research article a construction of multisender authentication … · 2019. 5. 12. · research...

Research ArticleA Construction of Multisender Authentication Codes withSequential Model from Symplectic Geometry over Finite Fields

Shangdi Chen1 and Chunli Yang2

1 College of Science, Civil Aviation University of China, Tianjin 300300, China2 Information Security Center, Beijing University of Posts and Telecommunications, P.O. Box 126, Beijing 100876, China

Correspondence should be addressed to Shangdi Chen; [email protected]

Received 26 August 2013; Revised 26 December 2013; Accepted 5 January 2014; Published 30 April 2014

Academic Editor: Francesco Pellicano

Copyright © 2014 S. Chen and C. Yang.This is an open access article distributed under the Creative CommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Multisender authentication codes allow a group of senders to construct an authenticatedmessage for a receiver such that the receivercan verify authenticity of the received message. In this paper, we construct multisender authentication codes with sequential modelfrom symplectic geometry over finite fields, and the parameters and the maximum probabilities of deceptions are also calculated.

1. Introduction

Information security consists of confidentiality and authen-tication. Confidentiality is to prevent the confidential infor-mation from decrypting by adversary. The purpose ofauthentication is to ensure the sender is real and to verifythat the information is integrated. Digital signature andauthentication codes are two important means of authen-ticating the information and provide good service in thenetwork. In practical, digital signature is computationallysecure assuming that the computing power of adversaryis limited and a mathematical problem is intractable andcomplex. However, authentication codes are generally safe(unconditional secure) and relatively simple. In the 1940s, C.E. Shannon first put forward the concept of perfect secrecyauthentication system using the information theory. In the1980s, information theory method had been applied to theproblem of authentication by G. J. Simmons; then authenti-cation codes became the foundation for constructing uncon-ditionally secure authentication system. In 1974, Gilbertet al. constructed the first authentication code [1], which isa landmark in the development of authentication theory.During the same period, Simmons independently studiedthe authentication theory and established three participantsand four participants certification models [2]. The famousmathematician Wan Zhexian constructed an authenticationcode without arbitration from the subspace of the classical

geometry [3]. In the case of transmitter and receiver beingnot honest, Ma et al. constructed a series of authentica-tion codes with arbitration [4–9]. Xing et al. constructedauthentication codes using algebraic curve and nonlinearfunctions, respectively [10, 11]. Safavi-Naini and Wang gavesome results onmultireceiver authentication codes [12]. Chenet al. made great contributions onmultisender authenticationcodes from polynomials and matrices [13–19].

With the rapid development of information science,the traditional one-to-one authentication codes have beenunable to meet the requirements of network communication,thus making the study of multiuser authentication codesparticularly important. Multiuser authentication code is ageneralization of traditional two-user authentication code.It can be divided into two cases: one is a sender andmany receivers authentication codes; the other one is manysenders and a receiver authentication codes. We call theformer as multireceiver authentication codes and the latteras multisender authentication codes. Safavi-Naini R gavesome results on multireceiver authentication codes usingthe subspace of the classical geometry, while there are onlysome multisender authentication codes using polynomialsand matrices to construct. We present the first constructionmultisender authentication code using the subspace of theclassical geometry, specifically symplectic geometry.

The main contribution of our paper is constructing amulti-sender authentication code using symplectic geometry.

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014, Article ID 102301, 7 pageshttp://dx.doi.org/10.1155/2014/102301

CORE Metadata, citation and similar papers at core.ac.uk

Provided by MUCC (Crossref)

https://core.ac.uk/display/192433756?utm_source=pdf&utm_medium=banner&utm_campaign=pdf-decoration-v1

2 Journal of Applied Mathematics

Furthermore, we calculate the corresponding parameters andthe maximum probabilities of deceptions.

The paper is organised as follows. Section 2 gives themodels of multisender authentication codes. In Section 3, weprovide the calculation formulas on probability of successin attacks by malicious groups of senders. In Section 4, wegive some definitions and properties on geometry of sym-plectic groups over finite fields. In Section 5, a constructionof multisender authentication codes with sequential modelfrom symplectic geometry over finite fields is given; thenthe parameters and the maximum probabilities of deceptionsare also calculated. We give a comparison with the otherconstruction of multisender authentication [19] in Section 6.

2. Models of Multisender AuthenticationCodes

We review the concepts of authentication codes which can beextracted from [20].

Definition 1 (see [20]). A systematic Cartesian authenticationcode 𝐶 is a 4-tuple (𝑆, 𝐸, 𝑇; 𝑓), where 𝑆 is the set of sourcestates, 𝐸 is the set of keys, 𝑇 is the set of authenticators, and𝑓 : 𝑆 × 𝐸 → 𝑇 is the authentication mapping. The messagespace𝑀 = 𝑆 × 𝑇 is the set of all possible messages.

In the actual computer network communications, mul-tisender authentication codes include sequential models andsimultaneousmodels. Sequential models are that each senderuses his own encoding rules to encode a source state orderly,and the last sender sends the encodedmessage to the receiver;then the receiver receives the message and verifies whetherthe message is legal or not. Simultaneous models are that allsenders use their own encoding rules to encode a source statesimultaneously; then the synthesizer forms an authenticatedmessage and sends it to the receiver; the receiver receives themessage and verifies whether the message is legal or not.

In the following we will give out the working principlesof two modes of multisender authentication codes and theprotocols that the participants should follow.

