash399 2006

[31] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[32] J Thompson and H Stewart Nonlinear Dynamics and ChaosJohn Wiley amp Sons 2002

[33] C E Shannon ldquoCommunication Theory of Secrecy SystemsrdquoBell Systems Technical Journal vol 28 pp 656ndash715 1949

[34] Y Wu Y Zhou G Saveriades S Agaian J P Noonan andP Natarajan ldquoLocal Shannon entropy measure with statisticaltests for image randomnessrdquo Information Sciences vol 222 no323 342 pages 2013

[35] Y Wu S Agaian and J P Noona ldquoA novel method of testingimage randomness with applications to image shuffling andencryptionrdquo in Defense Security and Sensing Proceedings ofthe SPIE pp 875507ndash875507 2013

[36] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI random-ness tests for image encryptionrdquo Cyber Journals pp 31ndash38 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

12

3

6

9

1

2

4

57

8

10

11

(a) Conventional Josephusproblem

P120587(3)

P120587(6)

P120587(9)

P120587(1)

P120587(2)

P120587(4)

P120587(5)P120587(7)

P120587(8)

P120587(10)

P120587(11)P120587(12)

(b) Chaotic Josephus problem

Figure 2 The chaotic Josephus problem

CJPS at length RCCJPM at size of R-by-C

Rearrangement

Figure 3 Rearranging a CJPS to a CJPM

3 Chaotic Josephus PermutationMatrix (CJPM)

31 Chaotic Josephus Permutation Sequence (CJPS) Fromprevious sections it is clear that the Josephus permutationsequence 119902

120587and the chaotic permutation sequence 119901

120587are

obtained from different mechanismsThe Josephus permuta-tion sequence 119902

120587is easy to obtain but not completely random-

like while the chaotic permutation sequence 119901120587is random-

like but requires a large amount of computations for main-taining accuracy It is desirable that a permutation sequencebe random-like and only costs moderate computations

In order to achieve the above objective the new Jose-phus permutation sequence based on a chaotic permutationsequence 119901

120587is defined whose initial positions on the circle

are not consecutive numbers like those in the conventionalJosephus problem This new problem can be restated as fol-lows (1) 119905 people with numbered tags stand in a circle (thesenumbers together form a permutation sequence accordingto sorting a chaotic sequence) (2) starting at predeterminedperson you count around the circle until you reach the 119899thperson take him out of the circle and have the members toclose the circle and record his tag number (3) repeat theprocess until only one person is left If we record the tags ofpeople who have been taken out of the circle as a sequencethen a chaotic Josephus permutation sequence is obtained

It is noticeable that the chaotic Josephus problem hasparameters for both the chaotic Logistic map and for theconventional Josephus problem In other words a chaoticJosephus permutation sequence 119888119902

120587is determined by (11)

where parameters 119905 119904 119899 have the same meanings as in (1)1198830

and 119903 are parameters in the Logistic map and 119898 determines

the number of thrown-away samples in the chaotic Logisticsequence Consider

119888119902120587= 119888119869 (119905 119904 119899 119883

0 119903 119898) (11)

In the conventional Josephus problem (controlled byparameters (119905 119904 119899) see (1)) a natural number sequence 1 to119905 is used to denote the tags for people standing in the circlewhile a permutated sequence 119901

120587(controlled by parameters

(1198830 119903119873119898) see (6)) is used in the chaotic Josephus problem

as the tags In order to match the sequence lengths of 119901120587

and 119902120587 119905 = 119873 is the condition that has to be satisfied

The conventional Josephus problem and the chaotic Josephusproblem for 119905 = 119873 = 12 are illustrated in Figure 2

Therefore the new Josephus permutation sequence 119888119902120587

can be denoted as a composed function of a conventionalJosephus permutation sequence 119902

120587and a chaotic permutation

sequence 119901120587

119888119902120587= 119901120587∘ 119902120587= 119901120587(119902120587) (12)

For example if 119902120587

= 119869(18 1 4) in (2) and119901120587

= 119871(04 38 18 0) in (10) are preknown 119888119902120587

=

119888119869(18 1 4 04 38 0) is obtained as (13) shows by using(12) Similarly (14)ndash(16) can be also obtained Consider

119888119902120587= 119888119869 (18 1 4 04 38 0)

= [13 18 4 2 7 15 14 16 1 10 3 8 11 6 12 17 9 5]

(13)

119888119902120587= 119888119869 (18 1 7 04 38 0)

= [15 10 3 8 17 9 7 14 1 16 2 12 6 11 4 13 18 5]

(14)

119888119902120587= 119888119869 (18 4 7 04 38 0)

= [9 6 1 10 13 14 11 2 5 3 17 16 7 18 12 15 8 4]

(15)

119888119902120587= 119888119869 (18 4 7 040001 38 0)

= [8 6 1 10 11 12 13 2 9 15 7 18 3 5 14 17 4 16]

(16)

Compared to the previous conventional Josephus permu-tation sequence (see (2)ndash(4)) the chaotic Josephus permuta-tion sequences (see (12)ndash(15)) are more random-like because(1) the difference between its very first elements is not relatedto its counting period anymore (2) the difference betweentwo 119888119902

120587s which are only different in their starting positions

is no longer equal to the difference of their starting positions(3) slight perturbations in chaotic map parameters lead tobig changes in resulting 119888119902

120587s (4) two neighbor elements in

119888119902120587may or may not be consecutive Therefore the chaotic

Josephus permutation sequence ismore random-like than theconventional Josephus permutation sequence

32 Chaotic Josephus Permutation Matrix (CJPM) Based onchaotic Josephus permutation sequence(s) a chaotic Jose-phus permutation matrix can be generated via various ways

Mathematical Problems in Engineering 5

1

2

3

4

5

6

7

8

9

10

20

30

40

50

60

2 4 6 8 0

(a) CJPM(119877 119862 1 4 1198830 119903 0)

2

3

4

5

6

7

8

9

2 4 6 81 0

10

20

30

40

50

60

(b) CJPM(119877 119862 1 7 1198830 119903 0)

2

3

4

5

6

7

8

9

2 4 6 81 0

10

20

30

40

50

60

(c) CJPM(119877 119862 4 7 1198830 119903 0)

1

2

3

4

5

6

7

8

9

2 4 6 80

10

20

30

40

50

60

(d) CJPM(119877 119862 1 4 1198830+ 119890 119903 0)

1

2

3

4

5

6

7

8

9

2 4 6 80

10

20

30

40

50

60

(e) CJPM(119877 119862 1 4 1198830+ 119890 119903 10)

2 4 6 81

2

3

4

5

6

7

8

9

0

10

20

30

40

50

60

(f) CJPM(119877 119862 1 4 1198830+ 119890 119903 + 119890 10)

Figure 4 Parametric CJPMs (Note 119877 = 8 119862 = 81198830= 04 119903 = 38 and 119890 = 00001)

PlaintextP

Initial key

CiphertextC

CJPMgenerator

Encryption key

M

One blockPixel

permutationPixel

substitutionS

Figure 5 Image encryption method based on CJPM

Among these methods Algorithm 2 illustrates a straightfor-ward method to obtain a CJPM via a CJPS by rearrangingCJPS elements to a matrix

It is noticeable that Algorithm 2 is equivalent to rearrang-ing a sequence of elements into a matrix following the orderillustrated in Figure 3

Therefore a CJPM is determined by the same set ofparameters controlling a CJPS In order to emphasize thematrix property the parameter 119905 in CJPS is replaced by twoparameters of height 119877 and width 119862 where 119905 = 119877119862 Similarlya CJPM is uniquely determined by a set of parameters(119877 119862 119904 119899 119883

0 119903 119898) as for a CJPS Mathematically this claim

can be denoted as

119872 = CJPM (119877 119862 119904 119899 1198830 119903 119898) (17)

tN

xr yr w

p = L(X0 s n)

cq = cJ(t s n X0 r m)

Initial key

q = J(t s n)

R C s nr m

X0

Figure 6 Key functions in CJPM

Figure 4 illustrates various CJPMs via Algorithm 2according to different parameter sets It is clear that CJPMis very sensitive to its parameters and small changes in theparameter set lead to distinct CJPMs

4 Image Encryption AlgorithmBased on CJPM

In 1949 Claude Shannon the father of ldquoInformationTheoryrdquoproposed that confusion and diffusion are two properties ofthe operation of a secure cipher where the term confusionrefers to making the relationship between the encryptionkey and the ciphertext a very complex and developed one

6 Mathematical Problems in Engineering

Center of gravity

PlaintextP

Reference

CJPM M

Initial key Encryption key

point (xr yr)R C s t r m

X0 = sin(arctan(ygxg))(xg yg)

Figure 7 The internal structure of CJPM generator

PlaintextP

M +M998400

P998400

SMod(P998400 F)Mod(MF)

Figure 8 The internal structure of CJPM generator

[33] and the term diffusion refers to the property that theredundancy in the statistics of the plaintext is ldquodissipatedrdquo inthe statistics of the ciphertext [33] In otherwords for a securecipher it has to have good confusion and diffusion properties(1) different ciphertexts are desired to have similar statistics(2) any slight change in a plaintext is desired to lead to bigdifference in its ciphertext The image encryption algorithmbased on CJPM is proposed in this section to meet these twocriteria

41 Flowchart of Image Encryption Algorithm Based on CJPMSince the CJPM is parametric and random-like it can beused for image encryption directly However considering therequirements from confusion and diffusion properties theencryption procedure can be described as Figure 5 showsThe plaintext image is first sent to the CJPM generator whichis a preparation stage for generating a CJPM119872 for future useLater this CJPM119872 is used as a reference matrix to permuteand substitute image pixels for each image block in the stagesof pixel permutation and pixel substitution respectively Thedecryption procedure is simply to reverse the encryptionprocedure

42 Key Schedule It is clear that the CJPM is the coreof the cipher and thus key is related to the used CJPMreference matrix 119872 Initial key is composed of parameters(119909119903 119910119903 119908 119903 119898 119877 119862 119904 119899) where (119909

119903 119910119903 119908) is used in CJPM

generator for obtaining plaintext-dependent parameter 1198830

used in the Logistic map (119877 119862) are used as the parameter 119905in (1) and the parameter119873 in (6) The functions of each partof the initial key are shown in Figure 6

The output encryption key is composed of (119877 119862

119904 119899 1198830 119903 119898) all of which are directly required for deter-

mining a CJPM according to (17) Among these parameters119877 119862 119904 119899 and 119898 are restricted to integers 119909

119903 119910119903 119903 and 119883

0

are decimals More specifically 119877 and 119862 should be positive

integers smaller than the plaintext image size 119904 and 119899 shouldbe positive integers below the product of 119877119862 119903 should be anumber in between [36 4] (119909

119903 119910119903) is an arbitrary point on

119909119910 plain with weight 119908 and119898 is a nonnegative integer

43 CJPM Generator In order to enhance the resistance todifferential attacks the CJPM generator used in Figure 5 isdesigned to be plaintext dependent Recall that a CJPM isdetermined by a set of parameters (119877 119862 119904 119899 119883

0 119903 119898) shown

in (17) In the CJPM generator for image encryption only theparameter119883

0is not directly given by the initial key but by the

plaintext and a reference point (119909119903 119910119903) controlling the weight

in calculating the center of gravity Once 1198830is generated

it is stored in the encryption key The whole procedure oftranslating the initial key to a plaintext-dependent CJPMmatrix119872 and encryption key is shown in Figure 7

A plaintext is considered as an object of pixels where itsupper-left corner pixel is the reference point located at (1 1)Correspondingly pixels next to it along 119909 and 119910 directionsare (2 1) and (1 2) respectively The center of gravity ofthis plaintext is calculated via (18) where 119875

119894denotes the

119894th pixel intensity value and 119909119894and 119910

119894denote the location

of the 119894th pixel in the image with respect to the upper-leftcorner Once the center of gravity (119909

119892 119910119892) is obtained the

initial value of Logistic map 1198830is also determined via (19)

where arctan(sdot) is the arc tangent function and sin(sdot) is thesine function It is easy to verify that the range of (19) is[0 1] which satisfies restrictions for the initial value119883

0in the

Logisticmap Finally all required parameters for aCJPM thatis (119877 119862 119904 119899 119883

0 119903 119898) are obtained and thus a CJPM 119872 is

generated Meanwhile the used parameters are stored as theencryption key which can be used in the decryption processConsider

119909119892=119909119903sdot 119908 + sum119909

119894119875119894

119908 + sum119875119894

119910119892=119910119903sdot 119908 + sum119910

119894119875119894

119908 + sum119875119894

(18)

1198830= 05 [sin(arctan(

119910119892

119909119892

)) + 1] (19)

It is worth noting that the plaintext-dependent CJPMgenerator guarantees that the proposed cipher has gooddiffusion property any slight changes in plaintext lead tobig difference in ciphertext This is because the resulting

Mathematical Problems in Engineering 7

(a) Original image (b) 32-by-32 (c) 64-by-64 (d) 128-by-128

0100200300400500600700800900

1000

0 50 100 150 200 250

(e) Histogram of (a)

0

100

200

300

400

500

600

0 50 100 150 200 250

(f) Histogram of (b)

0

100

200

300

400

500

600

0 50 100 150 200 250

(g) Histogram of (c)

0

100

200

300

400

500

600

0 50 100 150 200 250

(h) Histogram of (d)

Figure 9 Pixel substitution results for CJPM at various sizes

(a) Original image (b) 32-by-32 (c) 64-by-32 (d) 64-by-64

(e) 64-by-128 (f) 128-by-128 (g) 256-by-128 (h) 256-by-256

Figure 10 Pixel permutation results for CJPM at various sizes

CJPM matrix119872 is dependent on the parameter 1198830and the

parameter 1198830is dependent on the center of gravity for the

plaintext while the center of the plaintext gravity alters forany slight change in plaintext Furthermore this 119883

0is the

parameter in the chaotic map and thus any slight change

in initial value leads to a completely different trajectory asthe bifurcation diagram in Figure 1 shows Consequentlya completely different CJPM is obtained as the referencematrix Eventually this new reference matrix leads to adistinct ciphertext It can be demonstrated that without

8 Mathematical Problems in Engineering

Input 119888119902120587 a CJPS at length 119905 = 119877 times 119862

Output119872 a CJPM of size 119877-by-119862for 119894 = 1 119877

for 119895 = 1 119862119872(119894 119895) = 119888119902

120587((119894 minus 1) 119877 + 119895)

endend

Algorithm 2 Reshape a CJPS into a CJPM

preknowing the reference pointrsquos coordinate (119909119903 119910119903) and its

weight 119908 there is no way to slightly change plaintext imagepixels so that its gravity center is unchanged

44 Pixel Substitution Pixel substitution refers to the pro-cess of changing pixel values From the point of viewof statistics this process is to change the statistics of aplaintext image so that the statistics of resulting ciphertextimage is completely different Moreover it is desired thatdifferent ciphertext images have similar statistics whichimplies that ciphertext images tell little information aboutkeys and plaintext images As a result the confusion propertyis achieved

The proposed pixel substitution block is shown inFigure 8 where symbol 119865 = 2

119887 and 119887 denotes the numberof bits supported by the format of the plaintext image Forexample if plaintext 119875 is an 8-bit gray image then 119865 = 28 =256 if plaintext119875 is a binary image then 119865 = 21 = 2 It can benoticed that the reference CJPM119872 is first to convert to1198721015840whose format is compatible with the format of the plaintextimage later1198721015840 and 119875 are added over the space of 119865 finallythe encrypted image 119878 is obtained

Because elements in a CJPM matrix 119872 are uniformlydistributed on integer set [1 119905] after ldquoModrdquo operation 1198721015840still has a uniform-like distribution for its elements on[0 119865] As a result when this 1198721015840 is used to randomly shiftthe pixel value in plaintext the resulting pixel value inciphertext has an equal opportunity to be any value on [0 119865]As a result a uniform-like histogram is achieved in theciphertext

Pixel substitution results based on CJPMs are shown inFigure 9 It is clear that histograms before and after pixelsubstitution are very different and ciphertext histograms arevery flat compared to plaintext ones It is also noticeable thatas the size of the referenceCJPM increases the ciphertext has abetter encryption quality from the point view of human visualinspection

45 Pixel Permutation Pixel permutation refers to the pro-cess scrambling the positions of pixels in plaintext to disguiseinformation contained in an image Denote an image beforeand after pixel permutation as 119861 and 119860 respectively Assumethe way of indexing image pixels is the same as the order torearrange elements in a CJPM as Figure 3 shows Then thepixel permutation process can be mathematically defined asa permutation 119891

120587between domain 119861 and range 119860 forall119894 119895 isin

1 2 119905 exist119860119894= 119861119895= 119861119891120587(119894) where 119905 is the total

number of pixels in the image As we mentioned in previoussections a CJPM is generated from a CJPS which is apermutation sequence Therefore a CJPM can be directlyused for pixel permutation that is given a CJPM 119872 and itspixel permutation can be defined as 119891

120587(119894) = 119872(119894)

For example Figure 10 illustrates pixel permutationresults of the ldquoTuftsrdquo logo image for different CJPMsIt is clear that pixels are well shuffled within the imageblock As long as the size of CJPMprocessing imageblock increases the resulting shuffled image looksbetter and better When the block size reaches to orover 128-by-128 pixels in the plaintext are almost evenlyshuffled

It is clear that images after pixel permutation look verydifferent from the plaintext image It is also worth notingthat a CJPM also depends on a set of parameters besides thesize and that any change in other parameters will lead to acompletely different permutated image

5 Simulation Results and Security Analysis

An excellent encryption method should be both robustand effective Robustness means that the cipher should beapplicable to any plaintext image written in a supportedformat Effectiveness implies that the cipher is able to generateeligible ciphertext images which hide information frompossible intruders

In this section we focus on discussing the performanceof the CJPM based image cipher described in Section 4 It isworth noting that all following computer simulations are rununder MATLAB 2010a and Windows XP environment withCore 2 Quad 26GHz processors

51 Histogram Analysis Histogram analysis is one of themost straightforward evaluations for ciphertext quality for itdirectly analyzes the pixel distribution of a ciphertext image

Figure 11 illustrates image encryption results for variousplaintext images image ldquo113rdquo is a binary handwriting scannedimage selected from ICDAR 2009 database image ldquoLenardquois a commonly used image of gray type image ldquo5113rdquoand ldquotestpat1krdquo are used to mimic the possible complexpatterns in plaintext and they are both selected from USC-SIPI database It is worth noting that the used CJPM is at sizeof 256-by-256

Mathematical Problems in Engineering 9

0 10

051

152

253

354

0100200300400500600700800

0 50 100 150 200 2500

10002000300040005000600070008000

0 50 100 150 200 2500

051

152

253

354

455

0 50 100 150 200 250

0 10

05

1

15

2

25

0100200300400500600

0 50 100 150 200 2500

100200300400500600

0 50 100 150 200 2500

100020003000400050006000700080009000

10000

0 50 100 150 200 250

113 Lenna Testpat1k

times104

times106 times104

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Plai

ntex

tP

Plai

ntex

tP

Plai

ntex

tP

Plai

ntex

tP

Ciph

erte

xtC

Ciph

erte

xtC

Ciph

erte

xtC

Ciph

erte

xtC

times106

Hist

ogra

m o

fC

Hist

ogra

m o

fC

Hist

ogra

m o

fC

Hist

ogra

m o

fC

5113

Figure 11 Histogram analysis for image encryption based on CJPM

It is clear that no matter what histogram a plaintextimage has the histogram of its ciphertext image is flatwhich implies that the pixel distribution is almost uniformComplex patterns and large homogenous regions in plaintextimages are completely unintelligible and become random-like in ciphertext images These results imply that the pro-posed image encryption method based on CJPM is robustand effective for various image formats and contents

52 Adjacent Pixel Autocorrelation (APAC) Analysis Highcorrelations of adjacent pixels can be utilized to carryout cryptanalysis Therefore a secure encryption algorithmshould break the high correlation relationship between adja-cent pixels

In statistics the autocorrelation 119877119886of a random process

119883 describes the correlation between values of the process atdifferent points in time as a function of the two times or of thetime difference The autocorrelation coefficient 119877

119886is defined

in (20) where 119889 is the time difference 120583 is the mean valuedefined by (21) and 120590 is the standard deviation defined by(22) the definition of mathematical expectation is given in(23)

119877119886(119898) =

119864 [(119883119905minus 120583) (119883

119905+119889minus 120583)]

1205902

(20)

120583 = 119864 [119883] (21)

10 Mathematical Problems in Engineering

Plai

ntex

t

Plai

ntex

t

Ciph

erte

xt

Ciph

erte

xt

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 250

Plai

ntex

tCi

pher

text

0

50

100

150

200

250

0

50

100

150

200

250

DiagonalVerticalHorizontal

DiagonalVerticalHorizontal

Figure 12 APAC analysis for image encryption based on CJPM

Table 1 Adjacent pixel autocorrelation analysis

Correlation coefficients (10minus3) Horizontal Vertical DiagonalPlaintext Lena 9399652 9709000 9709894

Ciphertext

CJPM minus22281 minus03709 3265[6] 20970 161870 17805[8] minus25000 minus10000 minus93000[5] 57765 28434 20662[29] minus127212 minus602579 624427[9] minus134000 12000 398000[30] minus158900 minus653800 minus323100[16] 815860 minus400530 minus47150[10] 1257000 581000 504000

120590 = radic119864 [(119883 minus 120583)2

] (22)

119864 [119909] =

119873

sum

119894=1

119909119894

119873 (23)

The closer to zero this coefficient is the weaker the rela-tionship two different time functions have Specifically inadjacent pixel correlation test we let 119883 be the image pixelsequence and let 119889 be 1 that is compare to the adjacent pixelsequence

Based on the reference direction there are three waysof extracting a two-dimensional image to a one-dimensionalsequence and they are the horizontal adjacent correlation

coefficient the vertical adjacent correlation coefficient andthe diagonal adjacent correlation coefficient

ldquoLenardquo image in the 2nd column of Figure 11 is used asthe test plaintext image because its APAC is widely reportedby other encryption methods Peer comparison results ofthe proposed CJPM cipher and cited encryption methods onAPAC are shown in Table 1 (best results are bolded)

In addition Figure 12 shows the result of randomlyselected 1024 pairs of two adjacent pixels from the plaintextand the ciphertext along horizontal vertical and diagonaldirections where 119909- and 119910-axes denote the intensity values ofa randomly selected pixel and its adjacent pixel respectivelyIt is clear that after applying the CJPM based image cipherthe high correlations between adjacent pixels in plaintext arebroken

53 Plaintext Sensitivity Analysis In order to test the resis-tance of the cipher to differential attacks plaintext sensitivityanalysis is required for a secure cipher In differential attacksan adversary attempts to extract meaningful relationshipbetween a plaintext image and its ciphertext image bymakinga slight change usually only one pixel in the plaintext imagewhile encrypting the plaintext image with the same encryp-tion key By comparing the change in ciphertext imagesthe encryption key might be cracked and furthermore theinformation contained in ciphertext might be leaked

