Content Identification Based on digital fingerprint:what can be done if ML decoding fails?

Farzad Farhadzadeh, Sviatoslav Voloshynovskiy, Oleksiy Koval

Computer Science Department, University of Geneva7, route de Drize, Geneva, Switzerland

{farzad.farhadzadeh, svolos, oleksiy.koval}@unige.ch

Abstract—In this paper, the performance of the content iden-tification based on digital fingerprinting and order statistic listdecoding is analyzed by evaluating the probabilities of correctidentification, false acceptance and the probability mass functionof queried binary fingerprint position on the list of candidates.The particular attention is dedicated to the cases when traditionalmaximum likelihood decoder fails to produce the reliable contentidentification. The maximum likelihood decoding is shown tobe a particular case of order statistic list decoding for the listsize equals 1. We demonstrate the efficiency of the proposedcontent identification system performance by investigating theprobability mass function behavior and imposing the constrainton the cardinality of list size.

I. INTRODUCTION

The present work addresses a content identification problembased on digital fingerprints. In many applications, where thecontent modifications caused by watermark embedding areundesirable, content identification based on digital fingerprint-ing is the only possible solution. The content identificationbased on digital fingerprinting is an emerging thecnique invarious multimedia security and management applicationssuch as copyright protection, content filtering and automaticidentification, authentication, broadcast monitoring, contenttagging, etc.

A digital fingerprint represents a short, robust and distinc-tive content description allowing fast and privacy-preservingoperations. In this case, all operations are performed on thefingerprint instead of on the original large and privacysensitivedata.

Due to channel distortions, i.e., acquisition imperfection,compression etc., the content identification system should beable to cope with data variations. In classical content identi-fication setups the decoder estimate a unique index from thedatabase of records for a given query that has the constraint onfeasibility. Moreover, in the large scale database, the classicalapproach fails to act reliable. Another approach, which canbe considered as a generalization of the above mentioned one,was proposed by Elias [2] and Forney [3] in information theoryand is known as list decoding. The list decoder produces afixed/varying list size of the most likely candidates rather thana single one. However, contrarily to digital communications, in

MMSP’10, October 4-6, 2010, Saint-Malo, France. 978-1-4244-8112-5/10/$26.00 2010 IEEE.

the content identification setup the decoder should first of alldetermine whether a given query is related to some elements ofdatabase or not. Therefore, what we suggest is a list decodingwith the ranked (or scored) list of possible candidates.

The main contribution of this paper can be summarized asfollows. We introduce the content identification setup based onan order statistic list decoder (OSLD) of a fixed maximum listsize and analyze its performance by computing the probabilityof correct identification and false acceptance. Moreover, for thesake of deep investigation of the proposed approach, the PMFof the correct entry position behavior in different situationswill be considered. In performance analysis, we show that theclassical maximum likelihood (ML) decoding is a particularcase of our proposed decoder. We use random projectionsand binarization for binary fingerprint generation and channelstatistics conversion to a binary model [5].

Notations: We use capital letters X to denote scalar randomvariables and X to denote vector random variables. Corre-sponding small letters x and x denote the realizations of scalarand vector random variables. All vectors without sign tildeare assumed to be of the length N and with the sign tilde oflength L. B(N, p) denotes the Binomial distribution with Ntrials and probability of success p. V(r:M) stands for the r-thorder statistics of M i.i.d. random variables.

II. IDENTIFICATION SETUP

The identification setup under analysis shown in Fig. 1consists of two main phases: enrollment and identification.

In the enrollment phase, the digital fingerprints of biometricsor physical unclonable functions (PUFs) of items to be identi-fied x(m) ∈ XN , m = 1, . . . , M are extrated and stored in theDatabase. The digital fingerprint extraction is accomplished intwo steps. At the first step, by applying random projectors [5],which are a so-called approximate orthoprojector W ∈ R

L×N

with Wij ∼ N (0, 1N ), 1 ≤ i ≤ L and 1 ≤ j ≤ N , the

dimensionality is reduced from N to L. We assume, thatbiometrics or PUFs are drawn independently from a commonstationary distribution pX(x). At the second step, L-lengthbinary data are derived from the projected data by taking thesign of the reduced data, i.e., Bx = Q(X) ,where X = WX.