Definition 2 (see [17]). In sequential model, there are threeparticipants: a group of senders 𝑈 = {𝑈

1, 𝑈2, . . . , 𝑈

𝑛}; a

Key Distribution Center (KDC), for the distribution keys tosenders and receiver; a receiver who receives the authen-ticated message and verifies the message true or not. Thecode works as follows: each sender and receiver has theirown Cartesian authentication code, respectively. It is used togenerate part of the message and verify authenticity of thereceived message. Sender’s authentication codes are calledbranch authentication codes, and receiver’s authenticationcode is called channel authentication code. Let (𝑆

𝑖, 𝐸𝑖, 𝑇𝑖; 𝑓𝑖),

𝑖 = 1, 2, . . . , 𝑛, be the 𝑖th sender’s Cartesian authenticationcodes, and let 𝑇

𝑖−1⊂ 𝑆𝑖, 1 ≤ 𝑖 ≤ 𝑛, (𝑆, 𝐸, 𝑇; 𝑓) be the receiver’s

Cartesian authentication code, and let 𝑆 = 𝑆1, 𝑇 = 𝑇

𝑖, 𝜋𝑖:

𝐸 → 𝐸𝑖be a subkey generation algorithm. For authenticating

a message, the senders and the receiver should comply withprotocols:

(1) KDC randomly selects an 𝑒 ∈ 𝐸 and secretly sends itto the receiver 𝑅 and sends 𝑒

𝑖= 𝜋𝑖(𝑒) to the 𝑖th sender

𝑈𝑖, 𝑖 = 1, 2, . . . , 𝑛;

(2) if the senders would like to send a source state 𝑠 tothe receiver 𝑅, 𝑈

1calculates 𝑡

1= 𝑓1(𝑠, 𝑒1) and then

sends 𝑡1to 𝑈2through an open channel; 𝑈

2receives

𝑡1and calculates 𝑡

2= 𝑓2(𝑡1, 𝑒2) and then sends 𝑡

2to

𝑈3through an open channel. In general, 𝑈

𝑖receives

𝑡𝑖−1

and calculates 𝑡𝑖= 𝑓𝑖(𝑡𝑖−1, 𝑒𝑖) and then sends 𝑡

𝑖to

𝑈𝑖+1

through an open channel, 1 < 𝑖 < 𝑛. 𝑈𝑛receives

𝑡𝑛−1

and calculates 𝑡𝑛= 𝑓𝑛(𝑡𝑛−1, 𝑒𝑛) and then sends

𝑚 = (𝑠, 𝑡𝑛) through an open channel to the receiver 𝑅;

(3) when the receiver receives the message 𝑚 = (𝑠, 𝑡𝑛),

he checks the authenticity by verifying whether 𝑡𝑛=

𝑓(𝑠, 𝑒) or not. If the equality holds, the message isregarded as authentic and is accepted. Otherwise, themessage is rejected.

Definition 3 (see [17]). In simultaneous model of a multi-sender authentication code, there are four participants: agroup of senders 𝑈 = {𝑈

1, 𝑈2, . . . , 𝑈

𝑛}; a Key Distribution

Center (KDC), for the distribution keys to senders andreceiver; a synthesizer 𝐶 who only runs the trusted synthesisalgorithm; a receiver who receives the authenticated messageand verifies the message true or not. The code works asfollows: each sender and receiver has their own Cartesianauthentication code, respectively. It is used to generate partof the message and verify the received message. Sender’sauthentication codes are called branch authentication codes,and receiver’s authentication code is called channel authen-tication code. Let (𝑆

𝑖, 𝐸𝑖, 𝑇𝑖; 𝑓𝑖), 𝑖 = 1, 2, . . . , 𝑛, be the

sender’s Cartesian authentication codes, let (𝑆, 𝐸, 𝑇; 𝑓) be thereceiver’s Cartesian authentication code, let 𝑔 : 𝑇

1×𝑇2× ⋅ ⋅ ⋅ ×

𝑇𝑛→ 𝑇 be the synthesis algorithm, and let 𝜋

𝑖: 𝐸 → 𝐸

𝑖be

a subkey generation algorithm. For authenticating a message,the senders and the receiver should comply with protocols:

(1) KDC randomly selects a encoding rule 𝑒 ∈ 𝐸 andsecretly sends it to the receiver 𝑅 and sends 𝑒

𝑖= 𝜋𝑖(𝑒)

to the 𝑖th sender 𝑈𝑖, 𝑖 = 1, 2, . . . , 𝑛;

(2) if the senders would like to send a source state 𝑠to the receiver 𝑅, 𝑈

𝑖computes 𝑡

𝑖= 𝑓𝑖(𝑠, 𝑒𝑖), 𝑖 =

1, 2, . . . , 𝑛, and sends 𝑚𝑖= (𝑠, 𝑡

𝑖) (𝑖 = 1, 2, . . . , 𝑛) to

the synthesizer 𝐶 through an open channel;

(3) the synthesizer 𝐶 receives the messages 𝑚𝑖= (𝑠, 𝑡

𝑖),

𝑖 = 1, 2, . . . , 𝑛, and calculates 𝑡 = 𝑔(𝑡1, 𝑡2, . . . , 𝑡

𝑛) using

the synthesis algorithm 𝑔; then sends message 𝑚 =(𝑠, 𝑡) to the receiver 𝑅;

(4) when the receiver receives the message 𝑚 = (𝑠, 𝑡),he checks the authenticity by verifying whether 𝑡 =𝑓(𝑠, 𝑒) or not. If the equality holds, the message isregarded as authentic and is accepted. Otherwise, themessage is rejected.

Journal of Applied Mathematics 3

3. Probabilities of Deceptions

We assume that the arbitrator (KDC) and the synthesizer (C)are credible; though they know the senders’ and receiver’sencoding rules, they donot participate in any communicationactivities. When transmitter and receiver are disputing, thearbitrator settles it. At the same time, assume that the systemfollows Kerckhoff ’s principle which the other informationof the whole system is public except the actual used keys.Assume that the source state space 𝑆 and the receiver’s decod-ing rules space 𝐸

𝑅are according to a uniform probability

distribution; then the probability distribution of messagespace 𝑀 and tag space 𝑇 is determined by the probabilitydistribution of 𝑆 and 𝐸

𝑅. In a multisender authentication

system, assume that the whole senders cooperate to form avalid message; that is, all senders as a whole and receiver arereliable. But there are some malicious senders which theytogether cheat the receiver; the part of senders and receiverare not credible; they can take impersonation attack andsubstitution attack.

Assume that 𝑈1, 𝑈2, . . . , 𝑈

𝑛are senders, 𝑅 is a receiver,

and 𝐸𝑖is the encoding rules of 𝑈

𝑖, 1 ≤ 𝑖 ≤ 𝑛. 𝐸

𝑅is the

decoding rules of receiver 𝑅. 𝐿 = {𝑖1, 𝑖2, . . . , 𝑖

𝑙} ⊂ {1, 2, . . . , 𝑛},

𝑙 < 𝑛, 𝑈𝐿= {𝑈𝑖1

, 𝑈𝑖2

, . . . , 𝑈𝑖𝑙

}, 𝐸𝐿= {𝐸𝑖1

, 𝐸𝑖2

, . . . , 𝐸𝑖𝑙

}.