Although there are othermeasures [34 35] to evaluate theresistance of plaintext attacks two classic measures are theNumber of Pixel Change Rate (NPCR) and Unified Average

Mathematical Problems in Engineering 11

Before changing one pixel After changing one pixel DifferencePl

aint

ext i

mag

eCi

pher

text

imag

e

Figure 13 Plaintext sensitivity analysis (differential attacks) for image encryption based on CJPM

(a) Encrypted image (1 1 128128 1 1 4 0)

(b) Decrypted image (1 1 128128 1 1 4 0)

(c) Decrypted image (1 1 128128 1 2 4 0)

(d) Decrypted image (1 1 128128 1 1 39999 0)

Figure 14 Key sensitivity analysis

Changing Intensity (UACI) [5]TheNPCR is used tomeasurethe percentage of the number of pixels changed in ciphertextafter making a slight change in plaintext Therefore thetheoretical greatest upper-bound of the NPCR is 100 TheUACI is used to measure the averaged intensity change forpixels in ciphertext images after making a slight change ina plaintext image It is demonstrable that the UACI of an idealcipher for 8-bit gray images is about 03346 [36]

Suppose ciphertext images before and after one pixelchange in a plaintext image are 1198621 and 1198622 respectively thepixel values at grid (119894 119895) in 1198621 and 1198622 are denoted as 1198621(119894 119895)and 1198622(119894 119895) a bipolar array 119863 is defined as (24) then the

NPCR and UACI can be mathematically defined as (25) and(26) respectively where symbol119873 denotes the total numberof pixels in the ciphertext symbol 119865 denotes the largestsupported pixel value compatible with the ciphertext imageformat and | sdot | is the absolute value function Consider

119863(119894 119895) =

1 if 1198621 (119894 119895) = 1198622 (119894 119895)

0 if 1198621 (119894 119895) = 1198622

(119894 119895)

(24)

12 Mathematical Problems in Engineering

Table 2 NPCR and UACI analyses on ldquoLenardquo image

CPJMsize 16-by-16 32-by-32 64-by-64 128-by-128 256-by-256

NPCR 996170 996246 995895 995850 996338UACI 338205 338379 334048 334076 334040

NPCR = sum

119894119895

119863(119894 119895)

119873times 100 (25)

UACI = sum

119894119895

100381610038161003816100381610038161198621

(119894 119895) minus 1198622

(119894 119895)10038161003816100381610038161003816

(119865 sdot 119873)times 100 (26)

Table 2 shows the NPCR and UACI results for one pixelchange in the ldquoLenardquo image It is clear that for CJPMs atvarious sizes the proposed cipher has good performances inboth the NPCR and UACI analyses Simulation results fit theexpectations of the ideal cipher very well

Figure 13 shows a differential attack on ldquoLenardquo imagewhile keeping the encryption key unchanged It is noticeablethat the difference between two plaintext images is thepixel on Lenarsquos nose However the ciphertext images are sodifferent that the image of their difference is still random-like As a result the one pixel change in the plaintext image isldquodissipatedrdquo in ciphertext From this point view this plaintextsensitivity is closely related to the diffusion property of acipher In other words it is reasonable to claim that a cipherhas good NPCR and UACI results if it has good diffusionproperties

54 Key Space Analysis The encryption key in the proposedimage encryption method using CPJM is composed of a setof parameters (119909

119903 119910119903 119908 119877 119862 119904 119899 119903 119898) where (119909

119903 119910119903) is an

arbitrary point on 119909119910 plain119908 is a nonnegative decimal119877 and119862 should be positive integers smaller than the plaintext imagesize 119904 and 119899 should be positive integers below the productof 119877119862 119903 should be a number in between [36 4] and 119898 is anonnegative integer Therefore theoretically the key space ofthe proposed cipher is infinitely large

Because the chaotic Logistic map is used as the triggerfor pseudorandom sequences the proposed cipher has highkey sensitivities as well The results of key sensitivity analysisare shown in Figure 14 where the set of parameters writtenin parenthesis is the used key It is clear that unless thecorrect decryption key is applied a ciphertext image cannotbe restored

6 Conclusion

In this paper we discussed the generation of a chaotic Jose-phus permutation matrix by using the conventional Josephuspermutation sequences and the logistic chaotic map Theproposed CJPM is parametric and is uniquely dependent onthe set of parameters which is sufficiently large to providea secure size of key space As another heritage from thechaotic Logistic map the CJPM is highly sensitive to itsinitial values (parameters) Any slight change in parameters

leads to significant differences in resulting CJPM Simulationresults show that (1) the ciphertext image is random-likefrom the perspective of human visual inspection (2) theencryption quality is almost independent of the plaintextimage (3) the proposed encryptionmethod is able to encryptplaintext images with large homogeneous regions to secureciphertext images (4) histogram analysis also shows thatdifferent ciphertext images tend to have the uniform distri-bution on [0 119865] (5) adjacent pixel correlation analysis showsthat neighbor pixels in ciphertext have lower correlationsthan many existing encryption methods (6) the proposedcipher is highly sensitive to encryption key and plaintext (7)experimental UACI results of the proposed cipher are veryclose to those of the ideal one

The proposed cipher can be used for various image typesfor example binary images 8-bit gray images 16-bit grayimages RBG images and so forth The same encryption ideamay also be applied to audio video or other types of digitalformats

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Scientific Research Fundof Hunan Provincial Education Department under Grant(no 12B023) It is also supported by the National NaturalScience Foundation of China (nos 61204039 and 61106029)and Scientific Research Foundation for Returned ScholarsMinistry of Human Resources and Social Security of thePeoplersquos Republic of China

References

[1] D StinsonCryptographyTheory andPractice CRCPress 2006[2] M Yang N Bourbakis and S Li ldquoData-image-video encryp-

tionrdquo IEEE Potentials vol 23 no 3 pp 28ndash34 2004[3] FIPS PUB 46 Data Encryption Standard National Bureau of

Standards 1977[4] FIPS PUB 197 Avanced Encryption Standard Announcing the

Advanced Encryption Standard (AES) 2001[5] Y Mao and G Chen Chaos-Based Image Encryption Springer

Berlin Germany 2005[6] H Yang K-W Wong X Liao W Zhang and P Wei ldquoA fast

image encryption and authentication scheme based on chaoticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3507ndash3517 2010

[7] J Hu and F Han ldquoA pixel-based scrambling scheme for digitalmedical images protectionrdquo Journal of Network and ComputerApplications vol 32 no 4 pp 788ndash794 2009

[8] Z Shuo C Ruhua J Yingchun and G Shiping ldquoAn imageencryption algorithm based on multiple chaos and wavelettransformrdquo in Proceedings of the 2nd International CongressImage and Signal Processing (CISP rsquo09) pp 1ndash5 2009

Mathematical Problems in Engineering 13

[9] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010

[10] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009

[11] X Liao S Lai and Q Zhou ldquoA novel image encryptionalgorithm based on self-adaptive wave transmissionrdquo SignalProcessing vol 90 no 9 pp 2714ndash2722 2010

[12] S Fu-Yan L Shu-Tang and L Zong-Wang ldquoImage encryptionusing high-dimension chaotic systemrdquo Chinese Physics vol 16no 12 pp 3616ndash3623 2007

[13] V Patidar N K Pareek and K K Sud ldquoA new substitution-diffusion based image cipher using chaotic standard and logisticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 7 pp 3056ndash3075 2009

[14] Y Wu J P Noonan G Yang and H Jin ldquoImage encryptionusing the two-dimensional logistic chaotic maprdquo Journal ofElectronic Imaging vol 21 no 1 Article ID 013014 2012

[15] Y Wu J P Noonan and S Agaian ldquoA wheel-switch chaoticsystem for image encryptionrdquo in Proceedings of the InternationalConference on System Science and Engineering (ICSSE rsquo11) pp23ndash27 June 2011

[16] L Zhang X Liao andXWang ldquoAn image encryption approachbased on chaoticmapsrdquoChaos Solitons and Fractals vol 24 no3 pp 759ndash765 2005

[17] H S Kwok and W K S Tang ldquoA fast image encryption systembased on chaotic maps with finite precision representationrdquoChaos Solitons and Fractals vol 32 no 4 pp 1518ndash1529 2007

[18] J Jian Y Shi C Hu Q Ma and J Li ldquoEncryption of digitalimage based on chaos systemrdquo in Computer and ComputingTechnologies in Agriculture II vol 2 pp 1145ndash1151 2009

[19] P Van-Roy and S Haridi Concepts Techniques and Models ofComputer Programming MIT Press 2004

[20] P SchumerMathematical Journeys Wiley-Interscience 2004[21] G Ye X Huang and C Zhu ldquoImage encryption algorithm of

double scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007

[22] XDesheng andXYueshan ldquoDigital image scrambling based onjosephus traversingrdquo Computer Engineering and Applicationsno 10 pp 44ndash46 2005

[23] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2005

[24] Y-P Zhang Z-J Zhai W Liu X Nie S-P Cao and W-D Dai ldquoDigital image encryption algorithm based on chaosand improved DESrdquo in Proceedings of the IEEE InternationalConference on Systems Man and Cybernetics (SMC rsquo09) pp474ndash479 October 2009

[25] J Lang R Tao and Y Wang ldquoImage encryption based on themultiple-parameter discrete fractional Fourier transform andchaos functionrdquo Optics Communications vol 283 no 10 pp2092ndash2096 2010

[26] S Yang and S Sun ldquoVideo encryptionmethod based on chaoticmaps in DCT domainrdquo Progress in Natural Science vol 18 no10 pp 1299ndash1304 2008

[27] V Wu J P Noonan and S Agaian ldquoBinary data encryptionusing the Sudoku block cipherrdquo in Proceedings of the IEEEInternational Conference on SystemsMan andCybernetics (SMCrsquo10) pp 3915ndash3921 October 2010

[28] Y Wu Y Zhou J P Noonan K Panetta and S Agaian ldquoImageencryption using the Sudoku matrixrdquo in Proceedings of theMobileMultimediaImage Processing Security and ApplicationsApril 2010

[29] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004

[30] H Gao Y Zhang S Liang and D Li ldquoA new chaotic algorithmfor image encryptionrdquo Chaos Solitons and Fractals vol 29 no2 pp 393ndash399 2006

[31] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[32] J Thompson and H Stewart Nonlinear Dynamics and ChaosJohn Wiley amp Sons 2002

[33] C E Shannon ldquoCommunication Theory of Secrecy SystemsrdquoBell Systems Technical Journal vol 28 pp 656ndash715 1949

[34] Y Wu Y Zhou G Saveriades S Agaian J P Noonan andP Natarajan ldquoLocal Shannon entropy measure with statisticaltests for image randomnessrdquo Information Sciences vol 222 no323 342 pages 2013

[35] Y Wu S Agaian and J P Noona ldquoA novel method of testingimage randomness with applications to image shuffling andencryptionrdquo in Defense Security and Sensing Proceedings ofthe SPIE pp 875507ndash875507 2013

[36] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI random-ness tests for image encryptionrdquo Cyber Journals pp 31ndash38 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

1

2

3

4

5

6

7

8

9

10

20

30

40

50

60

2 4 6 8 0

(a) CJPM(119877 119862 1 4 1198830 119903 0)

2

3

4

5

6

7

8

9

2 4 6 81 0

10

20

30

40

50

60

(b) CJPM(119877 119862 1 7 1198830 119903 0)

2

3

4

5

6

7

8

9

2 4 6 81 0

10

20

30

40

50

60

(c) CJPM(119877 119862 4 7 1198830 119903 0)

1

2

3

4

5

6

7

8

9

2 4 6 80

10

20

30

40

50

60

(d) CJPM(119877 119862 1 4 1198830+ 119890 119903 0)

1

2

3

4

5

6

7

8

9

2 4 6 80

10

20

30

40

50

60

(e) CJPM(119877 119862 1 4 1198830+ 119890 119903 10)

2 4 6 81

2

3

4

5

6

7

8

9

0

10

20

30

40

50

60

(f) CJPM(119877 119862 1 4 1198830+ 119890 119903 + 119890 10)

Figure 4 Parametric CJPMs (Note 119877 = 8 119862 = 81198830= 04 119903 = 38 and 119890 = 00001)

PlaintextP

Initial key

CiphertextC

CJPMgenerator

Encryption key

M

One blockPixel

permutationPixel

substitutionS

Figure 5 Image encryption method based on CJPM

Among these methods Algorithm 2 illustrates a straightfor-ward method to obtain a CJPM via a CJPS by rearrangingCJPS elements to a matrix

It is noticeable that Algorithm 2 is equivalent to rearrang-ing a sequence of elements into a matrix following the orderillustrated in Figure 3

Therefore a CJPM is determined by the same set ofparameters controlling a CJPS In order to emphasize thematrix property the parameter 119905 in CJPS is replaced by twoparameters of height 119877 and width 119862 where 119905 = 119877119862 Similarlya CJPM is uniquely determined by a set of parameters(119877 119862 119904 119899 119883

0 119903 119898) as for a CJPS Mathematically this claim

can be denoted as

119872 = CJPM (119877 119862 119904 119899 1198830 119903 119898) (17)

tN

xr yr w

p = L(X0 s n)

cq = cJ(t s n X0 r m)

Initial key

q = J(t s n)

R C s nr m

X0

Figure 6 Key functions in CJPM

Figure 4 illustrates various CJPMs via Algorithm 2according to different parameter sets It is clear that CJPMis very sensitive to its parameters and small changes in theparameter set lead to distinct CJPMs

4 Image Encryption AlgorithmBased on CJPM

In 1949 Claude Shannon the father of ldquoInformationTheoryrdquoproposed that confusion and diffusion are two properties ofthe operation of a secure cipher where the term confusionrefers to making the relationship between the encryptionkey and the ciphertext a very complex and developed one

6 Mathematical Problems in Engineering

Center of gravity

PlaintextP

Reference

CJPM M

Initial key Encryption key

point (xr yr)R C s t r m

X0 = sin(arctan(ygxg))(xg yg)

Figure 7 The internal structure of CJPM generator

PlaintextP

M +M998400

P998400

SMod(P998400 F)Mod(MF)

Figure 8 The internal structure of CJPM generator

[33] and the term diffusion refers to the property that theredundancy in the statistics of the plaintext is ldquodissipatedrdquo inthe statistics of the ciphertext [33] In otherwords for a securecipher it has to have good confusion and diffusion properties(1) different ciphertexts are desired to have similar statistics(2) any slight change in a plaintext is desired to lead to bigdifference in its ciphertext The image encryption algorithmbased on CJPM is proposed in this section to meet these twocriteria

41 Flowchart of Image Encryption Algorithm Based on CJPMSince the CJPM is parametric and random-like it can beused for image encryption directly However considering therequirements from confusion and diffusion properties theencryption procedure can be described as Figure 5 showsThe plaintext image is first sent to the CJPM generator whichis a preparation stage for generating a CJPM119872 for future useLater this CJPM119872 is used as a reference matrix to permuteand substitute image pixels for each image block in the stagesof pixel permutation and pixel substitution respectively Thedecryption procedure is simply to reverse the encryptionprocedure

42 Key Schedule It is clear that the CJPM is the coreof the cipher and thus key is related to the used CJPMreference matrix 119872 Initial key is composed of parameters(119909119903 119910119903 119908 119903 119898 119877 119862 119904 119899) where (119909

119903 119910119903 119908) is used in CJPM

generator for obtaining plaintext-dependent parameter 1198830

used in the Logistic map (119877 119862) are used as the parameter 119905in (1) and the parameter119873 in (6) The functions of each partof the initial key are shown in Figure 6

The output encryption key is composed of (119877 119862

119904 119899 1198830 119903 119898) all of which are directly required for deter-

mining a CJPM according to (17) Among these parameters119877 119862 119904 119899 and 119898 are restricted to integers 119909

119903 119910119903 119903 and 119883

0

are decimals More specifically 119877 and 119862 should be positive

integers smaller than the plaintext image size 119904 and 119899 shouldbe positive integers below the product of 119877119862 119903 should be anumber in between [36 4] (119909

119903 119910119903) is an arbitrary point on

119909119910 plain with weight 119908 and119898 is a nonnegative integer

43 CJPM Generator In order to enhance the resistance todifferential attacks the CJPM generator used in Figure 5 isdesigned to be plaintext dependent Recall that a CJPM isdetermined by a set of parameters (119877 119862 119904 119899 119883

0 119903 119898) shown

in (17) In the CJPM generator for image encryption only theparameter119883

0is not directly given by the initial key but by the

plaintext and a reference point (119909119903 119910119903) controlling the weight

in calculating the center of gravity Once 1198830is generated

it is stored in the encryption key The whole procedure oftranslating the initial key to a plaintext-dependent CJPMmatrix119872 and encryption key is shown in Figure 7

A plaintext is considered as an object of pixels where itsupper-left corner pixel is the reference point located at (1 1)Correspondingly pixels next to it along 119909 and 119910 directionsare (2 1) and (1 2) respectively The center of gravity ofthis plaintext is calculated via (18) where 119875

119894denotes the

119894th pixel intensity value and 119909119894and 119910

119894denote the location

of the 119894th pixel in the image with respect to the upper-leftcorner Once the center of gravity (119909

119892 119910119892) is obtained the

initial value of Logistic map 1198830is also determined via (19)

where arctan(sdot) is the arc tangent function and sin(sdot) is thesine function It is easy to verify that the range of (19) is[0 1] which satisfies restrictions for the initial value119883

0in the

Logisticmap Finally all required parameters for aCJPM thatis (119877 119862 119904 119899 119883

0 119903 119898) are obtained and thus a CJPM 119872 is

generated Meanwhile the used parameters are stored as theencryption key which can be used in the decryption processConsider

119909119892=119909119903sdot 119908 + sum119909

119894119875119894

119908 + sum119875119894

119910119892=119910119903sdot 119908 + sum119910

119894119875119894

119908 + sum119875119894

(18)

1198830= 05 [sin(arctan(

119910119892

119909119892

)) + 1] (19)

It is worth noting that the plaintext-dependent CJPMgenerator guarantees that the proposed cipher has gooddiffusion property any slight changes in plaintext lead tobig difference in ciphertext This is because the resulting

Mathematical Problems in Engineering 7

(a) Original image (b) 32-by-32 (c) 64-by-64 (d) 128-by-128

0100200300400500600700800900

1000

0 50 100 150 200 250

(e) Histogram of (a)

0

100

200

300

400

500

600

0 50 100 150 200 250

(f) Histogram of (b)

0

100

200

300

400

500

600

0 50 100 150 200 250

(g) Histogram of (c)

0

100

200

300

400

500

600

0 50 100 150 200 250

(h) Histogram of (d)

Figure 9 Pixel substitution results for CJPM at various sizes

(a) Original image (b) 32-by-32 (c) 64-by-32 (d) 64-by-64

(e) 64-by-128 (f) 128-by-128 (g) 256-by-128 (h) 256-by-256

Figure 10 Pixel permutation results for CJPM at various sizes

CJPM matrix119872 is dependent on the parameter 1198830and the

parameter 1198830is dependent on the center of gravity for the

plaintext while the center of the plaintext gravity alters forany slight change in plaintext Furthermore this 119883

0is the

parameter in the chaotic map and thus any slight change

in initial value leads to a completely different trajectory asthe bifurcation diagram in Figure 1 shows Consequentlya completely different CJPM is obtained as the referencematrix Eventually this new reference matrix leads to adistinct ciphertext It can be demonstrated that without

8 Mathematical Problems in Engineering

Input 119888119902120587 a CJPS at length 119905 = 119877 times 119862

Output119872 a CJPM of size 119877-by-119862for 119894 = 1 119877

for 119895 = 1 119862119872(119894 119895) = 119888119902

120587((119894 minus 1) 119877 + 119895)

endend

Algorithm 2 Reshape a CJPS into a CJPM

preknowing the reference pointrsquos coordinate (119909119903 119910119903) and its

weight 119908 there is no way to slightly change plaintext imagepixels so that its gravity center is unchanged

44 Pixel Substitution Pixel substitution refers to the pro-cess of changing pixel values From the point of viewof statistics this process is to change the statistics of aplaintext image so that the statistics of resulting ciphertextimage is completely different Moreover it is desired thatdifferent ciphertext images have similar statistics whichimplies that ciphertext images tell little information aboutkeys and plaintext images As a result the confusion propertyis achieved

The proposed pixel substitution block is shown inFigure 8 where symbol 119865 = 2

119887 and 119887 denotes the numberof bits supported by the format of the plaintext image Forexample if plaintext 119875 is an 8-bit gray image then 119865 = 28 =256 if plaintext119875 is a binary image then 119865 = 21 = 2 It can benoticed that the reference CJPM119872 is first to convert to1198721015840whose format is compatible with the format of the plaintextimage later1198721015840 and 119875 are added over the space of 119865 finallythe encrypted image 119878 is obtained

Because elements in a CJPM matrix 119872 are uniformlydistributed on integer set [1 119905] after ldquoModrdquo operation 1198721015840still has a uniform-like distribution for its elements on[0 119865] As a result when this 1198721015840 is used to randomly shiftthe pixel value in plaintext the resulting pixel value inciphertext has an equal opportunity to be any value on [0 119865]As a result a uniform-like histogram is achieved in theciphertext

Pixel substitution results based on CJPMs are shown inFigure 9 It is clear that histograms before and after pixelsubstitution are very different and ciphertext histograms arevery flat compared to plaintext ones It is also noticeable thatas the size of the referenceCJPM increases the ciphertext has abetter encryption quality from the point view of human visualinspection

45 Pixel Permutation Pixel permutation refers to the pro-cess scrambling the positions of pixels in plaintext to disguiseinformation contained in an image Denote an image beforeand after pixel permutation as 119861 and 119860 respectively Assumethe way of indexing image pixels is the same as the order torearrange elements in a CJPM as Figure 3 shows Then thepixel permutation process can be mathematically defined asa permutation 119891

120587between domain 119861 and range 119860 forall119894 119895 isin

1 2 119905 exist119860119894= 119861119895= 119861119891120587(119894) where 119905 is the total

number of pixels in the image As we mentioned in previoussections a CJPM is generated from a CJPS which is apermutation sequence Therefore a CJPM can be directlyused for pixel permutation that is given a CJPM 119872 and itspixel permutation can be defined as 119891

120587(119894) = 119872(119894)

For example Figure 10 illustrates pixel permutationresults of the ldquoTuftsrdquo logo image for different CJPMsIt is clear that pixels are well shuffled within the imageblock As long as the size of CJPMprocessing imageblock increases the resulting shuffled image looksbetter and better When the block size reaches to orover 128-by-128 pixels in the plaintext are almost evenlyshuffled

It is clear that images after pixel permutation look verydifferent from the plaintext image It is also worth notingthat a CJPM also depends on a set of parameters besides thesize and that any change in other parameters will lead to acompletely different permutated image