In the identification phase; for a given query Y, which canbe a noisy version of the biometrics or PUFs, the digitalfingerprint is extracted the same way as in the enrollmentphase, i.e., By = Q(Y) ,where Y = WY. Afterwards, the

978-1-4244-8112-5/10/$26.00 ©2010 IEEE 64

010 010

011 101

Database

Identifier

X

Y

xB

yB

Hash Generation

|PY X

W Q

W Q

bP1 bP−1 bP−

xB

yB

Pr

l′

Ranked indices

PositionHash GenerationY

1 2 3 lN

{ } { } { }1, , , 1, , ,1ll N j l li i i M j N N′′ ′= ∈ ∅ ≤ ≤ ≤… … ∪

Fig. 1. Identification problem based on binary templates.

decoder detects that the query is related to some entries of thedatabase, and if so, identifies which ones.

The decoding process in the identification setup is accom-plished in two steps. At the first step, the primary candidatesare chosen by the OSLD. At the second step, a threshold isapplied to the chosen candidates, and the candidates whichsatisfy the constraint remain. The OSLD procedure is accom-plished as follows1:

1) The likelihood functions, p(y|x(m)), 1 ≤ m ≤ M , forall database entries are computed.

2) The computed likelihood functions are sorted in ascend-ing order.

3) The Nl indices with the largest likelihood functions arechosen, which form a primary list Nl. The parameter Nl

is referred as a primary list size.4) The final output set of the decoder is defined as N ′

l ={m ∈ Nl : p(y|x(m)) ≥ eNγ} , where the parameter γcontrols the number of final candidates.

The identification problem is considered as an M + 1-aryhypothesis testing. The probability of correct identification isPc = Pr{m ∈ N ′

l |Hm}, 1 ≤ m ≤ M , and the probability offalse acceptance is Pf = Pr{N ′

l �= ∅|H0}.In this paper, we analyze the performance of the OSLD and

investigate the distribution of decoded indices on the rankedlist of candidates.

III. ORDER STATITICS

Before considering error events, we will review funda-mentals of the Order Statistics which will be used in thecomputation of the probabilities of errors. We suppose thatV (1), V (2), . . . , V (M) are M i.i.d. random variables, eachwith a cumulative distribution function (CDF) F (v). LetF(r:M)(v) denote the CDF of the r-th order statistics V(r:M),which corresponds to the r-th position of v(1:M) ≤ v(2:M) ≤. . . ≤ v(r:M) ≤ . . . ≤ v(M :M) for a specific outcome. Then

1The low-complexity identification based on the OSLD decoding and theconcept of bit reliability is given in [6].

[4]:

F(r:M)(v)=Pr{V(r:M) ≤ v

}=Pr {at least r of Vi are less than or equal to v}

=M∑i=r

(M

i

)F i(v)[1 − F (v)]M−i, (1)

since the term in the summand is the binomial probability thatexactly i of V (1), V (2), . . . , V (M) are less than or equal tov.

IV. PROBABILITY OF CORRECT IDENTIFICATION

Once the list of primary candidates is selected by theOSLD, the final candidates are extracted by thresholding theirlikelihoods. A correct identification event occurs, when thequery related to the database entry is on the list of finalcandidates. The probability of correct identification, Pc, isgiven by:

Pc=M∑

m=1

Pr{(m ∈ Nl)⋂

(p(y|x(m)) ≥ eNγ)|Hm}

×Pr{Hm}, (2)

where Nl is the primary list of candidates. As the entries ofthe database are identically distributed and equally likely to bequeried, the overall probability of correct identification doesnot depend on the particular index and hence:

Pc = Pr{(1 ∈ Nl)⋂

(p(y|x(1)) ≥ eNγ)|H1}. (3)

After dimensionality reduction and binarization, we have bi-nary data of the length L, where L < N . In order to evaluatePc, we consider the BSC with a crossover probability Pb. Forany bx(m),by ∈ {0, 1}L, the likelihood function

p(by|bx(m)) = PdH(by,bx(m))b (1 − Pb)L−dH(by,bx(m)) (4)

is a decreasing function of the Hamming distance d(m) �dH(by,bx(m)) for 0 ≤ Pb ≤ 0.5. In the following, wewill consider the above Hamming distance as a realizationof a random variable D(m), where m refers to the index ofbx(m). Given a query related to the mth entry of the database,the event m ∈ Nl occurs, if d(m) is among the Nl smallest{d(1), d(2), . . . , d(M)}. Fig. 2 illustrates this event given thequery is related to the first entry of the database.

The dimensionality reduction and binarization modify thestatistics of the database generated by pX(x) to the Binomialdistribution, i.e., Bx ∼ B(L, 1/2) for any pX(x) with adiagonal covariance matrix.