Impersonation Attack. 𝑈𝐿, after receiving their secret keys,

sends a message 𝑚 to receiver. 𝑈𝐿

is successful if thereceiver accepts it as legitimate message. Denote 𝑃

𝐼[𝐿] as the

maximum probability of success of the impersonation attack.It can be expressed as

𝑃𝐼 [𝐿] = max

𝑒𝐿∈𝐸𝐿

max𝑚∈𝑀

𝑃 (𝑚 is accepted by 𝑅 | 𝑒𝐿) . (1)

Substitution Attack. 𝑈𝐿, after observing a legitimate message,

substitutes it with another message 𝑚. 𝑈𝐿is successful if

𝑚 is accepted by receiver as authentic. Denote 𝑃

𝑆[𝐿] as the

maximum probability of success of the substitution attack. Itcan be expressed as

𝑃𝑆 [𝐿] = max

𝑒𝐿∈𝐸𝐿

max𝑚∈𝑀

max𝑚̸= 𝑚∈𝑀

𝑃 (𝑚is accepted by 𝑅 | 𝑚, 𝑒

𝐿) .

(2)

4. Symplectic Geometry

In this section, we give some definitions and properties ongeometry of symplectic groups over finite fields, which canbe extracted from [20].

Let F𝑞be a finite field with 𝑞 elements, 𝑛 = 2] and define

the 2] × 2] alternate matrix

𝐾 = (0 𝐼(])

−𝐼(])

0) . (3)

The symplectic group of degree 2] over F𝑞, denoted by

𝑆𝑝2](F𝑞), is defined to be the set of matrices

𝑆𝑝2] (F𝑞) = {𝑇 | 𝑇𝐾

𝑡

𝑇 = 𝐾} , (4)

with matrix multiplication as its group operation. Let F (2])𝑞

bethe 2]-dimensional row vector space over F

𝑞. 𝑆𝑝2](F𝑞) has an

action on F (2])𝑞

defined as follows:

F(2])𝑞

× 𝑆𝑝2] (F𝑞) → F

(2])𝑞,

((𝑥1, 𝑥2, . . . , 𝑥

2]) , 𝑇) → (𝑥1, 𝑥2, . . . , 𝑥2]) 𝑇.

(5)

The vector space F (2])𝑞

together with this action of 𝑆𝑝2](F𝑞) is

called the symplectic space over F𝑞.

Let 𝑃 be an 𝑚-dimensional subspace of F (2])𝑞

. We use thesame latter 𝑃 to denote a matrix representation of 𝑃; that is, 𝑃is an𝑚×2]matrix of rank𝑚 such that its rows form a basis of𝑃. The 𝑃𝐾 𝑡𝑃 is alternate. Assume that it is of rank 2𝑠; then 𝑃is called a subspace of type (𝑚, 𝑠). It is known that subspacesof type (𝑚, 𝑠) exist in F (2])

𝑞if and only if

2𝑠 ≤ 𝑚 ≤ ] − 𝑠. (6)

It is also known that subspaces of the same type form anorbit under 𝑆𝑝

2](F𝑞). Denote by 𝑁(𝑚, 𝑠; 2]) the number ofsubspaces of type (𝑚, 𝑠) in F (2])

𝑞.

Denote by 𝑃⊥ the set of vectors which are orthogonal toevery vector of 𝑃; that is,

𝑃⊥= {𝑦 ∈ F

(2])𝑞

| 𝑦𝐾𝑡𝑥 = 0 for all 𝑥 ∈ 𝑃} . (7)

Obviously, 𝑃⊥ is a (2] − 𝑚)-dimensional subspace of F (2])𝑞

.Readers can refer to [15] for notations and terminology,

which are not explained, on symplectic geometry of classicalgroups over finite fields.

5. Construction

Let F𝑞be a finite field with 𝑞 elements. Assume that

1 < 𝑛 < 𝑟 < ]. 𝑈 = ⟨𝑒1, 𝑒2, . . . , 𝑒

𝑛⟩; then 𝑈⊥ =

⟨𝑒1, . . . , 𝑒], 𝑒]+𝑛+1, . . . , 𝑒2]⟩. Let𝑊𝑖 = ⟨𝑒1, . . . , 𝑒𝑖−1, 𝑒𝑖+1, . . . , 𝑒𝑛⟩;

then𝑊𝑖

⊥= ⟨𝑒1, . . . , 𝑒], 𝑒]+𝑖, 𝑒]+𝑛+1, . . . , 𝑒2]⟩. The set of source

states 𝑆 = {𝑠 | 𝑠 is a subspace of type (2𝑟 − 𝑛, 𝑟 − 𝑛) and𝑈 ⊂ 𝑠 ⊂ 𝑈

⊥}; the set of 𝑖th sender’s encoding rules𝐸

𝑖= {𝑒𝑖| 𝑒𝑖

is a subspace of type (𝑛 + 1, 1),𝑈 ⊂ 𝑒𝑖and 𝑒𝑖⊥ 𝑊𝑖}, 1 ≤ 𝑖 ≤ 𝑛;

the set of receiver’s decoding rules 𝐸𝑅= {𝑒𝑅| 𝑒𝑅is a subspace

of type (2𝑛, 𝑛) and 𝑈 ⊂ 𝑒𝑅}; the set of tags 𝑇

𝑖= {𝑡𝑖| 𝑡𝑖is a

subspace of type (2𝑟 − 𝑛 + 𝑖, 𝑟 − 𝑛 + 𝑖) and 𝑈 ⊂ 𝑡𝑖}, 1 ≤ 𝑖 ≤ 𝑛.

Define the encoding maps:

𝑓1: 𝑆 × 𝐸

1→ 𝑇1, 𝑓

1(𝑠, 𝑒1) = 𝑠 + 𝑒

1,

𝑓𝑖: 𝑇𝑖−1× 𝐸𝑖→ 𝑇𝑖, 𝑓𝑖(𝑡𝑖−1, 𝑒𝑖) = 𝑡𝑖−1+ 𝑒𝑖, 2 ≤ 𝑖 ≤ 𝑛.

(8)

Define the decoding map:

𝑓 : 𝑆 × 𝐸𝑅→ 𝑇𝑛, 𝑓 (𝑠, 𝑒

𝑅) = 𝑠 + 𝑒

𝑅. (9)


This code works as follows.