5 Simulation Results and Security Analysis

An excellent encryption method should be both robustand effective Robustness means that the cipher should beapplicable to any plaintext image written in a supportedformat Effectiveness implies that the cipher is able to generateeligible ciphertext images which hide information frompossible intruders

In this section we focus on discussing the performanceof the CJPM based image cipher described in Section 4 It isworth noting that all following computer simulations are rununder MATLAB 2010a and Windows XP environment withCore 2 Quad 26GHz processors

51 Histogram Analysis Histogram analysis is one of themost straightforward evaluations for ciphertext quality for itdirectly analyzes the pixel distribution of a ciphertext image

Figure 11 illustrates image encryption results for variousplaintext images image ldquo113rdquo is a binary handwriting scannedimage selected from ICDAR 2009 database image ldquoLenardquois a commonly used image of gray type image ldquo5113rdquoand ldquotestpat1krdquo are used to mimic the possible complexpatterns in plaintext and they are both selected from USC-SIPI database It is worth noting that the used CJPM is at sizeof 256-by-256

Mathematical Problems in Engineering 9

0 10

051

152

253

354

0100200300400500600700800

0 50 100 150 200 2500

10002000300040005000600070008000

0 50 100 150 200 2500

051

152

253

354

455

0 50 100 150 200 250

0 10

05

1

15

2

25

0100200300400500600

0 50 100 150 200 2500

100200300400500600

0 50 100 150 200 2500

100020003000400050006000700080009000

10000

0 50 100 150 200 250

113 Lenna Testpat1k

times104

times106 times104

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Plai

ntex

tP

Plai

ntex

tP

Plai

ntex

tP

Plai

ntex

tP

Ciph

erte

xtC

Ciph

erte

xtC

Ciph

erte

xtC

Ciph

erte

xtC

times106

Hist

ogra

m o

fC

Hist

ogra

m o

fC

Hist

ogra

m o

fC

Hist

ogra

m o

fC

5113

Figure 11 Histogram analysis for image encryption based on CJPM

It is clear that no matter what histogram a plaintextimage has the histogram of its ciphertext image is flatwhich implies that the pixel distribution is almost uniformComplex patterns and large homogenous regions in plaintextimages are completely unintelligible and become random-like in ciphertext images These results imply that the pro-posed image encryption method based on CJPM is robustand effective for various image formats and contents

52 Adjacent Pixel Autocorrelation (APAC) Analysis Highcorrelations of adjacent pixels can be utilized to carryout cryptanalysis Therefore a secure encryption algorithmshould break the high correlation relationship between adja-cent pixels

In statistics the autocorrelation 119877119886of a random process

119883 describes the correlation between values of the process atdifferent points in time as a function of the two times or of thetime difference The autocorrelation coefficient 119877

119886is defined

in (20) where 119889 is the time difference 120583 is the mean valuedefined by (21) and 120590 is the standard deviation defined by(22) the definition of mathematical expectation is given in(23)

119877119886(119898) =

119864 [(119883119905minus 120583) (119883

119905+119889minus 120583)]

1205902

(20)

120583 = 119864 [119883] (21)

10 Mathematical Problems in Engineering

Plai

ntex

t

Plai

ntex

t

Ciph

erte

xt

Ciph

erte

xt

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 250

Plai

ntex

tCi

pher

text

0

50

100

150

200

250

0

50

100

150

200

250

DiagonalVerticalHorizontal

DiagonalVerticalHorizontal

Figure 12 APAC analysis for image encryption based on CJPM

Table 1 Adjacent pixel autocorrelation analysis

Correlation coefficients (10minus3) Horizontal Vertical DiagonalPlaintext Lena 9399652 9709000 9709894

Ciphertext

CJPM minus22281 minus03709 3265[6] 20970 161870 17805[8] minus25000 minus10000 minus93000[5] 57765 28434 20662[29] minus127212 minus602579 624427[9] minus134000 12000 398000[30] minus158900 minus653800 minus323100[16] 815860 minus400530 minus47150[10] 1257000 581000 504000

120590 = radic119864 [(119883 minus 120583)2

] (22)

119864 [119909] =

119873

sum

119894=1

119909119894

119873 (23)

The closer to zero this coefficient is the weaker the rela-tionship two different time functions have Specifically inadjacent pixel correlation test we let 119883 be the image pixelsequence and let 119889 be 1 that is compare to the adjacent pixelsequence

Based on the reference direction there are three waysof extracting a two-dimensional image to a one-dimensionalsequence and they are the horizontal adjacent correlation

coefficient the vertical adjacent correlation coefficient andthe diagonal adjacent correlation coefficient

ldquoLenardquo image in the 2nd column of Figure 11 is used asthe test plaintext image because its APAC is widely reportedby other encryption methods Peer comparison results ofthe proposed CJPM cipher and cited encryption methods onAPAC are shown in Table 1 (best results are bolded)

In addition Figure 12 shows the result of randomlyselected 1024 pairs of two adjacent pixels from the plaintextand the ciphertext along horizontal vertical and diagonaldirections where 119909- and 119910-axes denote the intensity values ofa randomly selected pixel and its adjacent pixel respectivelyIt is clear that after applying the CJPM based image cipherthe high correlations between adjacent pixels in plaintext arebroken

53 Plaintext Sensitivity Analysis In order to test the resis-tance of the cipher to differential attacks plaintext sensitivityanalysis is required for a secure cipher In differential attacksan adversary attempts to extract meaningful relationshipbetween a plaintext image and its ciphertext image bymakinga slight change usually only one pixel in the plaintext imagewhile encrypting the plaintext image with the same encryp-tion key By comparing the change in ciphertext imagesthe encryption key might be cracked and furthermore theinformation contained in ciphertext might be leaked

Although there are othermeasures [34 35] to evaluate theresistance of plaintext attacks two classic measures are theNumber of Pixel Change Rate (NPCR) and Unified Average

Mathematical Problems in Engineering 11

Before changing one pixel After changing one pixel DifferencePl

aint

ext i

mag

eCi

pher

text

imag

e

Figure 13 Plaintext sensitivity analysis (differential attacks) for image encryption based on CJPM

(a) Encrypted image (1 1 128128 1 1 4 0)

(b) Decrypted image (1 1 128128 1 1 4 0)

(c) Decrypted image (1 1 128128 1 2 4 0)

(d) Decrypted image (1 1 128128 1 1 39999 0)

Figure 14 Key sensitivity analysis

Changing Intensity (UACI) [5]TheNPCR is used tomeasurethe percentage of the number of pixels changed in ciphertextafter making a slight change in plaintext Therefore thetheoretical greatest upper-bound of the NPCR is 100 TheUACI is used to measure the averaged intensity change forpixels in ciphertext images after making a slight change ina plaintext image It is demonstrable that the UACI of an idealcipher for 8-bit gray images is about 03346 [36]

Suppose ciphertext images before and after one pixelchange in a plaintext image are 1198621 and 1198622 respectively thepixel values at grid (119894 119895) in 1198621 and 1198622 are denoted as 1198621(119894 119895)and 1198622(119894 119895) a bipolar array 119863 is defined as (24) then the

NPCR and UACI can be mathematically defined as (25) and(26) respectively where symbol119873 denotes the total numberof pixels in the ciphertext symbol 119865 denotes the largestsupported pixel value compatible with the ciphertext imageformat and | sdot | is the absolute value function Consider

119863(119894 119895) =

1 if 1198621 (119894 119895) = 1198622 (119894 119895)

0 if 1198621 (119894 119895) = 1198622

(119894 119895)

(24)

12 Mathematical Problems in Engineering

Table 2 NPCR and UACI analyses on ldquoLenardquo image

CPJMsize 16-by-16 32-by-32 64-by-64 128-by-128 256-by-256

NPCR 996170 996246 995895 995850 996338UACI 338205 338379 334048 334076 334040

NPCR = sum

119894119895

119863(119894 119895)

119873times 100 (25)

UACI = sum

119894119895

100381610038161003816100381610038161198621

(119894 119895) minus 1198622

(119894 119895)10038161003816100381610038161003816

(119865 sdot 119873)times 100 (26)

Table 2 shows the NPCR and UACI results for one pixelchange in the ldquoLenardquo image It is clear that for CJPMs atvarious sizes the proposed cipher has good performances inboth the NPCR and UACI analyses Simulation results fit theexpectations of the ideal cipher very well

Figure 13 shows a differential attack on ldquoLenardquo imagewhile keeping the encryption key unchanged It is noticeablethat the difference between two plaintext images is thepixel on Lenarsquos nose However the ciphertext images are sodifferent that the image of their difference is still random-like As a result the one pixel change in the plaintext image isldquodissipatedrdquo in ciphertext From this point view this plaintextsensitivity is closely related to the diffusion property of acipher In other words it is reasonable to claim that a cipherhas good NPCR and UACI results if it has good diffusionproperties

54 Key Space Analysis The encryption key in the proposedimage encryption method using CPJM is composed of a setof parameters (119909

119903 119910119903 119908 119877 119862 119904 119899 119903 119898) where (119909

119903 119910119903) is an

arbitrary point on 119909119910 plain119908 is a nonnegative decimal119877 and119862 should be positive integers smaller than the plaintext imagesize 119904 and 119899 should be positive integers below the productof 119877119862 119903 should be a number in between [36 4] and 119898 is anonnegative integer Therefore theoretically the key space ofthe proposed cipher is infinitely large

Because the chaotic Logistic map is used as the triggerfor pseudorandom sequences the proposed cipher has highkey sensitivities as well The results of key sensitivity analysisare shown in Figure 14 where the set of parameters writtenin parenthesis is the used key It is clear that unless thecorrect decryption key is applied a ciphertext image cannotbe restored

6 Conclusion

In this paper we discussed the generation of a chaotic Jose-phus permutation matrix by using the conventional Josephuspermutation sequences and the logistic chaotic map Theproposed CJPM is parametric and is uniquely dependent onthe set of parameters which is sufficiently large to providea secure size of key space As another heritage from thechaotic Logistic map the CJPM is highly sensitive to itsinitial values (parameters) Any slight change in parameters

leads to significant differences in resulting CJPM Simulationresults show that (1) the ciphertext image is random-likefrom the perspective of human visual inspection (2) theencryption quality is almost independent of the plaintextimage (3) the proposed encryptionmethod is able to encryptplaintext images with large homogeneous regions to secureciphertext images (4) histogram analysis also shows thatdifferent ciphertext images tend to have the uniform distri-bution on [0 119865] (5) adjacent pixel correlation analysis showsthat neighbor pixels in ciphertext have lower correlationsthan many existing encryption methods (6) the proposedcipher is highly sensitive to encryption key and plaintext (7)experimental UACI results of the proposed cipher are veryclose to those of the ideal one

The proposed cipher can be used for various image typesfor example binary images 8-bit gray images 16-bit grayimages RBG images and so forth The same encryption ideamay also be applied to audio video or other types of digitalformats

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Scientific Research Fundof Hunan Provincial Education Department under Grant(no 12B023) It is also supported by the National NaturalScience Foundation of China (nos 61204039 and 61106029)and Scientific Research Foundation for Returned ScholarsMinistry of Human Resources and Social Security of thePeoplersquos Republic of China

References

[1] D StinsonCryptographyTheory andPractice CRCPress 2006[2] M Yang N Bourbakis and S Li ldquoData-image-video encryp-

tionrdquo IEEE Potentials vol 23 no 3 pp 28ndash34 2004[3] FIPS PUB 46 Data Encryption Standard National Bureau of

Standards 1977[4] FIPS PUB 197 Avanced Encryption Standard Announcing the

Advanced Encryption Standard (AES) 2001[5] Y Mao and G Chen Chaos-Based Image Encryption Springer

Berlin Germany 2005[6] H Yang K-W Wong X Liao W Zhang and P Wei ldquoA fast

image encryption and authentication scheme based on chaoticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3507ndash3517 2010

[7] J Hu and F Han ldquoA pixel-based scrambling scheme for digitalmedical images protectionrdquo Journal of Network and ComputerApplications vol 32 no 4 pp 788ndash794 2009

[8] Z Shuo C Ruhua J Yingchun and G Shiping ldquoAn imageencryption algorithm based on multiple chaos and wavelettransformrdquo in Proceedings of the 2nd International CongressImage and Signal Processing (CISP rsquo09) pp 1ndash5 2009

Mathematical Problems in Engineering 13

[9] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010

[10] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009

[11] X Liao S Lai and Q Zhou ldquoA novel image encryptionalgorithm based on self-adaptive wave transmissionrdquo SignalProcessing vol 90 no 9 pp 2714ndash2722 2010

[12] S Fu-Yan L Shu-Tang and L Zong-Wang ldquoImage encryptionusing high-dimension chaotic systemrdquo Chinese Physics vol 16no 12 pp 3616ndash3623 2007

[13] V Patidar N K Pareek and K K Sud ldquoA new substitution-diffusion based image cipher using chaotic standard and logisticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 7 pp 3056ndash3075 2009

[14] Y Wu J P Noonan G Yang and H Jin ldquoImage encryptionusing the two-dimensional logistic chaotic maprdquo Journal ofElectronic Imaging vol 21 no 1 Article ID 013014 2012

[15] Y Wu J P Noonan and S Agaian ldquoA wheel-switch chaoticsystem for image encryptionrdquo in Proceedings of the InternationalConference on System Science and Engineering (ICSSE rsquo11) pp23ndash27 June 2011

[16] L Zhang X Liao andXWang ldquoAn image encryption approachbased on chaoticmapsrdquoChaos Solitons and Fractals vol 24 no3 pp 759ndash765 2005

[17] H S Kwok and W K S Tang ldquoA fast image encryption systembased on chaotic maps with finite precision representationrdquoChaos Solitons and Fractals vol 32 no 4 pp 1518ndash1529 2007

[18] J Jian Y Shi C Hu Q Ma and J Li ldquoEncryption of digitalimage based on chaos systemrdquo in Computer and ComputingTechnologies in Agriculture II vol 2 pp 1145ndash1151 2009

[19] P Van-Roy and S Haridi Concepts Techniques and Models ofComputer Programming MIT Press 2004

[20] P SchumerMathematical Journeys Wiley-Interscience 2004[21] G Ye X Huang and C Zhu ldquoImage encryption algorithm of

double scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007

[22] XDesheng andXYueshan ldquoDigital image scrambling based onjosephus traversingrdquo Computer Engineering and Applicationsno 10 pp 44ndash46 2005

[23] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2005

[24] Y-P Zhang Z-J Zhai W Liu X Nie S-P Cao and W-D Dai ldquoDigital image encryption algorithm based on chaosand improved DESrdquo in Proceedings of the IEEE InternationalConference on Systems Man and Cybernetics (SMC rsquo09) pp474ndash479 October 2009

[25] J Lang R Tao and Y Wang ldquoImage encryption based on themultiple-parameter discrete fractional Fourier transform andchaos functionrdquo Optics Communications vol 283 no 10 pp2092ndash2096 2010

[26] S Yang and S Sun ldquoVideo encryptionmethod based on chaoticmaps in DCT domainrdquo Progress in Natural Science vol 18 no10 pp 1299ndash1304 2008

[27] V Wu J P Noonan and S Agaian ldquoBinary data encryptionusing the Sudoku block cipherrdquo in Proceedings of the IEEEInternational Conference on SystemsMan andCybernetics (SMCrsquo10) pp 3915ndash3921 October 2010

[28] Y Wu Y Zhou J P Noonan K Panetta and S Agaian ldquoImageencryption using the Sudoku matrixrdquo in Proceedings of theMobileMultimediaImage Processing Security and ApplicationsApril 2010

[29] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004

[30] H Gao Y Zhang S Liang and D Li ldquoA new chaotic algorithmfor image encryptionrdquo Chaos Solitons and Fractals vol 29 no2 pp 393ndash399 2006

[31] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[32] J Thompson and H Stewart Nonlinear Dynamics and ChaosJohn Wiley amp Sons 2002

[33] C E Shannon ldquoCommunication Theory of Secrecy SystemsrdquoBell Systems Technical Journal vol 28 pp 656ndash715 1949

[34] Y Wu Y Zhou G Saveriades S Agaian J P Noonan andP Natarajan ldquoLocal Shannon entropy measure with statisticaltests for image randomnessrdquo Information Sciences vol 222 no323 342 pages 2013

[35] Y Wu S Agaian and J P Noona ldquoA novel method of testingimage randomness with applications to image shuffling andencryptionrdquo in Defense Security and Sensing Proceedings ofthe SPIE pp 875507ndash875507 2013

[36] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI random-ness tests for image encryptionrdquo Cyber Journals pp 31ndash38 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Center of gravity

PlaintextP

Reference

CJPM M

Initial key Encryption key

point (xr yr)R C s t r m

X0 = sin(arctan(ygxg))(xg yg)

Figure 7 The internal structure of CJPM generator

PlaintextP

M +M998400

P998400

SMod(P998400 F)Mod(MF)

Figure 8 The internal structure of CJPM generator

[33] and the term diffusion refers to the property that theredundancy in the statistics of the plaintext is ldquodissipatedrdquo inthe statistics of the ciphertext [33] In otherwords for a securecipher it has to have good confusion and diffusion properties(1) different ciphertexts are desired to have similar statistics(2) any slight change in a plaintext is desired to lead to bigdifference in its ciphertext The image encryption algorithmbased on CJPM is proposed in this section to meet these twocriteria

41 Flowchart of Image Encryption Algorithm Based on CJPMSince the CJPM is parametric and random-like it can beused for image encryption directly However considering therequirements from confusion and diffusion properties theencryption procedure can be described as Figure 5 showsThe plaintext image is first sent to the CJPM generator whichis a preparation stage for generating a CJPM119872 for future useLater this CJPM119872 is used as a reference matrix to permuteand substitute image pixels for each image block in the stagesof pixel permutation and pixel substitution respectively Thedecryption procedure is simply to reverse the encryptionprocedure

42 Key Schedule It is clear that the CJPM is the coreof the cipher and thus key is related to the used CJPMreference matrix 119872 Initial key is composed of parameters(119909119903 119910119903 119908 119903 119898 119877 119862 119904 119899) where (119909

119903 119910119903 119908) is used in CJPM

generator for obtaining plaintext-dependent parameter 1198830

used in the Logistic map (119877 119862) are used as the parameter 119905in (1) and the parameter119873 in (6) The functions of each partof the initial key are shown in Figure 6

The output encryption key is composed of (119877 119862

119904 119899 1198830 119903 119898) all of which are directly required for deter-

mining a CJPM according to (17) Among these parameters119877 119862 119904 119899 and 119898 are restricted to integers 119909

119903 119910119903 119903 and 119883

0

are decimals More specifically 119877 and 119862 should be positive

integers smaller than the plaintext image size 119904 and 119899 shouldbe positive integers below the product of 119877119862 119903 should be anumber in between [36 4] (119909

119903 119910119903) is an arbitrary point on

119909119910 plain with weight 119908 and119898 is a nonnegative integer

43 CJPM Generator In order to enhance the resistance todifferential attacks the CJPM generator used in Figure 5 isdesigned to be plaintext dependent Recall that a CJPM isdetermined by a set of parameters (119877 119862 119904 119899 119883

0 119903 119898) shown

in (17) In the CJPM generator for image encryption only theparameter119883

0is not directly given by the initial key but by the

plaintext and a reference point (119909119903 119910119903) controlling the weight

in calculating the center of gravity Once 1198830is generated

it is stored in the encryption key The whole procedure oftranslating the initial key to a plaintext-dependent CJPMmatrix119872 and encryption key is shown in Figure 7

A plaintext is considered as an object of pixels where itsupper-left corner pixel is the reference point located at (1 1)Correspondingly pixels next to it along 119909 and 119910 directionsare (2 1) and (1 2) respectively The center of gravity ofthis plaintext is calculated via (18) where 119875

119894denotes the

119894th pixel intensity value and 119909119894and 119910

119894denote the location

of the 119894th pixel in the image with respect to the upper-leftcorner Once the center of gravity (119909

119892 119910119892) is obtained the

initial value of Logistic map 1198830is also determined via (19)

where arctan(sdot) is the arc tangent function and sin(sdot) is thesine function It is easy to verify that the range of (19) is[0 1] which satisfies restrictions for the initial value119883

0in the

Logisticmap Finally all required parameters for aCJPM thatis (119877 119862 119904 119899 119883

0 119903 119898) are obtained and thus a CJPM 119872 is

generated Meanwhile the used parameters are stored as theencryption key which can be used in the decryption processConsider

119909119892=119909119903sdot 119908 + sum119909

119894119875119894

119908 + sum119875119894

119910119892=119910119903sdot 119908 + sum119910

119894119875119894

119908 + sum119875119894

(18)

1198830= 05 [sin(arctan(

119910119892

119909119892

)) + 1] (19)

It is worth noting that the plaintext-dependent CJPMgenerator guarantees that the proposed cipher has gooddiffusion property any slight changes in plaintext lead tobig difference in ciphertext This is because the resulting

Mathematical Problems in Engineering 7

(a) Original image (b) 32-by-32 (c) 64-by-64 (d) 128-by-128

0100200300400500600700800900

1000

0 50 100 150 200 250

(e) Histogram of (a)

0

100

200

300

400

500

600

0 50 100 150 200 250

(f) Histogram of (b)

0

100

200

300

400

500

600

0 50 100 150 200 250

(g) Histogram of (c)

0

100

200

300

400

500

600

0 50 100 150 200 250

(h) Histogram of (d)

Figure 9 Pixel substitution results for CJPM at various sizes

(a) Original image (b) 32-by-32 (c) 64-by-32 (d) 64-by-64

(e) 64-by-128 (f) 128-by-128 (g) 256-by-128 (h) 256-by-256

Figure 10 Pixel permutation results for CJPM at various sizes

CJPM matrix119872 is dependent on the parameter 1198830and the

parameter 1198830is dependent on the center of gravity for the

plaintext while the center of the plaintext gravity alters forany slight change in plaintext Furthermore this 119883

0is the

parameter in the chaotic map and thus any slight change

in initial value leads to a completely different trajectory asthe bifurcation diagram in Figure 1 shows Consequentlya completely different CJPM is obtained as the referencematrix Eventually this new reference matrix leads to adistinct ciphertext It can be demonstrated that without

8 Mathematical Problems in Engineering

Input 119888119902120587 a CJPS at length 119905 = 119877 times 119862

Output119872 a CJPM of size 119877-by-119862for 119894 = 1 119877

for 119895 = 1 119862119872(119894 119895) = 119888119902

120587((119894 minus 1) 119877 + 119895)

endend

Algorithm 2 Reshape a CJPS into a CJPM

preknowing the reference pointrsquos coordinate (119909119903 119910119903) and its

weight 119908 there is no way to slightly change plaintext imagepixels so that its gravity center is unchanged