Conditioned on H1, the sufficient statistics can be expressedas follows:

D(m) ∼{B(L, Pb), for m = 1,

B(L, 12 ), for m �= 1.

(5)

From (1), the complement of the cumulative distributionfunction, F c

(Nl:M−1)(d), of N thl order statistics of the i.i.d

65

<>

1M −

(1) lD ∈

(1) lD ∉

(1)D ( : 1)lN MD −

(1: 1)MD − ( 1: 1)lN MD − − ( : 1)lN MD − ( 1: 1)M MD − −

Fig. 2. The illustration of the event that the first entry of database relatedto the query is not on the primary list.

random variables D(m), m �= 1, is given by:

F c(Nl:M−1)(d)=Pr{D(Nl:M−1) > d}

=Nl−1∑p=0

(M − 1

p

)s(d)ps(d)(M−1)−p, (6)

where s(d) �∑d

x=0

(Lx

) (12

)L. From (5) and (6), The

probability of correct identification (3) over the BSC can beexpressed by:

Pc(a)= Pr{(D(Nl:M−1) > D(1)) ∩ (D(1) ≤ η)|H1}

=η∑

d=0

Pr{D(Nl:M−1) > d|H1, D(1) = d}pD(1)(d)

=

{η∑

d=0

(L

d

)P d

b (1 − Pb)L−d

×Nl−1∑p=0

(M − 1

p

)s(d)p(1 − s(d))(M−1)−p

}

(7)

where from (3) and (4) η = L γ−ln(1−Pb)ln(Pb/(1−Pb))

, (a) follows from(3) and the fact that the likelihood function is a decreasingfunction of the Hamming distance, and pD(1)(d) denotes thePMF of D(1).

V. PROBABILITY OF FALSE ACCEPTANCE

The main reason to consider the probability of false ac-ceptance is to show the reliability of the decoding processwith respect to various attacking strategies. There are differ-ent scenarios to investigate the reliability of the decoder inidentification setups:

• The PDF of database generation is known.• The PDF of database generation is not known.• The database entries are partially known.• The database entries are totally known.

In this paper, we consider the scenario in which the PDFis fully known by the attacker. In the following, the falseacceptance event is considered over the BSC.

For this purpose, we define the following events:

ED(i:M) = {D(i:M) ≤ η|H0}, (8)

where 1 ≤ i ≤ Nl, D(m) ∼ B(L, 12 ), 1 ≤ m ≤ M , and

ED(i:M) is the event that the ith ascendingly ranked Hamming

()m

Dr

(1:

1)M

D−

(1:

1)k

MD

−−

(:

1)kM

D−

(2:

1)M

MD

−−

(1:

1)M

MD

−−

1r 1jr − jr 1jr + 1Mr − Mr

(1)

D

mrFig. 3. The illustration of the event that the first entry of database relatedto the query falls on the rth

j position.

distance between the query and an entry of the database issmaller than the threshold. The probability of false acceptanceis found as:

Pf =Pr{ Nl⋃

i=1

ED(i:M) |H0

}

=1 − Pr{ Nl⋂

i=1

EcD(i:M)

|H0

}(a)=1 − Pr

{Ec

D(1:M)|H0

}= Pr

{ED(1:M) |H0

}, (9)

where (a) follows from the fact that as the event EcD(1:M)

occurs the rest events occur. From (1), the probability of falseacceptance can be computed by:

Pf=Pr{

min1≤m≤M

D(m) ≤ η|H0

}

=1 −[1 −

η∑x=0

(L

x

) (12

)L]M

. (10)

The interesting point to note is that probability of falseacceptance is independent of the primary list size.

VI. PROBABILISTIC ANALYSIS OF LIST CONTENT

In this section, we consider the effect of list size on theidentification system performance by evaluating the PMF ofthe correct entry position. Conditioned on H1, the probabilityof D(1) to fall in the r th

j position of the ranked sufficientstatistics which satisfies the threshold is given by P (rj). From(5), all D(m), m ∈ {2, 3, . . . , M} are i.i.d. random variables,and as illustrated in Fig. 3, P (rj) can be expressed by:

P (rj) = Pr{(D(k:M−1) > D(1)) ∩ D(k−1:M−1) < D(1)∩(D(1) ≤ η)|H1}

(a)=

η∑d=0

Pr{D(k:M−1) > d|H1, D(1) = d}

×Pr{D(k−1:M−1) < d|H1, D(1) = d}pD(1)(d),(11)

where k ≡ rj+1, k − 1 ≡ rj−1 and (a) follows from the factthat the first two events are independent. From (1), P (rj) is