(1) Key Distribution. First, the KDC does a list 𝐿 ofsenders; assume that 𝐿 = {1, 2, . . . , 𝑛}. Then, the KDCrandomly chooses a subspace 𝑒

𝑅∈ 𝐸𝑅and privately

sends 𝑒𝑅to the receiver 𝑅. Last, the KDC randomly

chooses a subspace 𝑒𝑖∈ 𝐸𝑖and 𝑒𝑖⊂ 𝑒𝑅, then privately

sends 𝑒𝑖to the 𝑖th sender, 1 ≤ 𝑖 ≤ 𝑛.

(2) Broadcast. For a source state 𝑠 ∈ 𝑆, the sender 𝑈1

calculates 𝑡1= 𝑠 + 𝑒

1and sends (𝑠, 𝑡

1) to 𝑈

2. The

sender 𝑈2calculates 𝑡

2= 𝑡1+ 𝑒2and sends (𝑠, 𝑡

2) to

𝑈3. Finally, the sender𝑈

𝑛calculates 𝑡

𝑛= 𝑡𝑛−1+ 𝑒𝑛and

sends𝑚 = (𝑠, 𝑡𝑛) to the receiver 𝑅.

(3) Verification. Since the receiver 𝑅 holds the decodingrule 𝑒

𝑅, 𝑅 accepts 𝑚 as authentic if 𝑡

𝑛= 𝑠 + 𝑒

𝑅.

Otherwise, it is rejected by 𝑅.

Lemma 4. Let 𝐶 = (𝑆, 𝐸𝑅, 𝑇𝑛; 𝑓), 𝐶

1= (𝑆, 𝐸

1, 𝑇1; 𝑓1), 𝐶𝑖=

(𝑇𝑖−1, 𝐸𝑖, 𝑇𝑖; 𝑓𝑖) (2 ≤ 𝑖 ≤ 𝑛); then 𝐶, 𝐶

1, 𝐶𝑖are all Cartesian

authentication codes.

Proof. First, we show that 𝐶 is a Cartesian authenticationcode.

(1) For 𝑠 ∈ 𝑆, 𝑒𝑅∈ 𝐸𝑅. Let

𝑠 = (𝑈

𝑄)

𝑛

2 (𝑟 − 𝑛) ,

𝑒𝑅= (

𝑈

𝑉)𝑛

𝑛.

(10)

From the definition of 𝑠 and 𝑒𝑅, we can assume that

(𝑈

𝑄)𝐾

𝑡

(𝑈

𝑄) = (

0(𝑛)

0 0

0 0 𝐼(𝑟−𝑛)

0 −𝐼(𝑟−𝑛)

0

) ,

(𝑈

𝑉)𝐾

𝑡

(𝑈

𝑉) = (

0 𝐼(𝑛)

−𝐼(𝑛)

0) .

(11)

Obviously, we have V ∉ 𝑠 for any V ∈ 𝑉 and V ̸= 0. Therefore,

𝑡𝑛= 𝑠 + 𝑒

𝑅= (

𝑈

𝑉

𝑄

) ,

(

𝑈

𝑉

𝑄

)𝐾

𝑡

(

𝑈

𝑉

𝑄

) =(

0 𝐼(𝑛)

0 0

−𝐼(𝑛)

0 ∗ ∗

0 ∗ 0 𝐼(𝑟−𝑛)

0 ∗ −𝐼(𝑟−𝑛)

0

) .

(12)

From above, 𝑡𝑛is a subspace of type (2𝑟, 𝑟) and 𝑈 ⊂ 𝑡

𝑛; that

is, 𝑡𝑛∈ 𝑇𝑛.

(2) For 𝑡𝑛∈ 𝑇𝑛, 𝑡𝑛is a subspace of type (2𝑟, 𝑟) containing

𝑈. So there is subspace 𝑉 ⊂ 𝑡𝑛, satisfying

(𝑈

𝑉)𝐾

𝑡

(𝑈

𝑉) = (

0 𝐼(𝑛)

−𝐼(𝑛)

0) . (13)

Then, we can assume that 𝑡𝑛= (𝑈

𝑉

𝑄), satisfying

(

𝑈

𝑉

𝑄

)𝐾

𝑡

(

𝑈

𝑉

𝑄

) =(

0 𝐼(𝑛)

0 0

−𝐼(𝑛)

0 0 0

0 0 0 𝐼(𝑟−𝑛)

0 0 −𝐼(𝑟−𝑛)

0

) . (14)

Let 𝑠 = ( 𝑈𝑄); then 𝑠 is a subspace of type (2𝑟 − 𝑛, 𝑟 − 𝑛) and

𝑈 ⊂ 𝑠 ⊂ 𝑈⊥; that is, 𝑠 ∈ 𝑆 is a source state. For any V ∈ 𝑉 and

V ̸= 0, we have V ∉ 𝑠 and 𝑉 ∩ 𝑈⊥ = {0}. Therefore, 𝑡𝑛∩ 𝑈⊥=

(𝑈

𝑄) = 𝑠. Let 𝑒

𝑅= ( 𝑈𝑉); then 𝑒

𝑅is a transmitter’s encoding

rule satisfying 𝑡𝑛= 𝑠 + 𝑒

𝑅.

If 𝑠 is another source state contained in 𝑡𝑛, then𝑈 ⊂ 𝑠 ⊂

𝑈⊥. Therefore, 𝑠 ⊂ 𝑡

𝑛∩ 𝑈⊥= 𝑠, while dim 𝑠 = dim 𝑠, so

𝑠= 𝑠. That is, 𝑠 is the uniquely source state contained in 𝑡

𝑛.

Similarly, we can show that 𝐶1and 𝐶

𝑖(2 ≤ 𝑖 ≤ 𝑛) are also

Cartesian authentication code.

From Lemma 4, we know that such construction ofmultisender authentication codes is reasonable. Next wecompute the parameters of this code.

Lemma 5. The number of the source states is |𝑆| = 𝑁(2(𝑟 −𝑛), 𝑟 − 𝑛; 2(] − 𝑛)).