44 Pixel Substitution Pixel substitution refers to the pro-cess of changing pixel values From the point of viewof statistics this process is to change the statistics of aplaintext image so that the statistics of resulting ciphertextimage is completely different Moreover it is desired thatdifferent ciphertext images have similar statistics whichimplies that ciphertext images tell little information aboutkeys and plaintext images As a result the confusion propertyis achieved

The proposed pixel substitution block is shown inFigure 8 where symbol 119865 = 2

119887 and 119887 denotes the numberof bits supported by the format of the plaintext image Forexample if plaintext 119875 is an 8-bit gray image then 119865 = 28 =256 if plaintext119875 is a binary image then 119865 = 21 = 2 It can benoticed that the reference CJPM119872 is first to convert to1198721015840whose format is compatible with the format of the plaintextimage later1198721015840 and 119875 are added over the space of 119865 finallythe encrypted image 119878 is obtained

Because elements in a CJPM matrix 119872 are uniformlydistributed on integer set [1 119905] after ldquoModrdquo operation 1198721015840still has a uniform-like distribution for its elements on[0 119865] As a result when this 1198721015840 is used to randomly shiftthe pixel value in plaintext the resulting pixel value inciphertext has an equal opportunity to be any value on [0 119865]As a result a uniform-like histogram is achieved in theciphertext

Pixel substitution results based on CJPMs are shown inFigure 9 It is clear that histograms before and after pixelsubstitution are very different and ciphertext histograms arevery flat compared to plaintext ones It is also noticeable thatas the size of the referenceCJPM increases the ciphertext has abetter encryption quality from the point view of human visualinspection

45 Pixel Permutation Pixel permutation refers to the pro-cess scrambling the positions of pixels in plaintext to disguiseinformation contained in an image Denote an image beforeand after pixel permutation as 119861 and 119860 respectively Assumethe way of indexing image pixels is the same as the order torearrange elements in a CJPM as Figure 3 shows Then thepixel permutation process can be mathematically defined asa permutation 119891

120587between domain 119861 and range 119860 forall119894 119895 isin

1 2 119905 exist119860119894= 119861119895= 119861119891120587(119894) where 119905 is the total

number of pixels in the image As we mentioned in previoussections a CJPM is generated from a CJPS which is apermutation sequence Therefore a CJPM can be directlyused for pixel permutation that is given a CJPM 119872 and itspixel permutation can be defined as 119891

120587(119894) = 119872(119894)

For example Figure 10 illustrates pixel permutationresults of the ldquoTuftsrdquo logo image for different CJPMsIt is clear that pixels are well shuffled within the imageblock As long as the size of CJPMprocessing imageblock increases the resulting shuffled image looksbetter and better When the block size reaches to orover 128-by-128 pixels in the plaintext are almost evenlyshuffled

It is clear that images after pixel permutation look verydifferent from the plaintext image It is also worth notingthat a CJPM also depends on a set of parameters besides thesize and that any change in other parameters will lead to acompletely different permutated image

5 Simulation Results and Security Analysis

An excellent encryption method should be both robustand effective Robustness means that the cipher should beapplicable to any plaintext image written in a supportedformat Effectiveness implies that the cipher is able to generateeligible ciphertext images which hide information frompossible intruders

In this section we focus on discussing the performanceof the CJPM based image cipher described in Section 4 It isworth noting that all following computer simulations are rununder MATLAB 2010a and Windows XP environment withCore 2 Quad 26GHz processors

51 Histogram Analysis Histogram analysis is one of themost straightforward evaluations for ciphertext quality for itdirectly analyzes the pixel distribution of a ciphertext image

Figure 11 illustrates image encryption results for variousplaintext images image ldquo113rdquo is a binary handwriting scannedimage selected from ICDAR 2009 database image ldquoLenardquois a commonly used image of gray type image ldquo5113rdquoand ldquotestpat1krdquo are used to mimic the possible complexpatterns in plaintext and they are both selected from USC-SIPI database It is worth noting that the used CJPM is at sizeof 256-by-256

Mathematical Problems in Engineering 9

0 10

051

152

253

354

0100200300400500600700800

0 50 100 150 200 2500

10002000300040005000600070008000

0 50 100 150 200 2500

051

152

253

354

455

0 50 100 150 200 250

0 10

05

1

15

2

25

0100200300400500600

0 50 100 150 200 2500

100200300400500600

0 50 100 150 200 2500

100020003000400050006000700080009000

10000

0 50 100 150 200 250

113 Lenna Testpat1k

times104

times106 times104

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Plai

ntex

tP

Plai

ntex

tP

Plai

ntex

tP

Plai

ntex

tP

Ciph

erte

xtC

Ciph

erte

xtC

Ciph

erte

xtC

Ciph

erte

xtC

times106

Hist

ogra

m o

fC

Hist

ogra

m o

fC

Hist

ogra

m o

fC

Hist

ogra

m o

fC

5113

Figure 11 Histogram analysis for image encryption based on CJPM

It is clear that no matter what histogram a plaintextimage has the histogram of its ciphertext image is flatwhich implies that the pixel distribution is almost uniformComplex patterns and large homogenous regions in plaintextimages are completely unintelligible and become random-like in ciphertext images These results imply that the pro-posed image encryption method based on CJPM is robustand effective for various image formats and contents

52 Adjacent Pixel Autocorrelation (APAC) Analysis Highcorrelations of adjacent pixels can be utilized to carryout cryptanalysis Therefore a secure encryption algorithmshould break the high correlation relationship between adja-cent pixels

In statistics the autocorrelation 119877119886of a random process

119883 describes the correlation between values of the process atdifferent points in time as a function of the two times or of thetime difference The autocorrelation coefficient 119877

119886is defined

in (20) where 119889 is the time difference 120583 is the mean valuedefined by (21) and 120590 is the standard deviation defined by(22) the definition of mathematical expectation is given in(23)

119877119886(119898) =

119864 [(119883119905minus 120583) (119883

119905+119889minus 120583)]

1205902

(20)

120583 = 119864 [119883] (21)

10 Mathematical Problems in Engineering

Plai

ntex

t

Plai

ntex

t

Ciph

erte

xt

Ciph

erte

xt

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 250

Plai

ntex

tCi

pher

text

0

50

100

150

200

250

0

50

100

150

200

250

DiagonalVerticalHorizontal

DiagonalVerticalHorizontal

Figure 12 APAC analysis for image encryption based on CJPM

Table 1 Adjacent pixel autocorrelation analysis

Correlation coefficients (10minus3) Horizontal Vertical DiagonalPlaintext Lena 9399652 9709000 9709894

Ciphertext

CJPM minus22281 minus03709 3265[6] 20970 161870 17805[8] minus25000 minus10000 minus93000[5] 57765 28434 20662[29] minus127212 minus602579 624427[9] minus134000 12000 398000[30] minus158900 minus653800 minus323100[16] 815860 minus400530 minus47150[10] 1257000 581000 504000

120590 = radic119864 [(119883 minus 120583)2

] (22)

119864 [119909] =

119873

sum

119894=1

119909119894

119873 (23)

The closer to zero this coefficient is the weaker the rela-tionship two different time functions have Specifically inadjacent pixel correlation test we let 119883 be the image pixelsequence and let 119889 be 1 that is compare to the adjacent pixelsequence

Based on the reference direction there are three waysof extracting a two-dimensional image to a one-dimensionalsequence and they are the horizontal adjacent correlation

coefficient the vertical adjacent correlation coefficient andthe diagonal adjacent correlation coefficient

ldquoLenardquo image in the 2nd column of Figure 11 is used asthe test plaintext image because its APAC is widely reportedby other encryption methods Peer comparison results ofthe proposed CJPM cipher and cited encryption methods onAPAC are shown in Table 1 (best results are bolded)

In addition Figure 12 shows the result of randomlyselected 1024 pairs of two adjacent pixels from the plaintextand the ciphertext along horizontal vertical and diagonaldirections where 119909- and 119910-axes denote the intensity values ofa randomly selected pixel and its adjacent pixel respectivelyIt is clear that after applying the CJPM based image cipherthe high correlations between adjacent pixels in plaintext arebroken

53 Plaintext Sensitivity Analysis In order to test the resis-tance of the cipher to differential attacks plaintext sensitivityanalysis is required for a secure cipher In differential attacksan adversary attempts to extract meaningful relationshipbetween a plaintext image and its ciphertext image bymakinga slight change usually only one pixel in the plaintext imagewhile encrypting the plaintext image with the same encryp-tion key By comparing the change in ciphertext imagesthe encryption key might be cracked and furthermore theinformation contained in ciphertext might be leaked

Although there are othermeasures [34 35] to evaluate theresistance of plaintext attacks two classic measures are theNumber of Pixel Change Rate (NPCR) and Unified Average

Mathematical Problems in Engineering 11

Before changing one pixel After changing one pixel DifferencePl

aint

ext i

mag

eCi

pher

text

imag

e

Figure 13 Plaintext sensitivity analysis (differential attacks) for image encryption based on CJPM

(a) Encrypted image (1 1 128128 1 1 4 0)

(b) Decrypted image (1 1 128128 1 1 4 0)

(c) Decrypted image (1 1 128128 1 2 4 0)

(d) Decrypted image (1 1 128128 1 1 39999 0)

Figure 14 Key sensitivity analysis

Changing Intensity (UACI) [5]TheNPCR is used tomeasurethe percentage of the number of pixels changed in ciphertextafter making a slight change in plaintext Therefore thetheoretical greatest upper-bound of the NPCR is 100 TheUACI is used to measure the averaged intensity change forpixels in ciphertext images after making a slight change ina plaintext image It is demonstrable that the UACI of an idealcipher for 8-bit gray images is about 03346 [36]

Suppose ciphertext images before and after one pixelchange in a plaintext image are 1198621 and 1198622 respectively thepixel values at grid (119894 119895) in 1198621 and 1198622 are denoted as 1198621(119894 119895)and 1198622(119894 119895) a bipolar array 119863 is defined as (24) then the

NPCR and UACI can be mathematically defined as (25) and(26) respectively where symbol119873 denotes the total numberof pixels in the ciphertext symbol 119865 denotes the largestsupported pixel value compatible with the ciphertext imageformat and | sdot | is the absolute value function Consider

119863(119894 119895) =

1 if 1198621 (119894 119895) = 1198622 (119894 119895)

0 if 1198621 (119894 119895) = 1198622

(119894 119895)

(24)

12 Mathematical Problems in Engineering

Table 2 NPCR and UACI analyses on ldquoLenardquo image

CPJMsize 16-by-16 32-by-32 64-by-64 128-by-128 256-by-256

NPCR 996170 996246 995895 995850 996338UACI 338205 338379 334048 334076 334040

NPCR = sum

119894119895

119863(119894 119895)

119873times 100 (25)

UACI = sum

119894119895

100381610038161003816100381610038161198621

(119894 119895) minus 1198622

(119894 119895)10038161003816100381610038161003816

(119865 sdot 119873)times 100 (26)

Table 2 shows the NPCR and UACI results for one pixelchange in the ldquoLenardquo image It is clear that for CJPMs atvarious sizes the proposed cipher has good performances inboth the NPCR and UACI analyses Simulation results fit theexpectations of the ideal cipher very well

Figure 13 shows a differential attack on ldquoLenardquo imagewhile keeping the encryption key unchanged It is noticeablethat the difference between two plaintext images is thepixel on Lenarsquos nose However the ciphertext images are sodifferent that the image of their difference is still random-like As a result the one pixel change in the plaintext image isldquodissipatedrdquo in ciphertext From this point view this plaintextsensitivity is closely related to the diffusion property of acipher In other words it is reasonable to claim that a cipherhas good NPCR and UACI results if it has good diffusionproperties

54 Key Space Analysis The encryption key in the proposedimage encryption method using CPJM is composed of a setof parameters (119909

119903 119910119903 119908 119877 119862 119904 119899 119903 119898) where (119909

119903 119910119903) is an

arbitrary point on 119909119910 plain119908 is a nonnegative decimal119877 and119862 should be positive integers smaller than the plaintext imagesize 119904 and 119899 should be positive integers below the productof 119877119862 119903 should be a number in between [36 4] and 119898 is anonnegative integer Therefore theoretically the key space ofthe proposed cipher is infinitely large

Because the chaotic Logistic map is used as the triggerfor pseudorandom sequences the proposed cipher has highkey sensitivities as well The results of key sensitivity analysisare shown in Figure 14 where the set of parameters writtenin parenthesis is the used key It is clear that unless thecorrect decryption key is applied a ciphertext image cannotbe restored

6 Conclusion

In this paper we discussed the generation of a chaotic Jose-phus permutation matrix by using the conventional Josephuspermutation sequences and the logistic chaotic map Theproposed CJPM is parametric and is uniquely dependent onthe set of parameters which is sufficiently large to providea secure size of key space As another heritage from thechaotic Logistic map the CJPM is highly sensitive to itsinitial values (parameters) Any slight change in parameters

leads to significant differences in resulting CJPM Simulationresults show that (1) the ciphertext image is random-likefrom the perspective of human visual inspection (2) theencryption quality is almost independent of the plaintextimage (3) the proposed encryptionmethod is able to encryptplaintext images with large homogeneous regions to secureciphertext images (4) histogram analysis also shows thatdifferent ciphertext images tend to have the uniform distri-bution on [0 119865] (5) adjacent pixel correlation analysis showsthat neighbor pixels in ciphertext have lower correlationsthan many existing encryption methods (6) the proposedcipher is highly sensitive to encryption key and plaintext (7)experimental UACI results of the proposed cipher are veryclose to those of the ideal one

The proposed cipher can be used for various image typesfor example binary images 8-bit gray images 16-bit grayimages RBG images and so forth The same encryption ideamay also be applied to audio video or other types of digitalformats

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Scientific Research Fundof Hunan Provincial Education Department under Grant(no 12B023) It is also supported by the National NaturalScience Foundation of China (nos 61204039 and 61106029)and Scientific Research Foundation for Returned ScholarsMinistry of Human Resources and Social Security of thePeoplersquos Republic of China

References

[1] D StinsonCryptographyTheory andPractice CRCPress 2006[2] M Yang N Bourbakis and S Li ldquoData-image-video encryp-

tionrdquo IEEE Potentials vol 23 no 3 pp 28ndash34 2004[3] FIPS PUB 46 Data Encryption Standard National Bureau of

Standards 1977[4] FIPS PUB 197 Avanced Encryption Standard Announcing the

Advanced Encryption Standard (AES) 2001[5] Y Mao and G Chen Chaos-Based Image Encryption Springer

Berlin Germany 2005[6] H Yang K-W Wong X Liao W Zhang and P Wei ldquoA fast

image encryption and authentication scheme based on chaoticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3507ndash3517 2010

[7] J Hu and F Han ldquoA pixel-based scrambling scheme for digitalmedical images protectionrdquo Journal of Network and ComputerApplications vol 32 no 4 pp 788ndash794 2009

[8] Z Shuo C Ruhua J Yingchun and G Shiping ldquoAn imageencryption algorithm based on multiple chaos and wavelettransformrdquo in Proceedings of the 2nd International CongressImage and Signal Processing (CISP rsquo09) pp 1ndash5 2009

Mathematical Problems in Engineering 13

[9] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010

[10] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009

[11] X Liao S Lai and Q Zhou ldquoA novel image encryptionalgorithm based on self-adaptive wave transmissionrdquo SignalProcessing vol 90 no 9 pp 2714ndash2722 2010

[12] S Fu-Yan L Shu-Tang and L Zong-Wang ldquoImage encryptionusing high-dimension chaotic systemrdquo Chinese Physics vol 16no 12 pp 3616ndash3623 2007

[13] V Patidar N K Pareek and K K Sud ldquoA new substitution-diffusion based image cipher using chaotic standard and logisticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 7 pp 3056ndash3075 2009

[14] Y Wu J P Noonan G Yang and H Jin ldquoImage encryptionusing the two-dimensional logistic chaotic maprdquo Journal ofElectronic Imaging vol 21 no 1 Article ID 013014 2012

[15] Y Wu J P Noonan and S Agaian ldquoA wheel-switch chaoticsystem for image encryptionrdquo in Proceedings of the InternationalConference on System Science and Engineering (ICSSE rsquo11) pp23ndash27 June 2011

[16] L Zhang X Liao andXWang ldquoAn image encryption approachbased on chaoticmapsrdquoChaos Solitons and Fractals vol 24 no3 pp 759ndash765 2005

[17] H S Kwok and W K S Tang ldquoA fast image encryption systembased on chaotic maps with finite precision representationrdquoChaos Solitons and Fractals vol 32 no 4 pp 1518ndash1529 2007

[18] J Jian Y Shi C Hu Q Ma and J Li ldquoEncryption of digitalimage based on chaos systemrdquo in Computer and ComputingTechnologies in Agriculture II vol 2 pp 1145ndash1151 2009

[19] P Van-Roy and S Haridi Concepts Techniques and Models ofComputer Programming MIT Press 2004

[20] P SchumerMathematical Journeys Wiley-Interscience 2004[21] G Ye X Huang and C Zhu ldquoImage encryption algorithm of

double scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007

[22] XDesheng andXYueshan ldquoDigital image scrambling based onjosephus traversingrdquo Computer Engineering and Applicationsno 10 pp 44ndash46 2005

[23] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2005

[24] Y-P Zhang Z-J Zhai W Liu X Nie S-P Cao and W-D Dai ldquoDigital image encryption algorithm based on chaosand improved DESrdquo in Proceedings of the IEEE InternationalConference on Systems Man and Cybernetics (SMC rsquo09) pp474ndash479 October 2009

[25] J Lang R Tao and Y Wang ldquoImage encryption based on themultiple-parameter discrete fractional Fourier transform andchaos functionrdquo Optics Communications vol 283 no 10 pp2092ndash2096 2010

[26] S Yang and S Sun ldquoVideo encryptionmethod based on chaoticmaps in DCT domainrdquo Progress in Natural Science vol 18 no10 pp 1299ndash1304 2008

[27] V Wu J P Noonan and S Agaian ldquoBinary data encryptionusing the Sudoku block cipherrdquo in Proceedings of the IEEEInternational Conference on SystemsMan andCybernetics (SMCrsquo10) pp 3915ndash3921 October 2010

[28] Y Wu Y Zhou J P Noonan K Panetta and S Agaian ldquoImageencryption using the Sudoku matrixrdquo in Proceedings of theMobileMultimediaImage Processing Security and ApplicationsApril 2010

[29] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004

[30] H Gao Y Zhang S Liang and D Li ldquoA new chaotic algorithmfor image encryptionrdquo Chaos Solitons and Fractals vol 29 no2 pp 393ndash399 2006

[31] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[32] J Thompson and H Stewart Nonlinear Dynamics and ChaosJohn Wiley amp Sons 2002

[33] C E Shannon ldquoCommunication Theory of Secrecy SystemsrdquoBell Systems Technical Journal vol 28 pp 656ndash715 1949

[34] Y Wu Y Zhou G Saveriades S Agaian J P Noonan andP Natarajan ldquoLocal Shannon entropy measure with statisticaltests for image randomnessrdquo Information Sciences vol 222 no323 342 pages 2013

[35] Y Wu S Agaian and J P Noona ldquoA novel method of testingimage randomness with applications to image shuffling andencryptionrdquo in Defense Security and Sensing Proceedings ofthe SPIE pp 875507ndash875507 2013

[36] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI random-ness tests for image encryptionrdquo Cyber Journals pp 31ndash38 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

(a) Original image (b) 32-by-32 (c) 64-by-64 (d) 128-by-128

0100200300400500600700800900

1000

0 50 100 150 200 250

(e) Histogram of (a)

0

100

200

300

400

500

600

0 50 100 150 200 250

(f) Histogram of (b)

0

100

200

300

400

500

600

0 50 100 150 200 250

(g) Histogram of (c)

0

100

200

300

400

500

600

0 50 100 150 200 250

(h) Histogram of (d)

Figure 9 Pixel substitution results for CJPM at various sizes

(a) Original image (b) 32-by-32 (c) 64-by-32 (d) 64-by-64

(e) 64-by-128 (f) 128-by-128 (g) 256-by-128 (h) 256-by-256

Figure 10 Pixel permutation results for CJPM at various sizes

CJPM matrix119872 is dependent on the parameter 1198830and the

parameter 1198830is dependent on the center of gravity for the

plaintext while the center of the plaintext gravity alters forany slight change in plaintext Furthermore this 119883

0is the

parameter in the chaotic map and thus any slight change

in initial value leads to a completely different trajectory asthe bifurcation diagram in Figure 1 shows Consequentlya completely different CJPM is obtained as the referencematrix Eventually this new reference matrix leads to adistinct ciphertext It can be demonstrated that without

8 Mathematical Problems in Engineering

Input 119888119902120587 a CJPS at length 119905 = 119877 times 119862

Output119872 a CJPM of size 119877-by-119862for 119894 = 1 119877

for 119895 = 1 119862119872(119894 119895) = 119888119902

120587((119894 minus 1) 119877 + 119895)

endend

Algorithm 2 Reshape a CJPS into a CJPM

preknowing the reference pointrsquos coordinate (119909119903 119910119903) and its

weight 119908 there is no way to slightly change plaintext imagepixels so that its gravity center is unchanged

44 Pixel Substitution Pixel substitution refers to the pro-cess of changing pixel values From the point of viewof statistics this process is to change the statistics of aplaintext image so that the statistics of resulting ciphertextimage is completely different Moreover it is desired thatdifferent ciphertext images have similar statistics whichimplies that ciphertext images tell little information aboutkeys and plaintext images As a result the confusion propertyis achieved

The proposed pixel substitution block is shown inFigure 8 where symbol 119865 = 2

119887 and 119887 denotes the numberof bits supported by the format of the plaintext image Forexample if plaintext 119875 is an 8-bit gray image then 119865 = 28 =256 if plaintext119875 is a binary image then 119865 = 21 = 2 It can benoticed that the reference CJPM119872 is first to convert to1198721015840whose format is compatible with the format of the plaintextimage later1198721015840 and 119875 are added over the space of 119865 finallythe encrypted image 119878 is obtained

Because elements in a CJPM matrix 119872 are uniformlydistributed on integer set [1 119905] after ldquoModrdquo operation 1198721015840still has a uniform-like distribution for its elements on[0 119865] As a result when this 1198721015840 is used to randomly shiftthe pixel value in plaintext the resulting pixel value inciphertext has an equal opportunity to be any value on [0 119865]As a result a uniform-like histogram is achieved in theciphertext

Pixel substitution results based on CJPMs are shown inFigure 9 It is clear that histograms before and after pixelsubstitution are very different and ciphertext histograms arevery flat compared to plaintext ones It is also noticeable thatas the size of the referenceCJPM increases the ciphertext has abetter encryption quality from the point view of human visualinspection

45 Pixel Permutation Pixel permutation refers to the pro-cess scrambling the positions of pixels in plaintext to disguiseinformation contained in an image Denote an image beforeand after pixel permutation as 119861 and 119860 respectively Assumethe way of indexing image pixels is the same as the order torearrange elements in a CJPM as Figure 3 shows Then thepixel permutation process can be mathematically defined asa permutation 119891

