This is an accepted version of a paper published in IEEE Transactions on Information Forensics and Security.

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1 This is an accptd rsion of a papr publishd in I Transactions on Information Fornsics and Scurity. If you wish to cit this papr, plas us th following rfrnc: T. Murakami, K. Takahashi, K. Matsuura, Toward Optimal Fusion Algorithms with Scurity against Wols and Lambs in Biomtrics, I Transactions on Information Fornsics and Scurity, Vol.9, No.2, pp , I. Prsonal us of this matrial is prmittd. Prmission from I must b obtaind for all othr uss, in any currnt or futur mdia, including rprinting/ rpublishing this matrial for adrtising or promotional purposs, crating nw collcti works, for rsal or rdistribution to srrs or lists, or rus of any copyrightd componnt of this work in othr works.

2 I TRANSACTIONS ON. INFORMATION FORNSICS AND SCURITY, VOL. 9, NO. 2, FBRUARY Toward Optimal Fusion Algorithms with Scurity against Wols and Lambs in Biomtrics Takao Murakami, Mmbr, I, Knta Takahashi, Mmbr, I, Kanta Matsuura, Snior Mmbr, I, Abstract It is known that diffrnt usrs ha diffrnt dgrs of accuracy in biomtric authntication, and claimants and nrolls who caus fals accpts against many othrs ar rfrrd to as wols and lambs, rspctily. Th aim of this papr is to dlop a fusion algorithm which has scurity against both of th animals whil minimizing th numbr of qury sampls a gnuin claimant has to input. To achi our aim, w first introduc a taxonomy of wols and lambs, and propos a minimum log-liklihood ratio basd squntial fusion schm (MLR schm). W pro that this schm kps WAP (Wolf Attack Probability) and LAP (Lamb Accpt Probability), th maximum of th claimant-spcific FAP (Fals Accpt Probability) and th nroll-spcific FAP, lss than a dsird alu if logliklihood ratios ar prfctly stimatd, xcpt in th cas of adapti spoofing wols. W also pro that this schm is optimal with rgard to FRP (Fals Rjct Probability), and asymptotically optimal with rspct to th arag numbr of inputs (ANI) undr som conditions. W furthr propos an input ordr dcision schm basd on th KL (Kullback-Liblr) dirgnc which maximizs th xpctation of a gnuin logliklihood ratio, to furthr rduc ANI of th MLR schm in th cas whr th KL dirgnc diffrs from on modality to anothr. Th rsults of th xprimntal aluation using a irtual multi-modal (on fac and ight fingrprints) datast showd th ffctinss of our schms. Indx Trms biomtric zoo, wols, lambs, squntial fusion, FAP, FRP, WAP, LAP, ANI I. INTRODUCTION BIOMTRIC authntication systms rcogniz an indiidual basd on his/hr physiological charactristics (.g. fingrprint, fac) or bhaioral charactristics (.g. oic, signatur). Sinc biomtrics is not forgottn unlik passwords and is much hardr to stal than cards, thy ar now usd for arious kinds of applications (.g. computr login, physical accss control) as a mor connint and scur way of authntication. Thy rcogniz th indiidual using a scor (similarity or distanc) btwn his/hr biomtric sampl (rfrrd to as a qury sampl) and a biomtric sampl nrolld in adanc (rfrrd to as a tmplat). For xampl, thy comput th normalizd Hamming distanc btwn iriscods 2, th uclidan distanc btwn ignfacs 3, or th prcntag of matchd minutia 4 as a scor, and mak a dcision whthr th claimant is gnuin or not by comparing th scor to a A prliminary rsion of this papr was prsntd at th I Fifth Intrnational Confrnc on Biomtrics: Thory, Applications and Systms (BTAS 2012), Washington DC, USA, Sptmbr, T. Murakami is with Yokohama Rsarch Laboratory, Hitachi, Ltd., Yokohama , Japan and with Institut of Industrial Scinc, Th Unirsity of Tokyo, Tokyo , Japan (-mail: takao.murakami.nr@hitachi.com). K. Takahashi is with Yokohama Rsarch Laboratory, Hitachi, Ltd., Yokohama , Japan (-mail: knta.takahashi.bw@hitachi.com). K. Matsuura is with Institut of Industrial Scinc, Th Unirsity of Tokyo, Tokyo , Japan (-mail: kanta@iis.u-tokyo.ac.jp). prdtrmind thrshold. Th prformanc of thm is gnrally aluatd using FRR (Fals Rjct Rat) and FAR (Fals Accpt Rat), th rror rat that a gnuin indiidual is falsly rjctd and an impostor is falsly accptd, rspctily. Howr, FRR and FAR ar just th arag rror rats takn or all biomtric sampls, and do not masur th prformanc for a particular usr. Th rality is that th prformanc is diffrnt from usr to usr, and Doddington t al. 5 classifid usrs in spakr rcognition as follows: Shp: thos who ar asily rcognizd (dfault usrs); Goats: thos who ar particularly difficult to rcogniz; Lambs: thos who ar particularly asy to imitat; Wols: thos who ar particularly succssful at imitating othrs. This concpt is known as th biomtric zoo (or Doddington s zoo), and numrous studis ha bn mad on this issu (as dscribd in dtail in Sction III-A) 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16. Among th abo animals, wols and lambs ar particularly problmatic bcaus thy can caus fals accpts against many othrs and compromis th scurity of th systm. As it is oftn said that th orall scurity of th systm is dtrmind by th wakst link in th chain 17, w ha to tak countrmasurs against ths animals to incras th orall scurity of th biomtric systm. To ha scurity against wols and lambs, w ha to rduc th fals accpts causd by thm. To rduc th fals accpts, w can us multi-modal biomtric fusion schms 18 which combin multipl modalitis (.g. fingrprint, fac and oic; indx and middl fingrs). Howr, such a solution can mak th systm inconnint bcaus a claimant (a prson who attmpts authntication) has to input multipl qury sampls. That is, thr is a trad-off btwn scurity against wols and lambs and conninc in biomtrics. Th aim of this papr is to optimiz this trad-off. That is, to dlop a fusion algorithm which has scurity against wols and lambs whil minimizing th numbr of qury sampls th gnuin claimant has to input. Although a numbr of countrmasurs against wols or lambs ha bn proposd so far 12, 13, 14, 15, 16, this is th first attmpt to optimiz th abo trad-off, to th bst of our knowldg. To achi our aim, w first introduc a taxonomy of wols and lambs to clarify our targt, and dfin scurity masurs for th animals to nabl scurity aluation. W thn mo on to squntial fusion 19, 20, 21, which combins scors from multipl modalitis and maks a dcision ach tim th claimant inputs his/hr qury sampl, and propos a squntial fusion schm which intnds to optimiz th abo trad-off. Finally, w propos an input ordr dcision schm to furthr rduc th numbr of biomtric inputs. Mor spcifically, th