Proof. For any 𝑠 ∈ 𝑆, since 𝑈 ⊂ 𝑠 ⊂ 𝑈⊥, 𝑠 has the form

𝑠 = (𝐼(𝑛)

0 0 0

0 𝑃2

0 𝑃4

)𝑛

2 (𝑟 − 𝑛)

𝑛 ] − 𝑛 𝑛 ] − 𝑛,(15)

where (𝑃2, 𝑃4) is a subspace of type (2(𝑟 − 𝑛), 𝑟 − 𝑛) in the

symplectic space 𝐹2(]−𝑛)𝑞

. Therefore, the number of the sourcestates is |𝑆| = 𝑁(2(𝑟 − 𝑛), 𝑟 − 𝑛; 2(] − 𝑛)).

Lemma 6. The number of the 𝑖th sender’s encoding rules is|𝐸𝑖| = 𝑞2(]−𝑛).

Proof. For any 𝑒𝑖∈ 𝐸𝑖, 𝑒𝑖is a subspace of type (𝑛 +

1, 1) containing 𝑈 and 𝑒𝑖is orthogonal to 𝑊

𝑖. So we can

assume that 𝑒𝑖=𝑡(𝑒1, . . . , 𝑒

𝑛, 𝑢), where𝑢 = (𝑥

1𝑥2⋅ ⋅ ⋅ 𝑥

2]).Obviously, 𝑥

1= ⋅ ⋅ ⋅ = 𝑥

𝑛= 𝑥]+1 = ⋅ ⋅ ⋅ = 𝑥]+𝑖−1 = 𝑥]+𝑖+1 =

⋅ ⋅ ⋅ = 𝑥]+𝑛 = 0, 𝑥]+𝑖 = 1, and 𝑥𝑛+1, . . . , 𝑥], 𝑥]+𝑛+1, . . . , 𝑥2]arbitrarily. Therefore, |𝐸

𝑖| = 𝑞2(]−𝑛).

Lemma7. Thenumber of the receiver’s decoding rules is |𝐸𝑅| =

𝑞2𝑛(]−𝑛).

Proof. For any 𝑒𝑅∈ 𝐸𝑅, since 𝑒

𝑅is a subspace of type (2𝑛, 𝑛)

containing 𝑈, 𝑒𝑅has the form

𝑒𝑅= (

𝐼(𝑛)

0 0 0

0 𝑄2𝐼(𝑛)

𝑄4

)𝑛

𝑛

𝑛 ] − 𝑛 𝑛 ] − 𝑛,(16)

where 𝑄2, 𝑄4are arbitrary matrices. Therefore, |𝐸

𝑅| =

𝑞2𝑛(]−𝑛).


Lemma 8. (1)The number of decoding rules 𝑒𝑅contained in 𝑡

𝑛

is 𝑞2𝑛(𝑟−𝑛);(2) the number of the tags is |𝑇

𝑛| = 𝑞2𝑛(]−𝑟)

𝑁(2(𝑟 − 𝑛), 𝑟 −

𝑛; 2(] − 𝑛)).

Proof. (1) For any 𝑡𝑛∈ 𝑇𝑛, 𝑡𝑛is a subspace of type (2𝑟, 𝑟) and

𝑈 ⊂ 𝑡𝑛. We assume that 𝑡

𝑛has the form

𝑡𝑛

= (

𝐼(𝑛)

0 0 0 0 0

0 𝐼(𝑟−𝑛)

0 0 0 0

0 0 0 𝐼(𝑛)

0 0

0 0 0 0 𝐼(𝑟−𝑛)

0

)

𝑛

𝑟 − 𝑛

𝑛

𝑟 − 𝑛

𝑛 𝑟 − 𝑛 ] − 𝑟 𝑛 𝑟 − 𝑛 ] − 𝑟.(17)

If 𝑒𝑅⊂ 𝑡𝑛, then we can assume that

𝑒𝑅= (

𝐼(𝑛)

0 0 0 0 0

0 𝑅2

0 𝐼(𝑛)

𝑅5

0)𝑛

𝑛

𝑛 𝑟 − 𝑛 ] − 𝑟 𝑛 𝑟 − 𝑛 ] − 𝑟,(18)

where 𝑅2, 𝑅5are arbitrary matrices.Therefore, the number of

𝑒𝑅contained in 𝑡

𝑛is 𝑞2𝑛(𝑟−𝑛).

(2)We know that a tag contains only one source state andthe number of decoding rules 𝑒

𝑅contained in 𝑡

𝑛is 𝑞2𝑛(𝑟−𝑛).

Therefore, we have |𝑇𝑛| = |𝑆||𝐸

𝑅|/𝑞2𝑛(𝑟−𝑛)

= 𝑞2𝑛(]−𝑟)

𝑁(2(𝑟 −

𝑛), 𝑟 − 𝑛; 2(] − 𝑛)).

Theorem 9. The parameters of the above constructed multi-sender authentication code are

|𝑆| = 𝑁 (2 (𝑟 − 𝑛) , 𝑟 − 𝑛; 2 (] − 𝑛)) ;

𝐸𝑖 = 𝑞2(]−𝑛)

;

𝐸𝑅 = 𝑞2𝑛(]−𝑛)

;

𝑇𝑛 = 𝑞2𝑛(]−𝑟)

𝑁(2 (𝑟 − 𝑛) , 𝑟 − 𝑛; 2 (] − 𝑛)) .

(19)

Without loss of generality, we can assume that 𝑈𝐿=

{𝑈1, 𝑈2, . . . , 𝑈

𝑙}, 𝐸𝐿= {𝐸1× ⋅ ⋅ ⋅ × 𝐸

𝑙}, where 𝑙 < 𝑛.

Lemma 10. For any 𝑒𝐿= (𝑒1, 𝑒2, . . . , 𝑒

𝑙) ∈ 𝐸𝐿, the number of

𝑒𝑅containing 𝑒

𝐿is 𝑞2(𝑛−𝑙)(]−𝑛).

Proof. For any 𝑒𝐿= (𝑒1, 𝑒2, . . . , 𝑒

𝑙) ∈ 𝐸𝐿, we can assume that

𝑒𝐿= (

𝐼(𝑙)

0 0 0 0 0

0 𝐼(𝑛−𝑙)

0 0 0 0

0 0 𝑃3

𝐼(𝑙)0 𝑃

6

)

𝑙

𝑛 − 𝑙

𝑙

𝑙 𝑛 − 𝑙 ] − 𝑛 𝑙 𝑛 − 𝑙 ] − 𝑛.