120587between domain 119861 and range 119860 forall119894 119895 isin

1 2 119905 exist119860119894= 119861119895= 119861119891120587(119894) where 119905 is the total

number of pixels in the image As we mentioned in previoussections a CJPM is generated from a CJPS which is apermutation sequence Therefore a CJPM can be directlyused for pixel permutation that is given a CJPM 119872 and itspixel permutation can be defined as 119891

120587(119894) = 119872(119894)

For example Figure 10 illustrates pixel permutationresults of the ldquoTuftsrdquo logo image for different CJPMsIt is clear that pixels are well shuffled within the imageblock As long as the size of CJPMprocessing imageblock increases the resulting shuffled image looksbetter and better When the block size reaches to orover 128-by-128 pixels in the plaintext are almost evenlyshuffled

It is clear that images after pixel permutation look verydifferent from the plaintext image It is also worth notingthat a CJPM also depends on a set of parameters besides thesize and that any change in other parameters will lead to acompletely different permutated image

5 Simulation Results and Security Analysis

An excellent encryption method should be both robustand effective Robustness means that the cipher should beapplicable to any plaintext image written in a supportedformat Effectiveness implies that the cipher is able to generateeligible ciphertext images which hide information frompossible intruders

In this section we focus on discussing the performanceof the CJPM based image cipher described in Section 4 It isworth noting that all following computer simulations are rununder MATLAB 2010a and Windows XP environment withCore 2 Quad 26GHz processors

51 Histogram Analysis Histogram analysis is one of themost straightforward evaluations for ciphertext quality for itdirectly analyzes the pixel distribution of a ciphertext image

Figure 11 illustrates image encryption results for variousplaintext images image ldquo113rdquo is a binary handwriting scannedimage selected from ICDAR 2009 database image ldquoLenardquois a commonly used image of gray type image ldquo5113rdquoand ldquotestpat1krdquo are used to mimic the possible complexpatterns in plaintext and they are both selected from USC-SIPI database It is worth noting that the used CJPM is at sizeof 256-by-256

Mathematical Problems in Engineering 9

0 10

051

152

253

354

0100200300400500600700800

0 50 100 150 200 2500

10002000300040005000600070008000

0 50 100 150 200 2500

051

152

253

354

455

0 50 100 150 200 250

0 10

05

1

15

2

25

0100200300400500600

0 50 100 150 200 2500

100200300400500600

0 50 100 150 200 2500

100020003000400050006000700080009000

10000

0 50 100 150 200 250

113 Lenna Testpat1k

times104

times106 times104

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Plai

ntex

tP

Plai

ntex

tP

Plai

ntex

tP

Plai

ntex

tP

Ciph

erte

xtC

Ciph

erte

xtC

Ciph

erte

xtC

Ciph

erte

xtC

times106

Hist

ogra

m o

fC

Hist

ogra

m o

fC

Hist

ogra

m o

fC

Hist

ogra

m o

fC

5113

Figure 11 Histogram analysis for image encryption based on CJPM

It is clear that no matter what histogram a plaintextimage has the histogram of its ciphertext image is flatwhich implies that the pixel distribution is almost uniformComplex patterns and large homogenous regions in plaintextimages are completely unintelligible and become random-like in ciphertext images These results imply that the pro-posed image encryption method based on CJPM is robustand effective for various image formats and contents

52 Adjacent Pixel Autocorrelation (APAC) Analysis Highcorrelations of adjacent pixels can be utilized to carryout cryptanalysis Therefore a secure encryption algorithmshould break the high correlation relationship between adja-cent pixels

In statistics the autocorrelation 119877119886of a random process

119883 describes the correlation between values of the process atdifferent points in time as a function of the two times or of thetime difference The autocorrelation coefficient 119877

119886is defined

in (20) where 119889 is the time difference 120583 is the mean valuedefined by (21) and 120590 is the standard deviation defined by(22) the definition of mathematical expectation is given in(23)

119877119886(119898) =

119864 [(119883119905minus 120583) (119883

119905+119889minus 120583)]

1205902

(20)

120583 = 119864 [119883] (21)

10 Mathematical Problems in Engineering

Plai

ntex

t

Plai

ntex

t

Ciph

erte

xt

Ciph

erte

xt

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 250

Plai

ntex

tCi

pher

text

0

50

100

150

200

250

0

50

100

150

200

250

DiagonalVerticalHorizontal

DiagonalVerticalHorizontal

Figure 12 APAC analysis for image encryption based on CJPM

Table 1 Adjacent pixel autocorrelation analysis

Correlation coefficients (10minus3) Horizontal Vertical DiagonalPlaintext Lena 9399652 9709000 9709894

Ciphertext

CJPM minus22281 minus03709 3265[6] 20970 161870 17805[8] minus25000 minus10000 minus93000[5] 57765 28434 20662[29] minus127212 minus602579 624427[9] minus134000 12000 398000[30] minus158900 minus653800 minus323100[16] 815860 minus400530 minus47150[10] 1257000 581000 504000

120590 = radic119864 [(119883 minus 120583)2

] (22)

119864 [119909] =

119873

sum

119894=1

119909119894

119873 (23)

The closer to zero this coefficient is the weaker the rela-tionship two different time functions have Specifically inadjacent pixel correlation test we let 119883 be the image pixelsequence and let 119889 be 1 that is compare to the adjacent pixelsequence

Based on the reference direction there are three waysof extracting a two-dimensional image to a one-dimensionalsequence and they are the horizontal adjacent correlation

coefficient the vertical adjacent correlation coefficient andthe diagonal adjacent correlation coefficient

ldquoLenardquo image in the 2nd column of Figure 11 is used asthe test plaintext image because its APAC is widely reportedby other encryption methods Peer comparison results ofthe proposed CJPM cipher and cited encryption methods onAPAC are shown in Table 1 (best results are bolded)

In addition Figure 12 shows the result of randomlyselected 1024 pairs of two adjacent pixels from the plaintextand the ciphertext along horizontal vertical and diagonaldirections where 119909- and 119910-axes denote the intensity values ofa randomly selected pixel and its adjacent pixel respectivelyIt is clear that after applying the CJPM based image cipherthe high correlations between adjacent pixels in plaintext arebroken

53 Plaintext Sensitivity Analysis In order to test the resis-tance of the cipher to differential attacks plaintext sensitivityanalysis is required for a secure cipher In differential attacksan adversary attempts to extract meaningful relationshipbetween a plaintext image and its ciphertext image bymakinga slight change usually only one pixel in the plaintext imagewhile encrypting the plaintext image with the same encryp-tion key By comparing the change in ciphertext imagesthe encryption key might be cracked and furthermore theinformation contained in ciphertext might be leaked

Although there are othermeasures [34 35] to evaluate theresistance of plaintext attacks two classic measures are theNumber of Pixel Change Rate (NPCR) and Unified Average

Mathematical Problems in Engineering 11

Before changing one pixel After changing one pixel DifferencePl

aint

ext i

mag

eCi

pher

text

imag

e

Figure 13 Plaintext sensitivity analysis (differential attacks) for image encryption based on CJPM

(a) Encrypted image (1 1 128128 1 1 4 0)

(b) Decrypted image (1 1 128128 1 1 4 0)

(c) Decrypted image (1 1 128128 1 2 4 0)

(d) Decrypted image (1 1 128128 1 1 39999 0)

Figure 14 Key sensitivity analysis

Changing Intensity (UACI) [5]TheNPCR is used tomeasurethe percentage of the number of pixels changed in ciphertextafter making a slight change in plaintext Therefore thetheoretical greatest upper-bound of the NPCR is 100 TheUACI is used to measure the averaged intensity change forpixels in ciphertext images after making a slight change ina plaintext image It is demonstrable that the UACI of an idealcipher for 8-bit gray images is about 03346 [36]

Suppose ciphertext images before and after one pixelchange in a plaintext image are 1198621 and 1198622 respectively thepixel values at grid (119894 119895) in 1198621 and 1198622 are denoted as 1198621(119894 119895)and 1198622(119894 119895) a bipolar array 119863 is defined as (24) then the

NPCR and UACI can be mathematically defined as (25) and(26) respectively where symbol119873 denotes the total numberof pixels in the ciphertext symbol 119865 denotes the largestsupported pixel value compatible with the ciphertext imageformat and | sdot | is the absolute value function Consider

119863(119894 119895) =

1 if 1198621 (119894 119895) = 1198622 (119894 119895)

0 if 1198621 (119894 119895) = 1198622

(119894 119895)

(24)

12 Mathematical Problems in Engineering

Table 2 NPCR and UACI analyses on ldquoLenardquo image

CPJMsize 16-by-16 32-by-32 64-by-64 128-by-128 256-by-256

NPCR 996170 996246 995895 995850 996338UACI 338205 338379 334048 334076 334040

NPCR = sum

119894119895

119863(119894 119895)

119873times 100 (25)

UACI = sum

119894119895

100381610038161003816100381610038161198621

(119894 119895) minus 1198622

(119894 119895)10038161003816100381610038161003816

(119865 sdot 119873)times 100 (26)

Table 2 shows the NPCR and UACI results for one pixelchange in the ldquoLenardquo image It is clear that for CJPMs atvarious sizes the proposed cipher has good performances inboth the NPCR and UACI analyses Simulation results fit theexpectations of the ideal cipher very well

Figure 13 shows a differential attack on ldquoLenardquo imagewhile keeping the encryption key unchanged It is noticeablethat the difference between two plaintext images is thepixel on Lenarsquos nose However the ciphertext images are sodifferent that the image of their difference is still random-like As a result the one pixel change in the plaintext image isldquodissipatedrdquo in ciphertext From this point view this plaintextsensitivity is closely related to the diffusion property of acipher In other words it is reasonable to claim that a cipherhas good NPCR and UACI results if it has good diffusionproperties

54 Key Space Analysis The encryption key in the proposedimage encryption method using CPJM is composed of a setof parameters (119909

119903 119910119903 119908 119877 119862 119904 119899 119903 119898) where (119909

119903 119910119903) is an

arbitrary point on 119909119910 plain119908 is a nonnegative decimal119877 and119862 should be positive integers smaller than the plaintext imagesize 119904 and 119899 should be positive integers below the productof 119877119862 119903 should be a number in between [36 4] and 119898 is anonnegative integer Therefore theoretically the key space ofthe proposed cipher is infinitely large

Because the chaotic Logistic map is used as the triggerfor pseudorandom sequences the proposed cipher has highkey sensitivities as well The results of key sensitivity analysisare shown in Figure 14 where the set of parameters writtenin parenthesis is the used key It is clear that unless thecorrect decryption key is applied a ciphertext image cannotbe restored

6 Conclusion

In this paper we discussed the generation of a chaotic Jose-phus permutation matrix by using the conventional Josephuspermutation sequences and the logistic chaotic map Theproposed CJPM is parametric and is uniquely dependent onthe set of parameters which is sufficiently large to providea secure size of key space As another heritage from thechaotic Logistic map the CJPM is highly sensitive to itsinitial values (parameters) Any slight change in parameters

leads to significant differences in resulting CJPM Simulationresults show that (1) the ciphertext image is random-likefrom the perspective of human visual inspection (2) theencryption quality is almost independent of the plaintextimage (3) the proposed encryptionmethod is able to encryptplaintext images with large homogeneous regions to secureciphertext images (4) histogram analysis also shows thatdifferent ciphertext images tend to have the uniform distri-bution on [0 119865] (5) adjacent pixel correlation analysis showsthat neighbor pixels in ciphertext have lower correlationsthan many existing encryption methods (6) the proposedcipher is highly sensitive to encryption key and plaintext (7)experimental UACI results of the proposed cipher are veryclose to those of the ideal one

The proposed cipher can be used for various image typesfor example binary images 8-bit gray images 16-bit grayimages RBG images and so forth The same encryption ideamay also be applied to audio video or other types of digitalformats

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Scientific Research Fundof Hunan Provincial Education Department under Grant(no 12B023) It is also supported by the National NaturalScience Foundation of China (nos 61204039 and 61106029)and Scientific Research Foundation for Returned ScholarsMinistry of Human Resources and Social Security of thePeoplersquos Republic of China

References

[1] D StinsonCryptographyTheory andPractice CRCPress 2006[2] M Yang N Bourbakis and S Li ldquoData-image-video encryp-

tionrdquo IEEE Potentials vol 23 no 3 pp 28ndash34 2004[3] FIPS PUB 46 Data Encryption Standard National Bureau of

Standards 1977[4] FIPS PUB 197 Avanced Encryption Standard Announcing the

Advanced Encryption Standard (AES) 2001[5] Y Mao and G Chen Chaos-Based Image Encryption Springer

Berlin Germany 2005[6] H Yang K-W Wong X Liao W Zhang and P Wei ldquoA fast

image encryption and authentication scheme based on chaoticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3507ndash3517 2010

[7] J Hu and F Han ldquoA pixel-based scrambling scheme for digitalmedical images protectionrdquo Journal of Network and ComputerApplications vol 32 no 4 pp 788ndash794 2009

[8] Z Shuo C Ruhua J Yingchun and G Shiping ldquoAn imageencryption algorithm based on multiple chaos and wavelettransformrdquo in Proceedings of the 2nd International CongressImage and Signal Processing (CISP rsquo09) pp 1ndash5 2009

Mathematical Problems in Engineering 13

[9] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010

[10] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009

[11] X Liao S Lai and Q Zhou ldquoA novel image encryptionalgorithm based on self-adaptive wave transmissionrdquo SignalProcessing vol 90 no 9 pp 2714ndash2722 2010

[12] S Fu-Yan L Shu-Tang and L Zong-Wang ldquoImage encryptionusing high-dimension chaotic systemrdquo Chinese Physics vol 16no 12 pp 3616ndash3623 2007

[13] V Patidar N K Pareek and K K Sud ldquoA new substitution-diffusion based image cipher using chaotic standard and logisticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 7 pp 3056ndash3075 2009

[14] Y Wu J P Noonan G Yang and H Jin ldquoImage encryptionusing the two-dimensional logistic chaotic maprdquo Journal ofElectronic Imaging vol 21 no 1 Article ID 013014 2012

[15] Y Wu J P Noonan and S Agaian ldquoA wheel-switch chaoticsystem for image encryptionrdquo in Proceedings of the InternationalConference on System Science and Engineering (ICSSE rsquo11) pp23ndash27 June 2011

[16] L Zhang X Liao andXWang ldquoAn image encryption approachbased on chaoticmapsrdquoChaos Solitons and Fractals vol 24 no3 pp 759ndash765 2005

[17] H S Kwok and W K S Tang ldquoA fast image encryption systembased on chaotic maps with finite precision representationrdquoChaos Solitons and Fractals vol 32 no 4 pp 1518ndash1529 2007

[18] J Jian Y Shi C Hu Q Ma and J Li ldquoEncryption of digitalimage based on chaos systemrdquo in Computer and ComputingTechnologies in Agriculture II vol 2 pp 1145ndash1151 2009

[19] P Van-Roy and S Haridi Concepts Techniques and Models ofComputer Programming MIT Press 2004

[20] P SchumerMathematical Journeys Wiley-Interscience 2004[21] G Ye X Huang and C Zhu ldquoImage encryption algorithm of

double scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007

[22] XDesheng andXYueshan ldquoDigital image scrambling based onjosephus traversingrdquo Computer Engineering and Applicationsno 10 pp 44ndash46 2005

[23] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2005

[24] Y-P Zhang Z-J Zhai W Liu X Nie S-P Cao and W-D Dai ldquoDigital image encryption algorithm based on chaosand improved DESrdquo in Proceedings of the IEEE InternationalConference on Systems Man and Cybernetics (SMC rsquo09) pp474ndash479 October 2009

[25] J Lang R Tao and Y Wang ldquoImage encryption based on themultiple-parameter discrete fractional Fourier transform andchaos functionrdquo Optics Communications vol 283 no 10 pp2092ndash2096 2010

[26] S Yang and S Sun ldquoVideo encryptionmethod based on chaoticmaps in DCT domainrdquo Progress in Natural Science vol 18 no10 pp 1299ndash1304 2008

[27] V Wu J P Noonan and S Agaian ldquoBinary data encryptionusing the Sudoku block cipherrdquo in Proceedings of the IEEEInternational Conference on SystemsMan andCybernetics (SMCrsquo10) pp 3915ndash3921 October 2010

[28] Y Wu Y Zhou J P Noonan K Panetta and S Agaian ldquoImageencryption using the Sudoku matrixrdquo in Proceedings of theMobileMultimediaImage Processing Security and ApplicationsApril 2010

[29] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004

[30] H Gao Y Zhang S Liang and D Li ldquoA new chaotic algorithmfor image encryptionrdquo Chaos Solitons and Fractals vol 29 no2 pp 393ndash399 2006

[31] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[32] J Thompson and H Stewart Nonlinear Dynamics and ChaosJohn Wiley amp Sons 2002

[33] C E Shannon ldquoCommunication Theory of Secrecy SystemsrdquoBell Systems Technical Journal vol 28 pp 656ndash715 1949

[34] Y Wu Y Zhou G Saveriades S Agaian J P Noonan andP Natarajan ldquoLocal Shannon entropy measure with statisticaltests for image randomnessrdquo Information Sciences vol 222 no323 342 pages 2013

[35] Y Wu S Agaian and J P Noona ldquoA novel method of testingimage randomness with applications to image shuffling andencryptionrdquo in Defense Security and Sensing Proceedings ofthe SPIE pp 875507ndash875507 2013

[36] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI random-ness tests for image encryptionrdquo Cyber Journals pp 31ndash38 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Input 119888119902120587 a CJPS at length 119905 = 119877 times 119862

Output119872 a CJPM of size 119877-by-119862for 119894 = 1 119877

for 119895 = 1 119862119872(119894 119895) = 119888119902

120587((119894 minus 1) 119877 + 119895)

endend

Algorithm 2 Reshape a CJPS into a CJPM

preknowing the reference pointrsquos coordinate (119909119903 119910119903) and its

weight 119908 there is no way to slightly change plaintext imagepixels so that its gravity center is unchanged

44 Pixel Substitution Pixel substitution refers to the pro-cess of changing pixel values From the point of viewof statistics this process is to change the statistics of aplaintext image so that the statistics of resulting ciphertextimage is completely different Moreover it is desired thatdifferent ciphertext images have similar statistics whichimplies that ciphertext images tell little information aboutkeys and plaintext images As a result the confusion propertyis achieved

The proposed pixel substitution block is shown inFigure 8 where symbol 119865 = 2

119887 and 119887 denotes the numberof bits supported by the format of the plaintext image Forexample if plaintext 119875 is an 8-bit gray image then 119865 = 28 =256 if plaintext119875 is a binary image then 119865 = 21 = 2 It can benoticed that the reference CJPM119872 is first to convert to1198721015840whose format is compatible with the format of the plaintextimage later1198721015840 and 119875 are added over the space of 119865 finallythe encrypted image 119878 is obtained

Because elements in a CJPM matrix 119872 are uniformlydistributed on integer set [1 119905] after ldquoModrdquo operation 1198721015840still has a uniform-like distribution for its elements on[0 119865] As a result when this 1198721015840 is used to randomly shiftthe pixel value in plaintext the resulting pixel value inciphertext has an equal opportunity to be any value on [0 119865]As a result a uniform-like histogram is achieved in theciphertext

Pixel substitution results based on CJPMs are shown inFigure 9 It is clear that histograms before and after pixelsubstitution are very different and ciphertext histograms arevery flat compared to plaintext ones It is also noticeable thatas the size of the referenceCJPM increases the ciphertext has abetter encryption quality from the point view of human visualinspection

45 Pixel Permutation Pixel permutation refers to the pro-cess scrambling the positions of pixels in plaintext to disguiseinformation contained in an image Denote an image beforeand after pixel permutation as 119861 and 119860 respectively Assumethe way of indexing image pixels is the same as the order torearrange elements in a CJPM as Figure 3 shows Then thepixel permutation process can be mathematically defined asa permutation 119891

120587between domain 119861 and range 119860 forall119894 119895 isin

1 2 119905 exist119860119894= 119861119895= 119861119891120587(119894) where 119905 is the total

number of pixels in the image As we mentioned in previoussections a CJPM is generated from a CJPS which is apermutation sequence Therefore a CJPM can be directlyused for pixel permutation that is given a CJPM 119872 and itspixel permutation can be defined as 119891

120587(119894) = 119872(119894)

For example Figure 10 illustrates pixel permutationresults of the ldquoTuftsrdquo logo image for different CJPMsIt is clear that pixels are well shuffled within the imageblock As long as the size of CJPMprocessing imageblock increases the resulting shuffled image looksbetter and better When the block size reaches to orover 128-by-128 pixels in the plaintext are almost evenlyshuffled

It is clear that images after pixel permutation look verydifferent from the plaintext image It is also worth notingthat a CJPM also depends on a set of parameters besides thesize and that any change in other parameters will lead to acompletely different permutated image

5 Simulation Results and Security Analysis

An excellent encryption method should be both robustand effective Robustness means that the cipher should beapplicable to any plaintext image written in a supportedformat Effectiveness implies that the cipher is able to generateeligible ciphertext images which hide information frompossible intruders

In this section we focus on discussing the performanceof the CJPM based image cipher described in Section 4 It isworth noting that all following computer simulations are rununder MATLAB 2010a and Windows XP environment withCore 2 Quad 26GHz processors

51 Histogram Analysis Histogram analysis is one of themost straightforward evaluations for ciphertext quality for itdirectly analyzes the pixel distribution of a ciphertext image

Figure 11 illustrates image encryption results for variousplaintext images image ldquo113rdquo is a binary handwriting scannedimage selected from ICDAR 2009 database image ldquoLenardquois a commonly used image of gray type image ldquo5113rdquoand ldquotestpat1krdquo are used to mimic the possible complexpatterns in plaintext and they are both selected from USC-SIPI database It is worth noting that the used CJPM is at sizeof 256-by-256

Mathematical Problems in Engineering 9

0 10

051

152

253

354

0100200300400500600700800

0 50 100 150 200 2500

10002000300040005000600070008000

0 50 100 150 200 2500

051

152

253

354

455

0 50 100 150 200 250

0 10

05

1

15

2

25

0100200300400500600

0 50 100 150 200 2500

100200300400500600

0 50 100 150 200 2500

100020003000400050006000700080009000

10000

0 50 100 150 200 250

113 Lenna Testpat1k

times104

times106 times104

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Plai

ntex

tP

Plai

ntex

tP

Plai

ntex

tP

Plai

ntex

tP

Ciph

erte

xtC

Ciph

erte

xtC

Ciph

erte

xtC

Ciph

erte

xtC

times106

Hist

ogra

m o

fC

Hist

ogra

m o

fC

Hist

ogra

m o

fC

Hist

ogra

m o

fC

5113

Figure 11 Histogram analysis for image encryption based on CJPM

It is clear that no matter what histogram a plaintextimage has the histogram of its ciphertext image is flatwhich implies that the pixel distribution is almost uniformComplex patterns and large homogenous regions in plaintextimages are completely unintelligible and become random-like in ciphertext images These results imply that the pro-posed image encryption method based on CJPM is robustand effective for various image formats and contents