66

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ROC, L=1024, M=230

Pf

Pc

SNR=−5dB, Nl=1

SNR=−5dB, Nl=64

SNR=−8dB, Nl=1

SNR=−8dB, Nl=64

SNR=−10dB, Nl=1

SNR=−10dB, Nl=64

η increasing

Fig. 4. The OSLD performance improvement for the database with binaryentries.

given by:

P (rj) =η∑

d=0

(L

d

)P d

b (1 − Pb)L−d

×[ rj−1∑

p=0

(M − 1

p

)(1 − s(d))ps(d)(M−1)−p

]

×[

M−1∑q=rj−1

(M − 1

q

)(1 − s(d))ps(d)(M−1)−p

]. (12)

VII. SIMULATION RESULTS

The input to the identification system is obtained at theoutput of the BSC with a crossover probability Pb =1π arccosρXY [5], where ρXY is the cross-correlation coef-ficient between X and Y .

From (7) and (10), the probability of correct identificationand false acceptance over the BSC are computed and a receiveroperating characteristic (ROC) curves are shown for differentSNRs, database sizes M and primary list sizes Nl (Fig. 4 andFig. 5).

Fig. 4 shows that the OSLD improves the performancein the low SNR scenarios, while the probability of correctidentification Pc is not close to one.

Fig. 5 shows that increasing the primary list size Nl im-proves the identifier performance but after a certain value of Nl

it does not change significantly. On the other hand, increasingthe database size M decreases the performance.

Fig. 6 shows the behaviour of the PMF of the correct indexposition on the list. Decreasing the SNR increases the chanceof falling the correct entry in the other position instead ofthe first one that proves the inefficiency of the unique ML-based decoding in low SNR regimes justifying application ofthe proposed OSLD with a fixed maximum list size for thesesituations.

Fig. 7 shows that almost all positions of the list of candidatesare equal likely in very noisy environments.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ROC, L=1024, SNR=−10dB

Pf

Pc

M=210, Nl=64

M=220, Nl=64

M=230, Nl=128

M=230, Nl=64

M=230, Nl=1

M

Nl

Fig. 5. The effect of primary list size and database size on the OSLDperformance.

0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1PMF, L=1024, M=230

Position

Pr

SNR=−5dBSNR=−8dBSNR=−10dB

Fig. 6. The PMF of the correct entry posion in different SNRs.

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

−4 PMF, L=1024, M=230, SNR=−20

Position

Pr

Fig. 7. The unifrom behaviour of the PMF of the correct entry posion inlow SNR regim.

VIII. CONCLUSIONS

In this paper we investigated the performance of the maxi-mum fixed list size OSLD-based content identification systemin terms of the probability of correct identification, falseacceptance and the PMF of the correct entry position.

The obtained theoretical and simulation results show thaton the one hand, in low SNR regimes almost all position

67

in the list of candidates equally likely catch the correctentry. On the other hand, the OSLD can only improve theidentifier performance in very low SNR scenarios while thisimprovement is restricted by a certain range of list sizes.

ACKNOWLEDGMENTS

This paper was partially supported by SNF project 200021-119770.

REFERENCES

[1] P. Tuyls, B. Skoric, and T. Kevenaar, Security with noisy data: On PrivateBiometrics, Secure Key Storage and Anti-Counterfeiting, Springer, 2007.

[2] P. Elias, List decodeing for noisy channels, Tech. Rept. 335, ResearchLabratoary of Electronics, M.I.T, 1955.

[3] G. D. Forney, Jr, Exponential error bounds for erasure, list and decisionfeedback schemes, IEEETrans Inf. Theory, vol. IT-14, no. 2, pp. 206-220,Mar. 1968.

[4] H. A. David, H. N. Nagaraja, Order Statistics, 3rd ed, pp. 9, Wiley-Interscience, 2003.

[5] S. Voloshynovskiy, O. Koval, F. Beekhof, and T. Pun, Conception andlimits of robust perceptual hashing: toward side information assisted hashfunctions, SPIE Photonics West, San Jose, USA, 2009.

[6] F. Beekhof, S. Voloshynovskiy, O. Koval, and T. Holotyak, Fast Identifica-tion Algorithms for Forensic Applications, IEEE International Workshopon Information Forensics and Security, London, 2009.

68