(20)

If 𝑒𝐿⊂ 𝑒𝑅, then 𝑒

𝑅has the form

𝑒𝑅= (

𝐼(𝑙)

0 0 0 0 0

0 𝐼(𝑛−𝑙)

0 0 0 0

0 0 𝑃3

𝐼(𝑙)

0 𝑃6

0 0 𝑃

30 𝐼(𝑛−𝑙)

𝑃

6

)

𝑙

𝑛 − 𝑙

𝑙

𝑛 − 𝑙

𝑙 𝑛 − 𝑙 ] − 𝑛 𝑙 𝑛 − 𝑙 ] − 𝑛,

(21)

where 𝑃3, 𝑃6are arbitrary matrices. Therefore, the number of

𝑒𝑅containing 𝑒

𝐿is 𝑞2(𝑛−𝑙)(]−𝑛).

Lemma 11. For any 𝑡𝑛∈ 𝑇𝑛and 𝑒𝐿= (𝑒1, 𝑒2, . . . , 𝑒

𝑙) ∈ 𝐸𝐿, the

number of 𝑒𝑅contained in 𝑡

𝑛and containing 𝑒

𝐿is 𝑞2(𝑛−𝑙)(𝑟−𝑛).

Proof. For any 𝑡𝑛∈ 𝑇𝑛, 𝑡𝑛is a subspace of type (2𝑟, 𝑟) and

𝑈 ⊂ 𝑡𝑛. We assume that 𝑡

𝑛has the form

𝑡𝑛= (

𝐼(𝑛)

0 0 0 0 0

0 𝐼(𝑟−𝑛)

0 0 0 0

0 0 0 𝐼(𝑛)

0 0

0 0 0 0 𝐼(𝑟−𝑛)

0

)

𝑛

𝑟 − 𝑛

𝑛

𝑟 − 𝑛

𝑛 𝑟 − 𝑛 ] − 𝑟 𝑛 𝑟 − 𝑛 ] − 𝑟.

(22)

If 𝑒𝐿⊂ 𝑡𝑛, assume that 𝑒

𝐿has the form

𝑒𝐿= (

𝐼(𝑙)

0 0 0 0 0 0 0

0 𝐼(𝑛−𝑙)

0 0 0 0 0 0

0 0 𝑅3

0 𝐼(𝑙)

0 𝑅7

0

)

𝑙

𝑛 − 𝑙

𝑙

𝑙 𝑛 − 𝑙 𝑟 − 𝑛 V − 𝑟 𝑙 𝑛 − 𝑙 𝑟 − 𝑛 ] − 𝑟.

(23)

If 𝑒𝑅⊂ 𝑡𝑛and 𝑒𝐿⊂ 𝑒𝑅, then

𝑒𝑅= (

𝐼(𝑙)

0 0 0 0 0 0 0

0 𝐼(𝑛−𝑙)

0 0 0 0 0 0

0 0 𝑅3

0 𝐼(𝑙)

0 𝑅7

0

0 0 𝑅

30 0 𝐼

(𝑛−𝑙)𝑅

70

)

𝑙

𝑛 − 𝑙

𝑙

𝑛 − 𝑙

𝑙 𝑛 − 𝑙 𝑟 − 𝑛 V − 𝑟 𝑙 𝑛 − 𝑙 𝑟 − 𝑛 ] − 𝑟,

(24)


where 𝑅3, 𝑅7are arbitrary matrices.Therefore, the number of

𝑒𝑅contained in 𝑡

𝑛and containing 𝑒

𝐿is 𝑞2(𝑛−𝑙)(𝑟−𝑛).

Lemma 12. Assume that 𝑡𝑛∈ 𝑇𝑛and 𝑡𝑛∈ 𝑇𝑛are two distinct

tags which are decoded by receiver’s decoding rule 𝑒𝑅. 𝑠1and 𝑠2

contained in 𝑡𝑛and 𝑡𝑛, respectively. Let 𝑠

0= 𝑠1∩𝑠2, dim 𝑠

0= 𝑘;

then 𝑛 ≤ 𝑘 ≤ 2𝑟 − 𝑛 − 1; the number of 𝑒𝑅contained in 𝑡

𝑛∩ 𝑡

𝑛

and containing 𝑒𝐿is 𝑞(𝑛−𝑙)(𝑘−𝑛).

Proof. Since 𝑡𝑛= 𝑠1+ 𝑒𝑅, 𝑡𝑛= 𝑠2+ 𝑒𝑅and 𝑡𝑛̸= 𝑡

𝑛, then 𝑠

1̸= 𝑠2.

And for any 𝑠 ∈ 𝑆, 𝑈 ⊂ 𝑠, therefore, 𝑛 ≤ 𝑘 ≤ 2𝑟 − 𝑛 − 1.Assume that 𝑠

𝑖is the complementary subspace of 𝑠

0in the 𝑠

𝑖;

then 𝑠𝑖= 𝑠0+𝑠

𝑖(𝑖 = 1, 2). Because of 𝑡

𝑛= 𝑠1+𝑒𝑅= 𝑠0+𝑠

1+𝑒𝑅,

𝑡

𝑛= 𝑠2+𝑒𝑅= 𝑠0+𝑠

2+𝑒𝑅and 𝑠1= 𝑡𝑛∩𝑈⊥, 𝑠2= 𝑡

𝑛∩𝑈⊥, we know

𝑠0= (𝑡𝑛∩𝑈⊥)∩(𝑡

𝑛∩𝑈⊥) = 𝑡𝑛∩𝑡

𝑛∩𝑈⊥= 𝑠1∩𝑡

𝑛= 𝑠2∩𝑡𝑛, and

𝑡𝑛∩𝑡

𝑛= (𝑠1+𝑒𝑅)∩𝑡

𝑛= (𝑠0+𝑠

1+𝑒𝑅)∩𝑡

𝑛= ((𝑠0+𝑒𝑅)+𝑠

1)∩𝑡

𝑛.

Since 𝑠0+ 𝑒𝑅⊆ 𝑡

𝑛, then 𝑡

𝑛∩ 𝑡

𝑛= (𝑠0+ 𝑒𝑅) + (𝑠

1∩ 𝑡

𝑛), while

𝑠

1∩ 𝑡

𝑛⊆ 𝑠1∩ 𝑡

𝑛= 𝑠0, so 𝑡𝑛∩ 𝑡

𝑛= 𝑠0+ 𝑒𝑅.