52 Adjacent Pixel Autocorrelation (APAC) Analysis Highcorrelations of adjacent pixels can be utilized to carryout cryptanalysis Therefore a secure encryption algorithmshould break the high correlation relationship between adja-cent pixels

In statistics the autocorrelation 119877119886of a random process

119883 describes the correlation between values of the process atdifferent points in time as a function of the two times or of thetime difference The autocorrelation coefficient 119877

119886is defined

in (20) where 119889 is the time difference 120583 is the mean valuedefined by (21) and 120590 is the standard deviation defined by(22) the definition of mathematical expectation is given in(23)

119877119886(119898) =

119864 [(119883119905minus 120583) (119883

119905+119889minus 120583)]

1205902

(20)

120583 = 119864 [119883] (21)

10 Mathematical Problems in Engineering

Plai

ntex

t

Plai

ntex

t

Ciph

erte

xt

Ciph

erte

xt

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 250

Plai

ntex

tCi

pher

text

0

50

100

150

200

250

0

50

100

150

200

250

DiagonalVerticalHorizontal

DiagonalVerticalHorizontal

Figure 12 APAC analysis for image encryption based on CJPM

Table 1 Adjacent pixel autocorrelation analysis

Correlation coefficients (10minus3) Horizontal Vertical DiagonalPlaintext Lena 9399652 9709000 9709894

Ciphertext

CJPM minus22281 minus03709 3265[6] 20970 161870 17805[8] minus25000 minus10000 minus93000[5] 57765 28434 20662[29] minus127212 minus602579 624427[9] minus134000 12000 398000[30] minus158900 minus653800 minus323100[16] 815860 minus400530 minus47150[10] 1257000 581000 504000

120590 = radic119864 [(119883 minus 120583)2

] (22)

119864 [119909] =

119873

sum

119894=1

119909119894

119873 (23)

The closer to zero this coefficient is the weaker the rela-tionship two different time functions have Specifically inadjacent pixel correlation test we let 119883 be the image pixelsequence and let 119889 be 1 that is compare to the adjacent pixelsequence

Based on the reference direction there are three waysof extracting a two-dimensional image to a one-dimensionalsequence and they are the horizontal adjacent correlation

coefficient the vertical adjacent correlation coefficient andthe diagonal adjacent correlation coefficient

ldquoLenardquo image in the 2nd column of Figure 11 is used asthe test plaintext image because its APAC is widely reportedby other encryption methods Peer comparison results ofthe proposed CJPM cipher and cited encryption methods onAPAC are shown in Table 1 (best results are bolded)

In addition Figure 12 shows the result of randomlyselected 1024 pairs of two adjacent pixels from the plaintextand the ciphertext along horizontal vertical and diagonaldirections where 119909- and 119910-axes denote the intensity values ofa randomly selected pixel and its adjacent pixel respectivelyIt is clear that after applying the CJPM based image cipherthe high correlations between adjacent pixels in plaintext arebroken

53 Plaintext Sensitivity Analysis In order to test the resis-tance of the cipher to differential attacks plaintext sensitivityanalysis is required for a secure cipher In differential attacksan adversary attempts to extract meaningful relationshipbetween a plaintext image and its ciphertext image bymakinga slight change usually only one pixel in the plaintext imagewhile encrypting the plaintext image with the same encryp-tion key By comparing the change in ciphertext imagesthe encryption key might be cracked and furthermore theinformation contained in ciphertext might be leaked

Although there are othermeasures [34 35] to evaluate theresistance of plaintext attacks two classic measures are theNumber of Pixel Change Rate (NPCR) and Unified Average

Mathematical Problems in Engineering 11

Before changing one pixel After changing one pixel DifferencePl

aint

ext i

mag

eCi

pher

text

imag

e

Figure 13 Plaintext sensitivity analysis (differential attacks) for image encryption based on CJPM

(a) Encrypted image (1 1 128128 1 1 4 0)

(b) Decrypted image (1 1 128128 1 1 4 0)

(c) Decrypted image (1 1 128128 1 2 4 0)

(d) Decrypted image (1 1 128128 1 1 39999 0)

Figure 14 Key sensitivity analysis

Changing Intensity (UACI) [5]TheNPCR is used tomeasurethe percentage of the number of pixels changed in ciphertextafter making a slight change in plaintext Therefore thetheoretical greatest upper-bound of the NPCR is 100 TheUACI is used to measure the averaged intensity change forpixels in ciphertext images after making a slight change ina plaintext image It is demonstrable that the UACI of an idealcipher for 8-bit gray images is about 03346 [36]

Suppose ciphertext images before and after one pixelchange in a plaintext image are 1198621 and 1198622 respectively thepixel values at grid (119894 119895) in 1198621 and 1198622 are denoted as 1198621(119894 119895)and 1198622(119894 119895) a bipolar array 119863 is defined as (24) then the

NPCR and UACI can be mathematically defined as (25) and(26) respectively where symbol119873 denotes the total numberof pixels in the ciphertext symbol 119865 denotes the largestsupported pixel value compatible with the ciphertext imageformat and | sdot | is the absolute value function Consider

119863(119894 119895) =

1 if 1198621 (119894 119895) = 1198622 (119894 119895)

0 if 1198621 (119894 119895) = 1198622

(119894 119895)

(24)

12 Mathematical Problems in Engineering

Table 2 NPCR and UACI analyses on ldquoLenardquo image

CPJMsize 16-by-16 32-by-32 64-by-64 128-by-128 256-by-256

NPCR 996170 996246 995895 995850 996338UACI 338205 338379 334048 334076 334040

NPCR = sum

119894119895

119863(119894 119895)

119873times 100 (25)

UACI = sum

119894119895

100381610038161003816100381610038161198621

(119894 119895) minus 1198622

(119894 119895)10038161003816100381610038161003816

(119865 sdot 119873)times 100 (26)

Table 2 shows the NPCR and UACI results for one pixelchange in the ldquoLenardquo image It is clear that for CJPMs atvarious sizes the proposed cipher has good performances inboth the NPCR and UACI analyses Simulation results fit theexpectations of the ideal cipher very well

Figure 13 shows a differential attack on ldquoLenardquo imagewhile keeping the encryption key unchanged It is noticeablethat the difference between two plaintext images is thepixel on Lenarsquos nose However the ciphertext images are sodifferent that the image of their difference is still random-like As a result the one pixel change in the plaintext image isldquodissipatedrdquo in ciphertext From this point view this plaintextsensitivity is closely related to the diffusion property of acipher In other words it is reasonable to claim that a cipherhas good NPCR and UACI results if it has good diffusionproperties

54 Key Space Analysis The encryption key in the proposedimage encryption method using CPJM is composed of a setof parameters (119909

119903 119910119903 119908 119877 119862 119904 119899 119903 119898) where (119909

119903 119910119903) is an

arbitrary point on 119909119910 plain119908 is a nonnegative decimal119877 and119862 should be positive integers smaller than the plaintext imagesize 119904 and 119899 should be positive integers below the productof 119877119862 119903 should be a number in between [36 4] and 119898 is anonnegative integer Therefore theoretically the key space ofthe proposed cipher is infinitely large

Because the chaotic Logistic map is used as the triggerfor pseudorandom sequences the proposed cipher has highkey sensitivities as well The results of key sensitivity analysisare shown in Figure 14 where the set of parameters writtenin parenthesis is the used key It is clear that unless thecorrect decryption key is applied a ciphertext image cannotbe restored

6 Conclusion

In this paper we discussed the generation of a chaotic Jose-phus permutation matrix by using the conventional Josephuspermutation sequences and the logistic chaotic map Theproposed CJPM is parametric and is uniquely dependent onthe set of parameters which is sufficiently large to providea secure size of key space As another heritage from thechaotic Logistic map the CJPM is highly sensitive to itsinitial values (parameters) Any slight change in parameters

leads to significant differences in resulting CJPM Simulationresults show that (1) the ciphertext image is random-likefrom the perspective of human visual inspection (2) theencryption quality is almost independent of the plaintextimage (3) the proposed encryptionmethod is able to encryptplaintext images with large homogeneous regions to secureciphertext images (4) histogram analysis also shows thatdifferent ciphertext images tend to have the uniform distri-bution on [0 119865] (5) adjacent pixel correlation analysis showsthat neighbor pixels in ciphertext have lower correlationsthan many existing encryption methods (6) the proposedcipher is highly sensitive to encryption key and plaintext (7)experimental UACI results of the proposed cipher are veryclose to those of the ideal one

The proposed cipher can be used for various image typesfor example binary images 8-bit gray images 16-bit grayimages RBG images and so forth The same encryption ideamay also be applied to audio video or other types of digitalformats

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Scientific Research Fundof Hunan Provincial Education Department under Grant(no 12B023) It is also supported by the National NaturalScience Foundation of China (nos 61204039 and 61106029)and Scientific Research Foundation for Returned ScholarsMinistry of Human Resources and Social Security of thePeoplersquos Republic of China

References

[1] D StinsonCryptographyTheory andPractice CRCPress 2006[2] M Yang N Bourbakis and S Li ldquoData-image-video encryp-

tionrdquo IEEE Potentials vol 23 no 3 pp 28ndash34 2004[3] FIPS PUB 46 Data Encryption Standard National Bureau of

Standards 1977[4] FIPS PUB 197 Avanced Encryption Standard Announcing the

Advanced Encryption Standard (AES) 2001[5] Y Mao and G Chen Chaos-Based Image Encryption Springer

Berlin Germany 2005[6] H Yang K-W Wong X Liao W Zhang and P Wei ldquoA fast

image encryption and authentication scheme based on chaoticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3507ndash3517 2010

[7] J Hu and F Han ldquoA pixel-based scrambling scheme for digitalmedical images protectionrdquo Journal of Network and ComputerApplications vol 32 no 4 pp 788ndash794 2009

[8] Z Shuo C Ruhua J Yingchun and G Shiping ldquoAn imageencryption algorithm based on multiple chaos and wavelettransformrdquo in Proceedings of the 2nd International CongressImage and Signal Processing (CISP rsquo09) pp 1ndash5 2009

Mathematical Problems in Engineering 13

[9] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010

[10] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009

[11] X Liao S Lai and Q Zhou ldquoA novel image encryptionalgorithm based on self-adaptive wave transmissionrdquo SignalProcessing vol 90 no 9 pp 2714ndash2722 2010

[12] S Fu-Yan L Shu-Tang and L Zong-Wang ldquoImage encryptionusing high-dimension chaotic systemrdquo Chinese Physics vol 16no 12 pp 3616ndash3623 2007

[13] V Patidar N K Pareek and K K Sud ldquoA new substitution-diffusion based image cipher using chaotic standard and logisticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 7 pp 3056ndash3075 2009

[14] Y Wu J P Noonan G Yang and H Jin ldquoImage encryptionusing the two-dimensional logistic chaotic maprdquo Journal ofElectronic Imaging vol 21 no 1 Article ID 013014 2012

[15] Y Wu J P Noonan and S Agaian ldquoA wheel-switch chaoticsystem for image encryptionrdquo in Proceedings of the InternationalConference on System Science and Engineering (ICSSE rsquo11) pp23ndash27 June 2011

[16] L Zhang X Liao andXWang ldquoAn image encryption approachbased on chaoticmapsrdquoChaos Solitons and Fractals vol 24 no3 pp 759ndash765 2005

[17] H S Kwok and W K S Tang ldquoA fast image encryption systembased on chaotic maps with finite precision representationrdquoChaos Solitons and Fractals vol 32 no 4 pp 1518ndash1529 2007

[18] J Jian Y Shi C Hu Q Ma and J Li ldquoEncryption of digitalimage based on chaos systemrdquo in Computer and ComputingTechnologies in Agriculture II vol 2 pp 1145ndash1151 2009

[19] P Van-Roy and S Haridi Concepts Techniques and Models ofComputer Programming MIT Press 2004

[20] P SchumerMathematical Journeys Wiley-Interscience 2004[21] G Ye X Huang and C Zhu ldquoImage encryption algorithm of

double scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007

[22] XDesheng andXYueshan ldquoDigital image scrambling based onjosephus traversingrdquo Computer Engineering and Applicationsno 10 pp 44ndash46 2005

[23] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2005

[24] Y-P Zhang Z-J Zhai W Liu X Nie S-P Cao and W-D Dai ldquoDigital image encryption algorithm based on chaosand improved DESrdquo in Proceedings of the IEEE InternationalConference on Systems Man and Cybernetics (SMC rsquo09) pp474ndash479 October 2009

[25] J Lang R Tao and Y Wang ldquoImage encryption based on themultiple-parameter discrete fractional Fourier transform andchaos functionrdquo Optics Communications vol 283 no 10 pp2092ndash2096 2010

[26] S Yang and S Sun ldquoVideo encryptionmethod based on chaoticmaps in DCT domainrdquo Progress in Natural Science vol 18 no10 pp 1299ndash1304 2008

[27] V Wu J P Noonan and S Agaian ldquoBinary data encryptionusing the Sudoku block cipherrdquo in Proceedings of the IEEEInternational Conference on SystemsMan andCybernetics (SMCrsquo10) pp 3915ndash3921 October 2010

[28] Y Wu Y Zhou J P Noonan K Panetta and S Agaian ldquoImageencryption using the Sudoku matrixrdquo in Proceedings of theMobileMultimediaImage Processing Security and ApplicationsApril 2010

[29] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004

[30] H Gao Y Zhang S Liang and D Li ldquoA new chaotic algorithmfor image encryptionrdquo Chaos Solitons and Fractals vol 29 no2 pp 393ndash399 2006

[31] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[32] J Thompson and H Stewart Nonlinear Dynamics and ChaosJohn Wiley amp Sons 2002

[33] C E Shannon ldquoCommunication Theory of Secrecy SystemsrdquoBell Systems Technical Journal vol 28 pp 656ndash715 1949

[34] Y Wu Y Zhou G Saveriades S Agaian J P Noonan andP Natarajan ldquoLocal Shannon entropy measure with statisticaltests for image randomnessrdquo Information Sciences vol 222 no323 342 pages 2013

[35] Y Wu S Agaian and J P Noona ldquoA novel method of testingimage randomness with applications to image shuffling andencryptionrdquo in Defense Security and Sensing Proceedings ofthe SPIE pp 875507ndash875507 2013

[36] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI random-ness tests for image encryptionrdquo Cyber Journals pp 31ndash38 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0 10

051

152

253

354

0100200300400500600700800

0 50 100 150 200 2500

10002000300040005000600070008000

0 50 100 150 200 2500

051

152

253

354

455

0 50 100 150 200 250

0 10

05

1

15

2

25

0100200300400500600

0 50 100 150 200 2500

100200300400500600

0 50 100 150 200 2500

100020003000400050006000700080009000

10000

0 50 100 150 200 250

113 Lenna Testpat1k

times104

times106 times104

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Hist

ogra

m o

fP

Plai

ntex

tP

Plai

ntex

tP

Plai

ntex

tP

Plai

ntex

tP

Ciph

erte

xtC

Ciph

erte

xtC

Ciph

erte

xtC

Ciph

erte

xtC

times106

Hist

ogra

m o

fC

Hist

ogra

m o

fC

Hist

ogra

m o

fC

Hist

ogra

m o

fC

5113

Figure 11 Histogram analysis for image encryption based on CJPM

It is clear that no matter what histogram a plaintextimage has the histogram of its ciphertext image is flatwhich implies that the pixel distribution is almost uniformComplex patterns and large homogenous regions in plaintextimages are completely unintelligible and become random-like in ciphertext images These results imply that the pro-posed image encryption method based on CJPM is robustand effective for various image formats and contents

52 Adjacent Pixel Autocorrelation (APAC) Analysis Highcorrelations of adjacent pixels can be utilized to carryout cryptanalysis Therefore a secure encryption algorithmshould break the high correlation relationship between adja-cent pixels

In statistics the autocorrelation 119877119886of a random process

119883 describes the correlation between values of the process atdifferent points in time as a function of the two times or of thetime difference The autocorrelation coefficient 119877

119886is defined

in (20) where 119889 is the time difference 120583 is the mean valuedefined by (21) and 120590 is the standard deviation defined by(22) the definition of mathematical expectation is given in(23)

119877119886(119898) =

119864 [(119883119905minus 120583) (119883

119905+119889minus 120583)]

1205902

(20)

120583 = 119864 [119883] (21)

10 Mathematical Problems in Engineering

Plai

ntex

t

Plai

ntex

t

Ciph

erte

xt

Ciph

erte

xt

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 250

Plai

ntex

tCi

pher

text

0

50

100

150

200

250

0

50

100

150

200

250

DiagonalVerticalHorizontal

DiagonalVerticalHorizontal

Figure 12 APAC analysis for image encryption based on CJPM

Table 1 Adjacent pixel autocorrelation analysis

Correlation coefficients (10minus3) Horizontal Vertical DiagonalPlaintext Lena 9399652 9709000 9709894

Ciphertext

CJPM minus22281 minus03709 3265[6] 20970 161870 17805[8] minus25000 minus10000 minus93000[5] 57765 28434 20662[29] minus127212 minus602579 624427[9] minus134000 12000 398000[30] minus158900 minus653800 minus323100[16] 815860 minus400530 minus47150[10] 1257000 581000 504000

120590 = radic119864 [(119883 minus 120583)2

] (22)

119864 [119909] =

119873

sum

119894=1

119909119894

119873 (23)

The closer to zero this coefficient is the weaker the rela-tionship two different time functions have Specifically inadjacent pixel correlation test we let 119883 be the image pixelsequence and let 119889 be 1 that is compare to the adjacent pixelsequence

Based on the reference direction there are three waysof extracting a two-dimensional image to a one-dimensionalsequence and they are the horizontal adjacent correlation

coefficient the vertical adjacent correlation coefficient andthe diagonal adjacent correlation coefficient

ldquoLenardquo image in the 2nd column of Figure 11 is used asthe test plaintext image because its APAC is widely reportedby other encryption methods Peer comparison results ofthe proposed CJPM cipher and cited encryption methods onAPAC are shown in Table 1 (best results are bolded)

In addition Figure 12 shows the result of randomlyselected 1024 pairs of two adjacent pixels from the plaintextand the ciphertext along horizontal vertical and diagonaldirections where 119909- and 119910-axes denote the intensity values ofa randomly selected pixel and its adjacent pixel respectivelyIt is clear that after applying the CJPM based image cipherthe high correlations between adjacent pixels in plaintext arebroken

53 Plaintext Sensitivity Analysis In order to test the resis-tance of the cipher to differential attacks plaintext sensitivityanalysis is required for a secure cipher In differential attacksan adversary attempts to extract meaningful relationshipbetween a plaintext image and its ciphertext image bymakinga slight change usually only one pixel in the plaintext imagewhile encrypting the plaintext image with the same encryp-tion key By comparing the change in ciphertext imagesthe encryption key might be cracked and furthermore theinformation contained in ciphertext might be leaked

Although there are othermeasures [34 35] to evaluate theresistance of plaintext attacks two classic measures are theNumber of Pixel Change Rate (NPCR) and Unified Average

Mathematical Problems in Engineering 11

Before changing one pixel After changing one pixel DifferencePl

aint

ext i

mag

eCi

pher

text

imag

e

Figure 13 Plaintext sensitivity analysis (differential attacks) for image encryption based on CJPM

(a) Encrypted image (1 1 128128 1 1 4 0)

(b) Decrypted image (1 1 128128 1 1 4 0)

(c) Decrypted image (1 1 128128 1 2 4 0)

(d) Decrypted image (1 1 128128 1 1 39999 0)

Figure 14 Key sensitivity analysis

Changing Intensity (UACI) [5]TheNPCR is used tomeasurethe percentage of the number of pixels changed in ciphertextafter making a slight change in plaintext Therefore thetheoretical greatest upper-bound of the NPCR is 100 TheUACI is used to measure the averaged intensity change forpixels in ciphertext images after making a slight change ina plaintext image It is demonstrable that the UACI of an idealcipher for 8-bit gray images is about 03346 [36]

Suppose ciphertext images before and after one pixelchange in a plaintext image are 1198621 and 1198622 respectively thepixel values at grid (119894 119895) in 1198621 and 1198622 are denoted as 1198621(119894 119895)and 1198622(119894 119895) a bipolar array 119863 is defined as (24) then the

NPCR and UACI can be mathematically defined as (25) and(26) respectively where symbol119873 denotes the total numberof pixels in the ciphertext symbol 119865 denotes the largestsupported pixel value compatible with the ciphertext imageformat and | sdot | is the absolute value function Consider

119863(119894 119895) =

1 if 1198621 (119894 119895) = 1198622 (119894 119895)

0 if 1198621 (119894 119895) = 1198622

(119894 119895)

(24)

12 Mathematical Problems in Engineering

Table 2 NPCR and UACI analyses on ldquoLenardquo image

CPJMsize 16-by-16 32-by-32 64-by-64 128-by-128 256-by-256

NPCR 996170 996246 995895 995850 996338UACI 338205 338379 334048 334076 334040

NPCR = sum

119894119895

119863(119894 119895)

119873times 100 (25)

UACI = sum

119894119895

100381610038161003816100381610038161198621

(119894 119895) minus 1198622

(119894 119895)10038161003816100381610038161003816

(119865 sdot 119873)times 100 (26)

Table 2 shows the NPCR and UACI results for one pixelchange in the ldquoLenardquo image It is clear that for CJPMs atvarious sizes the proposed cipher has good performances inboth the NPCR and UACI analyses Simulation results fit theexpectations of the ideal cipher very well

Figure 13 shows a differential attack on ldquoLenardquo imagewhile keeping the encryption key unchanged It is noticeablethat the difference between two plaintext images is thepixel on Lenarsquos nose However the ciphertext images are sodifferent that the image of their difference is still random-like As a result the one pixel change in the plaintext image isldquodissipatedrdquo in ciphertext From this point view this plaintextsensitivity is closely related to the diffusion property of acipher In other words it is reasonable to claim that a cipherhas good NPCR and UACI results if it has good diffusionproperties

54 Key Space Analysis The encryption key in the proposedimage encryption method using CPJM is composed of a setof parameters (119909

119903 119910119903 119908 119877 119862 119904 119899 119903 119898) where (119909

119903 119910119903) is an

arbitrary point on 119909119910 plain119908 is a nonnegative decimal119877 and119862 should be positive integers smaller than the plaintext imagesize 119904 and 119899 should be positive integers below the productof 119877119862 119903 should be a number in between [36 4] and 119898 is anonnegative integer Therefore theoretically the key space ofthe proposed cipher is infinitely large

Because the chaotic Logistic map is used as the triggerfor pseudorandom sequences the proposed cipher has highkey sensitivities as well The results of key sensitivity analysisare shown in Figure 14 where the set of parameters writtenin parenthesis is the used key It is clear that unless thecorrect decryption key is applied a ciphertext image cannotbe restored

6 Conclusion

In this paper we discussed the generation of a chaotic Jose-phus permutation matrix by using the conventional Josephuspermutation sequences and the logistic chaotic map Theproposed CJPM is parametric and is uniquely dependent onthe set of parameters which is sufficiently large to providea secure size of key space As another heritage from thechaotic Logistic map the CJPM is highly sensitive to itsinitial values (parameters) Any slight change in parameters

leads to significant differences in resulting CJPM Simulationresults show that (1) the ciphertext image is random-likefrom the perspective of human visual inspection (2) theencryption quality is almost independent of the plaintextimage (3) the proposed encryptionmethod is able to encryptplaintext images with large homogeneous regions to secureciphertext images (4) histogram analysis also shows thatdifferent ciphertext images tend to have the uniform distri-bution on [0 119865] (5) adjacent pixel correlation analysis showsthat neighbor pixels in ciphertext have lower correlationsthan many existing encryption methods (6) the proposedcipher is highly sensitive to encryption key and plaintext (7)experimental UACI results of the proposed cipher are veryclose to those of the ideal one

The proposed cipher can be used for various image typesfor example binary images 8-bit gray images 16-bit grayimages RBG images and so forth The same encryption ideamay also be applied to audio video or other types of digitalformats

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Scientific Research Fundof Hunan Provincial Education Department under Grant(no 12B023) It is also supported by the National NaturalScience Foundation of China (nos 61204039 and 61106029)and Scientific Research Foundation for Returned ScholarsMinistry of Human Resources and Social Security of thePeoplersquos Republic of China

References

[1] D StinsonCryptographyTheory andPractice CRCPress 2006[2] M Yang N Bourbakis and S Li ldquoData-image-video encryp-

tionrdquo IEEE Potentials vol 23 no 3 pp 28ndash34 2004[3] FIPS PUB 46 Data Encryption Standard National Bureau of

Standards 1977[4] FIPS PUB 197 Avanced Encryption Standard Announcing the

Advanced Encryption Standard (AES) 2001[5] Y Mao and G Chen Chaos-Based Image Encryption Springer

Berlin Germany 2005[6] H Yang K-W Wong X Liao W Zhang and P Wei ldquoA fast

image encryption and authentication scheme based on chaoticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3507ndash3517 2010

[7] J Hu and F Han ldquoA pixel-based scrambling scheme for digitalmedical images protectionrdquo Journal of Network and ComputerApplications vol 32 no 4 pp 788ndash794 2009

[8] Z Shuo C Ruhua J Yingchun and G Shiping ldquoAn imageencryption algorithm based on multiple chaos and wavelettransformrdquo in Proceedings of the 2nd International CongressImage and Signal Processing (CISP rsquo09) pp 1ndash5 2009

Mathematical Problems in Engineering 13

[9] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010

[10] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009

[11] X Liao S Lai and Q Zhou ldquoA novel image encryptionalgorithm based on self-adaptive wave transmissionrdquo SignalProcessing vol 90 no 9 pp 2714ndash2722 2010

[12] S Fu-Yan L Shu-Tang and L Zong-Wang ldquoImage encryptionusing high-dimension chaotic systemrdquo Chinese Physics vol 16no 12 pp 3616ndash3623 2007

[13] V Patidar N K Pareek and K K Sud ldquoA new substitution-diffusion based image cipher using chaotic standard and logisticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 7 pp 3056ndash3075 2009

[14] Y Wu J P Noonan G Yang and H Jin ldquoImage encryptionusing the two-dimensional logistic chaotic maprdquo Journal ofElectronic Imaging vol 21 no 1 Article ID 013014 2012

[15] Y Wu J P Noonan and S Agaian ldquoA wheel-switch chaoticsystem for image encryptionrdquo in Proceedings of the InternationalConference on System Science and Engineering (ICSSE rsquo11) pp23ndash27 June 2011

[16] L Zhang X Liao andXWang ldquoAn image encryption approachbased on chaoticmapsrdquoChaos Solitons and Fractals vol 24 no3 pp 759ndash765 2005

[17] H S Kwok and W K S Tang ldquoA fast image encryption systembased on chaotic maps with finite precision representationrdquoChaos Solitons and Fractals vol 32 no 4 pp 1518ndash1529 2007

[18] J Jian Y Shi C Hu Q Ma and J Li ldquoEncryption of digitalimage based on chaos systemrdquo in Computer and ComputingTechnologies in Agriculture II vol 2 pp 1145ndash1151 2009

[19] P Van-Roy and S Haridi Concepts Techniques and Models ofComputer Programming MIT Press 2004

[20] P SchumerMathematical Journeys Wiley-Interscience 2004[21] G Ye X Huang and C Zhu ldquoImage encryption algorithm of

double scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007

[22] XDesheng andXYueshan ldquoDigital image scrambling based onjosephus traversingrdquo Computer Engineering and Applicationsno 10 pp 44ndash46 2005

[23] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2005

[24] Y-P Zhang Z-J Zhai W Liu X Nie S-P Cao and W-D Dai ldquoDigital image encryption algorithm based on chaosand improved DESrdquo in Proceedings of the IEEE InternationalConference on Systems Man and Cybernetics (SMC rsquo09) pp474ndash479 October 2009

[25] J Lang R Tao and Y Wang ldquoImage encryption based on themultiple-parameter discrete fractional Fourier transform andchaos functionrdquo Optics Communications vol 283 no 10 pp2092ndash2096 2010

[26] S Yang and S Sun ldquoVideo encryptionmethod based on chaoticmaps in DCT domainrdquo Progress in Natural Science vol 18 no10 pp 1299ndash1304 2008

[27] V Wu J P Noonan and S Agaian ldquoBinary data encryptionusing the Sudoku block cipherrdquo in Proceedings of the IEEEInternational Conference on SystemsMan andCybernetics (SMCrsquo10) pp 3915ndash3921 October 2010

[28] Y Wu Y Zhou J P Noonan K Panetta and S Agaian ldquoImageencryption using the Sudoku matrixrdquo in Proceedings of theMobileMultimediaImage Processing Security and ApplicationsApril 2010

[29] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004

[30] H Gao Y Zhang S Liang and D Li ldquoA new chaotic algorithmfor image encryptionrdquo Chaos Solitons and Fractals vol 29 no2 pp 393ndash399 2006

[31] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[32] J Thompson and H Stewart Nonlinear Dynamics and ChaosJohn Wiley amp Sons 2002

[33] C E Shannon ldquoCommunication Theory of Secrecy SystemsrdquoBell Systems Technical Journal vol 28 pp 656ndash715 1949

[34] Y Wu Y Zhou G Saveriades S Agaian J P Noonan andP Natarajan ldquoLocal Shannon entropy measure with statisticaltests for image randomnessrdquo Information Sciences vol 222 no323 342 pages 2013

[35] Y Wu S Agaian and J P Noona ldquoA novel method of testingimage randomness with applications to image shuffling andencryptionrdquo in Defense Security and Sensing Proceedings ofthe SPIE pp 875507ndash875507 2013

[36] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI random-ness tests for image encryptionrdquo Cyber Journals pp 31ndash38 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Plai

ntex

t

Plai

ntex

t

Ciph

erte

xt

Ciph

erte

xt

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 2500

50

100

150

200

250

0 50 100 150 200 250

Plai

ntex

tCi

pher

text

0

50

100

150

200

250

0

50

100

150

200

250

DiagonalVerticalHorizontal

DiagonalVerticalHorizontal

Figure 12 APAC analysis for image encryption based on CJPM

Table 1 Adjacent pixel autocorrelation analysis

Correlation coefficients (10minus3) Horizontal Vertical DiagonalPlaintext Lena 9399652 9709000 9709894

Ciphertext

CJPM minus22281 minus03709 3265[6] 20970 161870 17805[8] minus25000 minus10000 minus93000[5] 57765 28434 20662[29] minus127212 minus602579 624427[9] minus134000 12000 398000[30] minus158900 minus653800 minus323100[16] 815860 minus400530 minus47150[10] 1257000 581000 504000

120590 = radic119864 [(119883 minus 120583)2

] (22)

119864 [119909] =

119873

sum

119894=1

119909119894

119873 (23)

The closer to zero this coefficient is the weaker the rela-tionship two different time functions have Specifically inadjacent pixel correlation test we let 119883 be the image pixelsequence and let 119889 be 1 that is compare to the adjacent pixelsequence

Based on the reference direction there are three waysof extracting a two-dimensional image to a one-dimensionalsequence and they are the horizontal adjacent correlation

coefficient the vertical adjacent correlation coefficient andthe diagonal adjacent correlation coefficient

ldquoLenardquo image in the 2nd column of Figure 11 is used asthe test plaintext image because its APAC is widely reportedby other encryption methods Peer comparison results ofthe proposed CJPM cipher and cited encryption methods onAPAC are shown in Table 1 (best results are bolded)

In addition Figure 12 shows the result of randomlyselected 1024 pairs of two adjacent pixels from the plaintextand the ciphertext along horizontal vertical and diagonaldirections where 119909- and 119910-axes denote the intensity values ofa randomly selected pixel and its adjacent pixel respectivelyIt is clear that after applying the CJPM based image cipherthe high correlations between adjacent pixels in plaintext arebroken

53 Plaintext Sensitivity Analysis In order to test the resis-tance of the cipher to differential attacks plaintext sensitivityanalysis is required for a secure cipher In differential attacksan adversary attempts to extract meaningful relationshipbetween a plaintext image and its ciphertext image bymakinga slight change usually only one pixel in the plaintext imagewhile encrypting the plaintext image with the same encryp-tion key By comparing the change in ciphertext imagesthe encryption key might be cracked and furthermore theinformation contained in ciphertext might be leaked

Although there are othermeasures [34 35] to evaluate theresistance of plaintext attacks two classic measures are theNumber of Pixel Change Rate (NPCR) and Unified Average

Mathematical Problems in Engineering 11

Before changing one pixel After changing one pixel DifferencePl

aint

ext i

mag

eCi

pher

text

imag

e

Figure 13 Plaintext sensitivity analysis (differential attacks) for image encryption based on CJPM

(a) Encrypted image (1 1 128128 1 1 4 0)

(b) Decrypted image (1 1 128128 1 1 4 0)

(c) Decrypted image (1 1 128128 1 2 4 0)

(d) Decrypted image (1 1 128128 1 1 39999 0)

Figure 14 Key sensitivity analysis

Changing Intensity (UACI) [5]TheNPCR is used tomeasurethe percentage of the number of pixels changed in ciphertextafter making a slight change in plaintext Therefore thetheoretical greatest upper-bound of the NPCR is 100 TheUACI is used to measure the averaged intensity change forpixels in ciphertext images after making a slight change ina plaintext image It is demonstrable that the UACI of an idealcipher for 8-bit gray images is about 03346 [36]

Suppose ciphertext images before and after one pixelchange in a plaintext image are 1198621 and 1198622 respectively thepixel values at grid (119894 119895) in 1198621 and 1198622 are denoted as 1198621(119894 119895)and 1198622(119894 119895) a bipolar array 119863 is defined as (24) then the

NPCR and UACI can be mathematically defined as (25) and(26) respectively where symbol119873 denotes the total numberof pixels in the ciphertext symbol 119865 denotes the largestsupported pixel value compatible with the ciphertext imageformat and | sdot | is the absolute value function Consider

119863(119894 119895) =

1 if 1198621 (119894 119895) = 1198622 (119894 119895)

0 if 1198621 (119894 119895) = 1198622

(119894 119895)

(24)

12 Mathematical Problems in Engineering

Table 2 NPCR and UACI analyses on ldquoLenardquo image

CPJMsize 16-by-16 32-by-32 64-by-64 128-by-128 256-by-256

NPCR 996170 996246 995895 995850 996338UACI 338205 338379 334048 334076 334040

NPCR = sum

119894119895

119863(119894 119895)

119873times 100 (25)

UACI = sum

119894119895

100381610038161003816100381610038161198621

(119894 119895) minus 1198622

(119894 119895)10038161003816100381610038161003816

(119865 sdot 119873)times 100 (26)

Table 2 shows the NPCR and UACI results for one pixelchange in the ldquoLenardquo image It is clear that for CJPMs atvarious sizes the proposed cipher has good performances inboth the NPCR and UACI analyses Simulation results fit theexpectations of the ideal cipher very well

Figure 13 shows a differential attack on ldquoLenardquo imagewhile keeping the encryption key unchanged It is noticeablethat the difference between two plaintext images is thepixel on Lenarsquos nose However the ciphertext images are sodifferent that the image of their difference is still random-like As a result the one pixel change in the plaintext image isldquodissipatedrdquo in ciphertext From this point view this plaintextsensitivity is closely related to the diffusion property of acipher In other words it is reasonable to claim that a cipherhas good NPCR and UACI results if it has good diffusionproperties

54 Key Space Analysis The encryption key in the proposedimage encryption method using CPJM is composed of a setof parameters (119909

119903 119910119903 119908 119877 119862 119904 119899 119903 119898) where (119909

119903 119910119903) is an

arbitrary point on 119909119910 plain119908 is a nonnegative decimal119877 and119862 should be positive integers smaller than the plaintext imagesize 119904 and 119899 should be positive integers below the productof 119877119862 119903 should be a number in between [36 4] and 119898 is anonnegative integer Therefore theoretically the key space ofthe proposed cipher is infinitely large

Because the chaotic Logistic map is used as the triggerfor pseudorandom sequences the proposed cipher has highkey sensitivities as well The results of key sensitivity analysisare shown in Figure 14 where the set of parameters writtenin parenthesis is the used key It is clear that unless thecorrect decryption key is applied a ciphertext image cannotbe restored

6 Conclusion

In this paper we discussed the generation of a chaotic Jose-phus permutation matrix by using the conventional Josephuspermutation sequences and the logistic chaotic map Theproposed CJPM is parametric and is uniquely dependent onthe set of parameters which is sufficiently large to providea secure size of key space As another heritage from thechaotic Logistic map the CJPM is highly sensitive to itsinitial values (parameters) Any slight change in parameters

leads to significant differences in resulting CJPM Simulationresults show that (1) the ciphertext image is random-likefrom the perspective of human visual inspection (2) theencryption quality is almost independent of the plaintextimage (3) the proposed encryptionmethod is able to encryptplaintext images with large homogeneous regions to secureciphertext images (4) histogram analysis also shows thatdifferent ciphertext images tend to have the uniform distri-bution on [0 119865] (5) adjacent pixel correlation analysis showsthat neighbor pixels in ciphertext have lower correlationsthan many existing encryption methods (6) the proposedcipher is highly sensitive to encryption key and plaintext (7)experimental UACI results of the proposed cipher are veryclose to those of the ideal one

The proposed cipher can be used for various image typesfor example binary images 8-bit gray images 16-bit grayimages RBG images and so forth The same encryption ideamay also be applied to audio video or other types of digitalformats

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Scientific Research Fundof Hunan Provincial Education Department under Grant(no 12B023) It is also supported by the National NaturalScience Foundation of China (nos 61204039 and 61106029)and Scientific Research Foundation for Returned ScholarsMinistry of Human Resources and Social Security of thePeoplersquos Republic of China

References

[1] D StinsonCryptographyTheory andPractice CRCPress 2006[2] M Yang N Bourbakis and S Li ldquoData-image-video encryp-

tionrdquo IEEE Potentials vol 23 no 3 pp 28ndash34 2004[3] FIPS PUB 46 Data Encryption Standard National Bureau of

Standards 1977[4] FIPS PUB 197 Avanced Encryption Standard Announcing the

Advanced Encryption Standard (AES) 2001[5] Y Mao and G Chen Chaos-Based Image Encryption Springer

Berlin Germany 2005[6] H Yang K-W Wong X Liao W Zhang and P Wei ldquoA fast

image encryption and authentication scheme based on chaoticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3507ndash3517 2010

[7] J Hu and F Han ldquoA pixel-based scrambling scheme for digitalmedical images protectionrdquo Journal of Network and ComputerApplications vol 32 no 4 pp 788ndash794 2009

[8] Z Shuo C Ruhua J Yingchun and G Shiping ldquoAn imageencryption algorithm based on multiple chaos and wavelettransformrdquo in Proceedings of the 2nd International CongressImage and Signal Processing (CISP rsquo09) pp 1ndash5 2009

Mathematical Problems in Engineering 13

[9] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010

[10] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009

[11] X Liao S Lai and Q Zhou ldquoA novel image encryptionalgorithm based on self-adaptive wave transmissionrdquo SignalProcessing vol 90 no 9 pp 2714ndash2722 2010

[12] S Fu-Yan L Shu-Tang and L Zong-Wang ldquoImage encryptionusing high-dimension chaotic systemrdquo Chinese Physics vol 16no 12 pp 3616ndash3623 2007

[13] V Patidar N K Pareek and K K Sud ldquoA new substitution-diffusion based image cipher using chaotic standard and logisticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 7 pp 3056ndash3075 2009

[14] Y Wu J P Noonan G Yang and H Jin ldquoImage encryptionusing the two-dimensional logistic chaotic maprdquo Journal ofElectronic Imaging vol 21 no 1 Article ID 013014 2012

[15] Y Wu J P Noonan and S Agaian ldquoA wheel-switch chaoticsystem for image encryptionrdquo in Proceedings of the InternationalConference on System Science and Engineering (ICSSE rsquo11) pp23ndash27 June 2011

[16] L Zhang X Liao andXWang ldquoAn image encryption approachbased on chaoticmapsrdquoChaos Solitons and Fractals vol 24 no3 pp 759ndash765 2005

[17] H S Kwok and W K S Tang ldquoA fast image encryption systembased on chaotic maps with finite precision representationrdquoChaos Solitons and Fractals vol 32 no 4 pp 1518ndash1529 2007

[18] J Jian Y Shi C Hu Q Ma and J Li ldquoEncryption of digitalimage based on chaos systemrdquo in Computer and ComputingTechnologies in Agriculture II vol 2 pp 1145ndash1151 2009

[19] P Van-Roy and S Haridi Concepts Techniques and Models ofComputer Programming MIT Press 2004

[20] P SchumerMathematical Journeys Wiley-Interscience 2004[21] G Ye X Huang and C Zhu ldquoImage encryption algorithm of

double scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007

[22] XDesheng andXYueshan ldquoDigital image scrambling based onjosephus traversingrdquo Computer Engineering and Applicationsno 10 pp 44ndash46 2005

[23] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2005

[24] Y-P Zhang Z-J Zhai W Liu X Nie S-P Cao and W-D Dai ldquoDigital image encryption algorithm based on chaosand improved DESrdquo in Proceedings of the IEEE InternationalConference on Systems Man and Cybernetics (SMC rsquo09) pp474ndash479 October 2009

[25] J Lang R Tao and Y Wang ldquoImage encryption based on themultiple-parameter discrete fractional Fourier transform andchaos functionrdquo Optics Communications vol 283 no 10 pp2092ndash2096 2010

[26] S Yang and S Sun ldquoVideo encryptionmethod based on chaoticmaps in DCT domainrdquo Progress in Natural Science vol 18 no10 pp 1299ndash1304 2008

[27] V Wu J P Noonan and S Agaian ldquoBinary data encryptionusing the Sudoku block cipherrdquo in Proceedings of the IEEEInternational Conference on SystemsMan andCybernetics (SMCrsquo10) pp 3915ndash3921 October 2010

[28] Y Wu Y Zhou J P Noonan K Panetta and S Agaian ldquoImageencryption using the Sudoku matrixrdquo in Proceedings of theMobileMultimediaImage Processing Security and ApplicationsApril 2010

[29] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004

[30] H Gao Y Zhang S Liang and D Li ldquoA new chaotic algorithmfor image encryptionrdquo Chaos Solitons and Fractals vol 29 no2 pp 393ndash399 2006

[31] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[32] J Thompson and H Stewart Nonlinear Dynamics and ChaosJohn Wiley amp Sons 2002

[33] C E Shannon ldquoCommunication Theory of Secrecy SystemsrdquoBell Systems Technical Journal vol 28 pp 656ndash715 1949

[34] Y Wu Y Zhou G Saveriades S Agaian J P Noonan andP Natarajan ldquoLocal Shannon entropy measure with statisticaltests for image randomnessrdquo Information Sciences vol 222 no323 342 pages 2013

[35] Y Wu S Agaian and J P Noona ldquoA novel method of testingimage randomness with applications to image shuffling andencryptionrdquo in Defense Security and Sensing Proceedings ofthe SPIE pp 875507ndash875507 2013

[36] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI random-ness tests for image encryptionrdquo Cyber Journals pp 31ndash38 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Before changing one pixel After changing one pixel DifferencePl

aint

ext i

mag

eCi

pher

text

imag

e

Figure 13 Plaintext sensitivity analysis (differential attacks) for image encryption based on CJPM

(a) Encrypted image (1 1 128128 1 1 4 0)

(b) Decrypted image (1 1 128128 1 1 4 0)

(c) Decrypted image (1 1 128128 1 2 4 0)

(d) Decrypted image (1 1 128128 1 1 39999 0)

Figure 14 Key sensitivity analysis

Changing Intensity (UACI) [5]TheNPCR is used tomeasurethe percentage of the number of pixels changed in ciphertextafter making a slight change in plaintext Therefore thetheoretical greatest upper-bound of the NPCR is 100 TheUACI is used to measure the averaged intensity change forpixels in ciphertext images after making a slight change ina plaintext image It is demonstrable that the UACI of an idealcipher for 8-bit gray images is about 03346 [36]

Suppose ciphertext images before and after one pixelchange in a plaintext image are 1198621 and 1198622 respectively thepixel values at grid (119894 119895) in 1198621 and 1198622 are denoted as 1198621(119894 119895)and 1198622(119894 119895) a bipolar array 119863 is defined as (24) then the

NPCR and UACI can be mathematically defined as (25) and(26) respectively where symbol119873 denotes the total numberof pixels in the ciphertext symbol 119865 denotes the largestsupported pixel value compatible with the ciphertext imageformat and | sdot | is the absolute value function Consider

119863(119894 119895) =

1 if 1198621 (119894 119895) = 1198622 (119894 119895)

0 if 1198621 (119894 119895) = 1198622

(119894 119895)

(24)

12 Mathematical Problems in Engineering

Table 2 NPCR and UACI analyses on ldquoLenardquo image

CPJMsize 16-by-16 32-by-32 64-by-64 128-by-128 256-by-256

NPCR 996170 996246 995895 995850 996338UACI 338205 338379 334048 334076 334040

NPCR = sum

119894119895

119863(119894 119895)

119873times 100 (25)

UACI = sum

119894119895

100381610038161003816100381610038161198621

(119894 119895) minus 1198622

(119894 119895)10038161003816100381610038161003816

(119865 sdot 119873)times 100 (26)