From the definition of the 𝑡𝑛and 𝑡𝑛, we assume that

𝑡𝑛= (

𝐼(𝑛)

0 0 0

0 𝑃22

0 0

0 0 𝐼(𝑛)

0

0 0 0 𝑃44

)

𝑛

𝑟 − 𝑛

𝑛

𝑟 − 𝑛

𝑛 ] − 𝑛 𝑛 ] − 𝑛,

𝑡

𝑛= (

𝐼(𝑛)

0 0 0

0 𝑃

220 0

0 0 𝐼(𝑛)

0

0 0 0 𝑃

44

)

𝑛

𝑟 − 𝑛

𝑛

𝑟 − 𝑛

𝑛 ] − 𝑛 𝑛 ] − 𝑛.

(25)

Let

𝑡𝑛∩ 𝑡

𝑛= (

𝐼(𝑛)

0 0 0

0 𝑃2

0 0

0 0 𝐼(𝑛)

0

0 0 0 𝑃4

)

𝑛

𝑟 − 𝑛

𝑛

𝑟 − 𝑛

𝑛 ] − 𝑛 𝑛 ] − 𝑛.

(26)

And from above we know that 𝑡𝑛∩𝑡

𝑛= 𝑠0+𝑒𝑅; then dim (𝑡

𝑛∩

𝑡

𝑛) = 𝑘 + 𝑛; therefore,

dim(0 𝑃2 0 00 0 0 𝑃

4

) = 𝑘 − 𝑛. (27)

For any 𝑒𝐿⊂ 𝑡𝑛∩ 𝑡

𝑛, we assume that

𝑒𝐿= (

𝐼(𝑙)

0 0 0 0 0

0 𝐼(𝑛−𝑙)

0 0 0 0

0 0 𝑅3

𝐼(𝑙)0 𝑅

6

)

𝑙

𝑛 − 𝑙

𝑙

𝑙 𝑛 − 𝑙 ] − 𝑛 𝑙 𝑛 − 𝑙 ] − 𝑛.

(28)

If 𝑒𝑅⊂ 𝑡𝑛∩ 𝑡

𝑛and 𝑒𝐿⊂ 𝑒𝑅, then 𝑒

𝑅has the form

𝑒𝑅= (

𝐼(𝑙)

0 0 0 0 0

0 𝐼(𝑛−𝑙)

0 0 0 0

0 0 𝑅3

𝐼(𝑙)

0 𝑅6

0 0 𝑅

30 𝐼(𝑛−𝑙)

𝑅

6

)

𝑙

𝑛 − 𝑙

𝑙

𝑛 − 𝑙

𝑙 𝑛 − 𝑙 ] − 𝑛 𝑙 𝑛 − 𝑙 ] − 𝑛.

(29)

So, every row of (0 𝑅30 𝑅

6) is the linear combination of

(0 𝑃20 0

0 0 0 𝑃4

). Therefore, the number of 𝑒𝑅contained in 𝑡

𝑛∩ 𝑡

𝑛

and containing 𝑒𝐿is 𝑞(𝑛−𝑙)(𝑘−𝑛).

Theorem 13. In the constructed multi-sender authenticationcodes, the maximum probabilities of success for impersonationattack and substitution attack from 𝑈

𝐿on the receiver 𝑅 are

𝑃𝐼(𝐿) =

1

𝑞2(𝑛−𝑙)(]−𝑟), 𝑃

𝑆(𝐿) =

1

𝑞(𝑛−𝑙). (30)

Proof. (1) Impersonation Attack. 𝑈𝐿, after receiving their

secret keys, sends a message 𝑚 to 𝑅. 𝑈𝐿is successful if the

receiver accepts it as authentic. Therefore,

𝑃𝐼(𝐿) = max

𝑒𝐿∈𝐸𝐿

max𝑚∈𝑀

{

{𝑒𝑅 ∈ 𝐸𝑅 | 𝑒𝐿 ⊂ 𝑒𝑅, 𝑒𝑅 ⊂ 𝑡}

{𝑒𝑅 ∈ 𝐸𝑅 | 𝑒𝐿 ⊂ 𝑒𝑅}

}

=𝑞2(𝑛−𝑙)(𝑟−𝑛)

𝑞2(𝑛−𝑙)(]−𝑛)

=1

𝑞2(𝑛−𝑙)(]−𝑟).

(31)

(2) Substitution Attack. 𝑅𝐿, after observing a message 𝑚

that is transmitted by the sender, replaces 𝑚 with anothermessage 𝑚. 𝑅

𝐿is successful if 𝑚 is accepted by 𝑅 as

authentic. Therefore,𝑃𝑆(𝐿)

=max𝑒𝐿∈𝐸𝐿

max𝑚∈𝑀

max𝑚 ̸=𝑚

∈𝑀

{

{𝑒𝑅∈ 𝐸𝑅| 𝑒𝐿⊂ 𝑒𝑅, 𝑒𝑅⊂ 𝑡, 𝑒𝑅⊂ 𝑡}

{𝑒𝑅 ∈ 𝐸𝑅 | 𝑒𝐿 ⊂ 𝑒𝑅, 𝑒𝑅 ⊂ 𝑡}

}

= max𝑛≤𝑘≤2𝑟−𝑛−1

𝑞(𝑛−𝑙)(𝑘−𝑛)

𝑞2(𝑛−𝑙)(𝑟−𝑛)

=1

𝑞(𝑛−𝑙).

(32)

6. The Advantage of the ConstructedAuthentication Code

The security of an authentication code could be measuredby the maximum probabilities of deceptions. The smaller theprobability of successful attack, the higher the security of theauthentication codes. Now let us compare the security of ourconstructed authentication code with the known one [19].

The constructed authentication code in [19] is also amultisender authentication code from symplectic geometryover finite fields, but which is in simultaneous model. Ifwe choose the parameters 𝑛, 𝑛, 𝑟, and ] with 1 < 𝑛 <𝑛< 𝑟 < ], 𝑛 > (𝑟/2), and 𝑛 − 𝑛 > ] − 𝑟, from

Table 1 we see that the maximum probabilities of deceptionsof our construction are smaller than the construction in[19]. Therefore, compared with the construction in [19], ourconstruction is more efficient.