Table 2 shows the NPCR and UACI results for one pixelchange in the ldquoLenardquo image It is clear that for CJPMs atvarious sizes the proposed cipher has good performances inboth the NPCR and UACI analyses Simulation results fit theexpectations of the ideal cipher very well

Figure 13 shows a differential attack on ldquoLenardquo imagewhile keeping the encryption key unchanged It is noticeablethat the difference between two plaintext images is thepixel on Lenarsquos nose However the ciphertext images are sodifferent that the image of their difference is still random-like As a result the one pixel change in the plaintext image isldquodissipatedrdquo in ciphertext From this point view this plaintextsensitivity is closely related to the diffusion property of acipher In other words it is reasonable to claim that a cipherhas good NPCR and UACI results if it has good diffusionproperties

54 Key Space Analysis The encryption key in the proposedimage encryption method using CPJM is composed of a setof parameters (119909

119903 119910119903 119908 119877 119862 119904 119899 119903 119898) where (119909

119903 119910119903) is an

arbitrary point on 119909119910 plain119908 is a nonnegative decimal119877 and119862 should be positive integers smaller than the plaintext imagesize 119904 and 119899 should be positive integers below the productof 119877119862 119903 should be a number in between [36 4] and 119898 is anonnegative integer Therefore theoretically the key space ofthe proposed cipher is infinitely large

Because the chaotic Logistic map is used as the triggerfor pseudorandom sequences the proposed cipher has highkey sensitivities as well The results of key sensitivity analysisare shown in Figure 14 where the set of parameters writtenin parenthesis is the used key It is clear that unless thecorrect decryption key is applied a ciphertext image cannotbe restored

6 Conclusion

In this paper we discussed the generation of a chaotic Jose-phus permutation matrix by using the conventional Josephuspermutation sequences and the logistic chaotic map Theproposed CJPM is parametric and is uniquely dependent onthe set of parameters which is sufficiently large to providea secure size of key space As another heritage from thechaotic Logistic map the CJPM is highly sensitive to itsinitial values (parameters) Any slight change in parameters

leads to significant differences in resulting CJPM Simulationresults show that (1) the ciphertext image is random-likefrom the perspective of human visual inspection (2) theencryption quality is almost independent of the plaintextimage (3) the proposed encryptionmethod is able to encryptplaintext images with large homogeneous regions to secureciphertext images (4) histogram analysis also shows thatdifferent ciphertext images tend to have the uniform distri-bution on [0 119865] (5) adjacent pixel correlation analysis showsthat neighbor pixels in ciphertext have lower correlationsthan many existing encryption methods (6) the proposedcipher is highly sensitive to encryption key and plaintext (7)experimental UACI results of the proposed cipher are veryclose to those of the ideal one

The proposed cipher can be used for various image typesfor example binary images 8-bit gray images 16-bit grayimages RBG images and so forth The same encryption ideamay also be applied to audio video or other types of digitalformats

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Scientific Research Fundof Hunan Provincial Education Department under Grant(no 12B023) It is also supported by the National NaturalScience Foundation of China (nos 61204039 and 61106029)and Scientific Research Foundation for Returned ScholarsMinistry of Human Resources and Social Security of thePeoplersquos Republic of China

References

[1] D StinsonCryptographyTheory andPractice CRCPress 2006[2] M Yang N Bourbakis and S Li ldquoData-image-video encryp-

tionrdquo IEEE Potentials vol 23 no 3 pp 28ndash34 2004[3] FIPS PUB 46 Data Encryption Standard National Bureau of

Standards 1977[4] FIPS PUB 197 Avanced Encryption Standard Announcing the

Advanced Encryption Standard (AES) 2001[5] Y Mao and G Chen Chaos-Based Image Encryption Springer

Berlin Germany 2005[6] H Yang K-W Wong X Liao W Zhang and P Wei ldquoA fast

image encryption and authentication scheme based on chaoticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3507ndash3517 2010

[7] J Hu and F Han ldquoA pixel-based scrambling scheme for digitalmedical images protectionrdquo Journal of Network and ComputerApplications vol 32 no 4 pp 788ndash794 2009

[8] Z Shuo C Ruhua J Yingchun and G Shiping ldquoAn imageencryption algorithm based on multiple chaos and wavelettransformrdquo in Proceedings of the 2nd International CongressImage and Signal Processing (CISP rsquo09) pp 1ndash5 2009

Mathematical Problems in Engineering 13

[9] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010

[10] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009

[11] X Liao S Lai and Q Zhou ldquoA novel image encryptionalgorithm based on self-adaptive wave transmissionrdquo SignalProcessing vol 90 no 9 pp 2714ndash2722 2010

[12] S Fu-Yan L Shu-Tang and L Zong-Wang ldquoImage encryptionusing high-dimension chaotic systemrdquo Chinese Physics vol 16no 12 pp 3616ndash3623 2007

[13] V Patidar N K Pareek and K K Sud ldquoA new substitution-diffusion based image cipher using chaotic standard and logisticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 7 pp 3056ndash3075 2009

[14] Y Wu J P Noonan G Yang and H Jin ldquoImage encryptionusing the two-dimensional logistic chaotic maprdquo Journal ofElectronic Imaging vol 21 no 1 Article ID 013014 2012

[15] Y Wu J P Noonan and S Agaian ldquoA wheel-switch chaoticsystem for image encryptionrdquo in Proceedings of the InternationalConference on System Science and Engineering (ICSSE rsquo11) pp23ndash27 June 2011

[16] L Zhang X Liao andXWang ldquoAn image encryption approachbased on chaoticmapsrdquoChaos Solitons and Fractals vol 24 no3 pp 759ndash765 2005

[17] H S Kwok and W K S Tang ldquoA fast image encryption systembased on chaotic maps with finite precision representationrdquoChaos Solitons and Fractals vol 32 no 4 pp 1518ndash1529 2007

[18] J Jian Y Shi C Hu Q Ma and J Li ldquoEncryption of digitalimage based on chaos systemrdquo in Computer and ComputingTechnologies in Agriculture II vol 2 pp 1145ndash1151 2009

[19] P Van-Roy and S Haridi Concepts Techniques and Models ofComputer Programming MIT Press 2004

[20] P SchumerMathematical Journeys Wiley-Interscience 2004[21] G Ye X Huang and C Zhu ldquoImage encryption algorithm of

double scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007

[22] XDesheng andXYueshan ldquoDigital image scrambling based onjosephus traversingrdquo Computer Engineering and Applicationsno 10 pp 44ndash46 2005

[23] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2005

[24] Y-P Zhang Z-J Zhai W Liu X Nie S-P Cao and W-D Dai ldquoDigital image encryption algorithm based on chaosand improved DESrdquo in Proceedings of the IEEE InternationalConference on Systems Man and Cybernetics (SMC rsquo09) pp474ndash479 October 2009

[25] J Lang R Tao and Y Wang ldquoImage encryption based on themultiple-parameter discrete fractional Fourier transform andchaos functionrdquo Optics Communications vol 283 no 10 pp2092ndash2096 2010

[26] S Yang and S Sun ldquoVideo encryptionmethod based on chaoticmaps in DCT domainrdquo Progress in Natural Science vol 18 no10 pp 1299ndash1304 2008

[27] V Wu J P Noonan and S Agaian ldquoBinary data encryptionusing the Sudoku block cipherrdquo in Proceedings of the IEEEInternational Conference on SystemsMan andCybernetics (SMCrsquo10) pp 3915ndash3921 October 2010

[28] Y Wu Y Zhou J P Noonan K Panetta and S Agaian ldquoImageencryption using the Sudoku matrixrdquo in Proceedings of theMobileMultimediaImage Processing Security and ApplicationsApril 2010

[29] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004

[30] H Gao Y Zhang S Liang and D Li ldquoA new chaotic algorithmfor image encryptionrdquo Chaos Solitons and Fractals vol 29 no2 pp 393ndash399 2006

[31] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[32] J Thompson and H Stewart Nonlinear Dynamics and ChaosJohn Wiley amp Sons 2002

[33] C E Shannon ldquoCommunication Theory of Secrecy SystemsrdquoBell Systems Technical Journal vol 28 pp 656ndash715 1949

[34] Y Wu Y Zhou G Saveriades S Agaian J P Noonan andP Natarajan ldquoLocal Shannon entropy measure with statisticaltests for image randomnessrdquo Information Sciences vol 222 no323 342 pages 2013

[35] Y Wu S Agaian and J P Noona ldquoA novel method of testingimage randomness with applications to image shuffling andencryptionrdquo in Defense Security and Sensing Proceedings ofthe SPIE pp 875507ndash875507 2013

[36] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI random-ness tests for image encryptionrdquo Cyber Journals pp 31ndash38 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

Table 2 NPCR and UACI analyses on ldquoLenardquo image

CPJMsize 16-by-16 32-by-32 64-by-64 128-by-128 256-by-256

NPCR 996170 996246 995895 995850 996338UACI 338205 338379 334048 334076 334040

NPCR = sum

119894119895

119863(119894 119895)

119873times 100 (25)

UACI = sum

119894119895

100381610038161003816100381610038161198621

(119894 119895) minus 1198622

(119894 119895)10038161003816100381610038161003816

(119865 sdot 119873)times 100 (26)

Table 2 shows the NPCR and UACI results for one pixelchange in the ldquoLenardquo image It is clear that for CJPMs atvarious sizes the proposed cipher has good performances inboth the NPCR and UACI analyses Simulation results fit theexpectations of the ideal cipher very well

Figure 13 shows a differential attack on ldquoLenardquo imagewhile keeping the encryption key unchanged It is noticeablethat the difference between two plaintext images is thepixel on Lenarsquos nose However the ciphertext images are sodifferent that the image of their difference is still random-like As a result the one pixel change in the plaintext image isldquodissipatedrdquo in ciphertext From this point view this plaintextsensitivity is closely related to the diffusion property of acipher In other words it is reasonable to claim that a cipherhas good NPCR and UACI results if it has good diffusionproperties

54 Key Space Analysis The encryption key in the proposedimage encryption method using CPJM is composed of a setof parameters (119909

119903 119910119903 119908 119877 119862 119904 119899 119903 119898) where (119909

119903 119910119903) is an

arbitrary point on 119909119910 plain119908 is a nonnegative decimal119877 and119862 should be positive integers smaller than the plaintext imagesize 119904 and 119899 should be positive integers below the productof 119877119862 119903 should be a number in between [36 4] and 119898 is anonnegative integer Therefore theoretically the key space ofthe proposed cipher is infinitely large

Because the chaotic Logistic map is used as the triggerfor pseudorandom sequences the proposed cipher has highkey sensitivities as well The results of key sensitivity analysisare shown in Figure 14 where the set of parameters writtenin parenthesis is the used key It is clear that unless thecorrect decryption key is applied a ciphertext image cannotbe restored

6 Conclusion

In this paper we discussed the generation of a chaotic Jose-phus permutation matrix by using the conventional Josephuspermutation sequences and the logistic chaotic map Theproposed CJPM is parametric and is uniquely dependent onthe set of parameters which is sufficiently large to providea secure size of key space As another heritage from thechaotic Logistic map the CJPM is highly sensitive to itsinitial values (parameters) Any slight change in parameters

leads to significant differences in resulting CJPM Simulationresults show that (1) the ciphertext image is random-likefrom the perspective of human visual inspection (2) theencryption quality is almost independent of the plaintextimage (3) the proposed encryptionmethod is able to encryptplaintext images with large homogeneous regions to secureciphertext images (4) histogram analysis also shows thatdifferent ciphertext images tend to have the uniform distri-bution on [0 119865] (5) adjacent pixel correlation analysis showsthat neighbor pixels in ciphertext have lower correlationsthan many existing encryption methods (6) the proposedcipher is highly sensitive to encryption key and plaintext (7)experimental UACI results of the proposed cipher are veryclose to those of the ideal one

The proposed cipher can be used for various image typesfor example binary images 8-bit gray images 16-bit grayimages RBG images and so forth The same encryption ideamay also be applied to audio video or other types of digitalformats

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Scientific Research Fundof Hunan Provincial Education Department under Grant(no 12B023) It is also supported by the National NaturalScience Foundation of China (nos 61204039 and 61106029)and Scientific Research Foundation for Returned ScholarsMinistry of Human Resources and Social Security of thePeoplersquos Republic of China

References

[1] D StinsonCryptographyTheory andPractice CRCPress 2006[2] M Yang N Bourbakis and S Li ldquoData-image-video encryp-

tionrdquo IEEE Potentials vol 23 no 3 pp 28ndash34 2004[3] FIPS PUB 46 Data Encryption Standard National Bureau of

Standards 1977[4] FIPS PUB 197 Avanced Encryption Standard Announcing the

Advanced Encryption Standard (AES) 2001[5] Y Mao and G Chen Chaos-Based Image Encryption Springer

Berlin Germany 2005[6] H Yang K-W Wong X Liao W Zhang and P Wei ldquoA fast

image encryption and authentication scheme based on chaoticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3507ndash3517 2010

[7] J Hu and F Han ldquoA pixel-based scrambling scheme for digitalmedical images protectionrdquo Journal of Network and ComputerApplications vol 32 no 4 pp 788ndash794 2009

[8] Z Shuo C Ruhua J Yingchun and G Shiping ldquoAn imageencryption algorithm based on multiple chaos and wavelettransformrdquo in Proceedings of the 2nd International CongressImage and Signal Processing (CISP rsquo09) pp 1ndash5 2009

Mathematical Problems in Engineering 13

[9] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010

[10] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009

[11] X Liao S Lai and Q Zhou ldquoA novel image encryptionalgorithm based on self-adaptive wave transmissionrdquo SignalProcessing vol 90 no 9 pp 2714ndash2722 2010

[12] S Fu-Yan L Shu-Tang and L Zong-Wang ldquoImage encryptionusing high-dimension chaotic systemrdquo Chinese Physics vol 16no 12 pp 3616ndash3623 2007

[13] V Patidar N K Pareek and K K Sud ldquoA new substitution-diffusion based image cipher using chaotic standard and logisticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 7 pp 3056ndash3075 2009

[14] Y Wu J P Noonan G Yang and H Jin ldquoImage encryptionusing the two-dimensional logistic chaotic maprdquo Journal ofElectronic Imaging vol 21 no 1 Article ID 013014 2012

[15] Y Wu J P Noonan and S Agaian ldquoA wheel-switch chaoticsystem for image encryptionrdquo in Proceedings of the InternationalConference on System Science and Engineering (ICSSE rsquo11) pp23ndash27 June 2011

[16] L Zhang X Liao andXWang ldquoAn image encryption approachbased on chaoticmapsrdquoChaos Solitons and Fractals vol 24 no3 pp 759ndash765 2005

[17] H S Kwok and W K S Tang ldquoA fast image encryption systembased on chaotic maps with finite precision representationrdquoChaos Solitons and Fractals vol 32 no 4 pp 1518ndash1529 2007

[18] J Jian Y Shi C Hu Q Ma and J Li ldquoEncryption of digitalimage based on chaos systemrdquo in Computer and ComputingTechnologies in Agriculture II vol 2 pp 1145ndash1151 2009

[19] P Van-Roy and S Haridi Concepts Techniques and Models ofComputer Programming MIT Press 2004

[20] P SchumerMathematical Journeys Wiley-Interscience 2004[21] G Ye X Huang and C Zhu ldquoImage encryption algorithm of

double scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007

[22] XDesheng andXYueshan ldquoDigital image scrambling based onjosephus traversingrdquo Computer Engineering and Applicationsno 10 pp 44ndash46 2005

[23] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2005

[24] Y-P Zhang Z-J Zhai W Liu X Nie S-P Cao and W-D Dai ldquoDigital image encryption algorithm based on chaosand improved DESrdquo in Proceedings of the IEEE InternationalConference on Systems Man and Cybernetics (SMC rsquo09) pp474ndash479 October 2009

[25] J Lang R Tao and Y Wang ldquoImage encryption based on themultiple-parameter discrete fractional Fourier transform andchaos functionrdquo Optics Communications vol 283 no 10 pp2092ndash2096 2010

[26] S Yang and S Sun ldquoVideo encryptionmethod based on chaoticmaps in DCT domainrdquo Progress in Natural Science vol 18 no10 pp 1299ndash1304 2008

[27] V Wu J P Noonan and S Agaian ldquoBinary data encryptionusing the Sudoku block cipherrdquo in Proceedings of the IEEEInternational Conference on SystemsMan andCybernetics (SMCrsquo10) pp 3915ndash3921 October 2010

[28] Y Wu Y Zhou J P Noonan K Panetta and S Agaian ldquoImageencryption using the Sudoku matrixrdquo in Proceedings of theMobileMultimediaImage Processing Security and ApplicationsApril 2010

[29] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004

[30] H Gao Y Zhang S Liang and D Li ldquoA new chaotic algorithmfor image encryptionrdquo Chaos Solitons and Fractals vol 29 no2 pp 393ndash399 2006

[31] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[32] J Thompson and H Stewart Nonlinear Dynamics and ChaosJohn Wiley amp Sons 2002

[33] C E Shannon ldquoCommunication Theory of Secrecy SystemsrdquoBell Systems Technical Journal vol 28 pp 656ndash715 1949

[34] Y Wu Y Zhou G Saveriades S Agaian J P Noonan andP Natarajan ldquoLocal Shannon entropy measure with statisticaltests for image randomnessrdquo Information Sciences vol 222 no323 342 pages 2013

[35] Y Wu S Agaian and J P Noona ldquoA novel method of testingimage randomness with applications to image shuffling andencryptionrdquo in Defense Security and Sensing Proceedings ofthe SPIE pp 875507ndash875507 2013

[36] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI random-ness tests for image encryptionrdquo Cyber Journals pp 31ndash38 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

[9] G Ye ldquoImage scrambling encryption algorithm of pixel bitbased on chaos maprdquo Pattern Recognition Letters vol 31 no 5pp 347ndash354 2010

[10] C K Huang andH H Nien ldquoMulti chaotic systems based pixelshuffle for image encryptionrdquoOptics Communications vol 282no 11 pp 2123ndash2127 2009

[11] X Liao S Lai and Q Zhou ldquoA novel image encryptionalgorithm based on self-adaptive wave transmissionrdquo SignalProcessing vol 90 no 9 pp 2714ndash2722 2010

[12] S Fu-Yan L Shu-Tang and L Zong-Wang ldquoImage encryptionusing high-dimension chaotic systemrdquo Chinese Physics vol 16no 12 pp 3616ndash3623 2007

[13] V Patidar N K Pareek and K K Sud ldquoA new substitution-diffusion based image cipher using chaotic standard and logisticmapsrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 7 pp 3056ndash3075 2009

[14] Y Wu J P Noonan G Yang and H Jin ldquoImage encryptionusing the two-dimensional logistic chaotic maprdquo Journal ofElectronic Imaging vol 21 no 1 Article ID 013014 2012

[15] Y Wu J P Noonan and S Agaian ldquoA wheel-switch chaoticsystem for image encryptionrdquo in Proceedings of the InternationalConference on System Science and Engineering (ICSSE rsquo11) pp23ndash27 June 2011

[16] L Zhang X Liao andXWang ldquoAn image encryption approachbased on chaoticmapsrdquoChaos Solitons and Fractals vol 24 no3 pp 759ndash765 2005

[17] H S Kwok and W K S Tang ldquoA fast image encryption systembased on chaotic maps with finite precision representationrdquoChaos Solitons and Fractals vol 32 no 4 pp 1518ndash1529 2007

[18] J Jian Y Shi C Hu Q Ma and J Li ldquoEncryption of digitalimage based on chaos systemrdquo in Computer and ComputingTechnologies in Agriculture II vol 2 pp 1145ndash1151 2009

[19] P Van-Roy and S Haridi Concepts Techniques and Models ofComputer Programming MIT Press 2004

[20] P SchumerMathematical Journeys Wiley-Interscience 2004[21] G Ye X Huang and C Zhu ldquoImage encryption algorithm of

double scrambling based on ASCII code of matrix elementrdquo inProceedings of the International Conference on ComputationalIntelligence and Security (CIS rsquo07) pp 843ndash847 December 2007

[22] XDesheng andXYueshan ldquoDigital image scrambling based onjosephus traversingrdquo Computer Engineering and Applicationsno 10 pp 44ndash46 2005

[23] B Furht and D KirovskiMultimedia Security Handbook CRCPress 2005

[24] Y-P Zhang Z-J Zhai W Liu X Nie S-P Cao and W-D Dai ldquoDigital image encryption algorithm based on chaosand improved DESrdquo in Proceedings of the IEEE InternationalConference on Systems Man and Cybernetics (SMC rsquo09) pp474ndash479 October 2009

[25] J Lang R Tao and Y Wang ldquoImage encryption based on themultiple-parameter discrete fractional Fourier transform andchaos functionrdquo Optics Communications vol 283 no 10 pp2092ndash2096 2010

[26] S Yang and S Sun ldquoVideo encryptionmethod based on chaoticmaps in DCT domainrdquo Progress in Natural Science vol 18 no10 pp 1299ndash1304 2008

[27] V Wu J P Noonan and S Agaian ldquoBinary data encryptionusing the Sudoku block cipherrdquo in Proceedings of the IEEEInternational Conference on SystemsMan andCybernetics (SMCrsquo10) pp 3915ndash3921 October 2010

[28] Y Wu Y Zhou J P Noonan K Panetta and S Agaian ldquoImageencryption using the Sudoku matrixrdquo in Proceedings of theMobileMultimediaImage Processing Security and ApplicationsApril 2010

[29] G Chen Y Mao and C K Chui ldquoA symmetric image encryp-tion scheme based on 3D chaotic cat mapsrdquo Chaos Solitons andFractals vol 21 no 3 pp 749ndash761 2004

[30] H Gao Y Zhang S Liang and D Li ldquoA new chaotic algorithmfor image encryptionrdquo Chaos Solitons and Fractals vol 29 no2 pp 393ndash399 2006

[31] RMMay ldquoSimplemathematicalmodels with very complicateddynamicsrdquo Nature vol 261 no 5560 pp 459ndash467 1976

[32] J Thompson and H Stewart Nonlinear Dynamics and ChaosJohn Wiley amp Sons 2002

[33] C E Shannon ldquoCommunication Theory of Secrecy SystemsrdquoBell Systems Technical Journal vol 28 pp 656ndash715 1949

[34] Y Wu Y Zhou G Saveriades S Agaian J P Noonan andP Natarajan ldquoLocal Shannon entropy measure with statisticaltests for image randomnessrdquo Information Sciences vol 222 no323 342 pages 2013

[35] Y Wu S Agaian and J P Noona ldquoA novel method of testingimage randomness with applications to image shuffling andencryptionrdquo in Defense Security and Sensing Proceedings ofthe SPIE pp 875507ndash875507 2013

[36] Y Wu J P Noonan and S Agaian ldquoNPCR and UACI random-ness tests for image encryptionrdquo Cyber Journals pp 31ndash38 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of