Table 1: (𝑛 > 𝑟/2, 𝑛 − 𝑛 > ] − 𝑟).

Constructions [19] Size relation OursThe number of senders 𝑛 = 𝑛The number of attackers 𝑙, 1 ≤ 𝑙 < 𝑛 = 𝑙, 1 ≤ 𝑙 < 𝑛The parameters of codes|𝑆| 𝑁(2(𝑟 − 𝑛), 𝑟 − 𝑛; 2(] − 𝑛)) = 𝑁(2(𝑟 − 𝑛), 𝑟 − 𝑛; 2(] − 𝑛))𝐸𝑖 𝑞

2(]−𝑛)= 𝑞

2(]−𝑛)

𝐸𝑅 𝑞

2𝑛(]−𝑛)

> 𝑞2𝑛(]−𝑛)

|𝑇| 𝑁(2(𝑟 − 𝑛), 𝑟 − 𝑛; 2(] − 𝑛))𝑞2𝑛(]−𝑟−𝑛+𝑛)

< 𝑁(2(𝑟 − 𝑛), 𝑟 − 𝑛; 2(] − 𝑛))𝑞2𝑛(]−𝑟)

The probabilities of deceptions𝑃𝐼(𝐿)

1

𝑞2(𝑛−𝑙)(]+𝑛−𝑛−𝑟)−(𝑛−𝑛)(𝑛−𝑙)

>1

𝑞2(𝑛−𝑙)(]−𝑟)

𝑃𝑆(𝐿)

1

𝑞(𝑛−𝑙)(2𝑛−2𝑛

+1)+(𝑛

−𝑛)(𝑛−𝑙)

>1

𝑞𝑛−𝑙

Conflict of Interests

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

Acknowledgments

The Project is sponsored by the National Natural Sci-ence Foundation of China (no. 61179026) and the Fun-damental Research Funds of the Central Universities (no.3122013 K001).

References

[1] E. N. Gilbert, F. J. MacWilliams, and N. J. A. Sloane, “Codeswhich detect deception,”The Bell System Technical Journal, vol.53, pp. 405–424, 1974.

[2] G. J. Simmons, “Message authentication with arbitration oftransmitter/receiver disputes,” in Proceedings of the 6th AnnualInternational Conference on Theory and Application of Crypto-graphic Techniques (Eurocrypt ’87), vol. 304, pp. 151–165, 1988.

[3] Z. X. Wan, “Construction of cartesian authentication codesfrom unitary geometry,” Designs, Codes and Cryptography, vol.2, no. 4, pp. 333–356, 1992.

[4] W. P. Ma and X. M. Wang, “A construction of authenticationcodes with arbitration based on symplectic spaces,” ChineseJournal of Computers, vol. 22, no. 9, pp. 949–952, 1999.

[5] G. You, S. Xinhua, and W. Hongli, “Construction of authen-tication codes with arbitration from symplectic geometryover finite fields,” Acta Scientiarum Naturalium UniversitatisNankaiensis, vol. 41, no. 6, pp. 72–77, 2008.

[6] S. Chen and D. Zhao, “New construction of authenticationcodes with arbitration from pseudo-symplectic geometry overfinite fields,” Ars Combinatoria, vol. 97, pp. 453–465, 2010.

[7] S. Chen and D. Zhao, “Two constructions of optimal Cartesianauthentication codes from unitary geometry over finite fields,”ActaMathematicae Applicatae Sinica, vol. 29, no. 4, pp. 829–836,2013.

[8] S. Chen and D. Zhao, “Construction of multi-receiver multi-fold authentication codes from singular symplectic geometryover finite fields,” Algebra Colloquium, vol. 20, no. 4, pp. 701–710, 2013.

[9] L. Ruihu and L. Zunxian, “Construction of A2-codes fromsymplectic geometry,” Journal of Shanxi Normal University, vol.26, no. 4, pp. 10–15, 1998.

[10] C. Xing, H. Wang, and K.-Y. Lam, “Constructions of authen-tication codes from algebraic curves over finite fields,” IEEETransactions on InformationTheory, vol. 46, no. 3, pp. 886–892,2000.

[11] C. Carlet, C. Ding, and H. Niederreiter, “Authenticationschemes from highly nonlinear functions,” Designs, Codes andCryptography, vol. 40, no. 1, pp. 71–79, 2006.

[12] R. Safavi-Naini and H. Wang, “Multireceiver authenticationcodes: models, bounds, constructions, and extensions,” Infor-mation and Computation, vol. 151, no. 1-2, pp. 148–172, 1999.

[13] S. Chen and D. Zhao, “Construction of multi-receiver multi-fold authentication codes from singular symplectic geometryover finite fields,” Algebra Colloquium, vol. 20, no. 4, pp. 701–710, 2013.

[14] S. Chen and D. Zhao, “Two constructions of multireceiverauthentication codes from symplectic geometry over finitefields,” Ars Combinatoria, vol. 99, pp. 193–203, 2011.

[15] Y. Desmedt, Y. Frankel, and M. Yung, “Multi-receiver/multi-sender network security: efficient authenticatedmulticast/feedback,” in Proceedings of the the 11th AnnualConference of the IEEE Computer and CommunicationsSocieties, vol. 3, pp. 2045–2054, May 1992.

[16] Q. Yingchun and Z. Tong, “Multiple authentication Code withmulti-transmitter and its constructions,” Journal of ZhongzhouUniversity, vol. 20, no. 1, pp. 118–120, 2003.

[17] M. Wen-Ping and W. Xin-Mei, “Several new constructionson multitransmitters authentication codes,” Acta ElectronicaSinica, vol. 28, no. 4, pp. 117–119, 2000.

[18] D. Qingling and L. Shuwang, “Bounds and construction formulti-sender authentication code,” Computer Engineering andApplications, vol. 10, pp. 9–10, 2004.

[19] S. Chen and C. Yang, “A new construction of multisenderauthentication codes from symplectic geometry over finitefields,” Ars Combinatoria, vol. 106, pp. 353–366, 2012.

[20] Z. Wan, Geometry of Classical Groups over Finite Fields, SciencePress, Beijing, China, 2nd edition, 2002.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

research article a construction of multisender authentication … · 2019. 5. 12. · research...

Documents