3 I TRANSACTIONS ON. INFORMATION FORNSICS AND SCURITY, VOL. 9, NO. 2, FBRUARY contributions of our work (which ar common to this papr and its prious rsion 1) ar as follows: W first introduc a taxonomy which classifis wols into thr catgoris (zro-ffort wols, non-adapti spoofing wols, and adapti spoofing wols) and lambs into two catgoris (zro-ffort lambs and spoofing lambs). This taxonomy clarifis th dfinition of wols and lambs, and our targt as wll. W also dfin LAP (Lamb Accpt Probability), th maximum of th nrollspcific FAP (Fals Accpt Probability) 1, as a scurity masur for lambs in th sam way as WAP (Wolf Attack Probability) 22, th maximum of th claimant-spcific FAP, to nabl scurity aluation. W thn propos a minimum log-liklihood ratio basd squntial fusion schm (rfrrd to as th MLR schm). W pro that this schm kps WAP and LAP lss than a dsird alu xcpt in th cas of adapti spoofing wols, if log-liklihood ratios ar prfctly stimatd. W also pro that th arag numbr of inputs (rfrrd to as ANI) rquird in this schm can achi almost th minimum alu undr som conditions. W finally propos an input ordr dcision schm basd on th KL (Kullback-Liblr) dirgnc 23. This schm furthr rducs ANI of th MLR schm in th cas whr th KL dirgnc diffrs from on modality to anothr, by maximizing th xpctation of a gnuin log-liklihood ratio at any numbr of biomtric inputs. This papr is an xtnsion of a priously publishd confrnc papr 1. Th main nhancmnts ar as follows: 1) W pro th optimality of th MLR schm with rgard to not only ANI but also FRP (Fals Rjct Probability) (Thorm 2 in Sction IV-C2). Sinc fals rjcts caus much gratr inconninc than th incras of biomtric inputs, this thorm significantly contributs to th ffctinss of th MLR schm. W also show th alidity of this thorm through xprimntal aluation (Sction VI). 2) In 1, w only aluatd our schms using th CASIA- FingrprintV5 24 which contains multipl fingrprint imags. In this papr, w show th ffctinss of our schms using a irtual multi-modal (on fac and ight fingrprints) datast by combining th abo datast with th NIST BSSR1 St3 datast 25 (Sction VI). This papr is organizd as follows. Sction II introducs a taxonomy of wols and lambs, and scurity masurs for th animals. Sction III dscribs th prious work on th biomtric zoo and squntial fusion. Sction IV proposs th MLR schm, and pros its scurity and optimality with rgard to FRP and ANI. Sction V proposs th input ordr dcision schm using th KL dirgnc. Sction VI shows th xprimntal rsults. Finally, Sction VII concluds this papr with dirctions for th futur. 1 In this papr, in addition to th conntional FRR and FAR, w dfin FRP (Fals Rjct Probability) and FAP (Fals Accpt Probability) to diffrntiat rror probabilitis from rror rats (s Sction II-B for dtails). II. WOLVS AND LAMBS A. Taxonomy of wols and lambs Thr ar two diffrnt ways of biomtric authntication: rification and idntification 26, 27. In rification, a claimant claims an idntity along with a qury sampl, and th systm computs a scor btwn th qury sampl and a tmplat corrsponding to th claimd idntity, making a dcision whthr th claimant is gnuin or not. In idntification, a claimant only inputs a qury sampl, and th systm computs scors btwn th qury sampl and all tmplats in th databas (i.. on-to-many matching), making a dcision who th claimant is. Although lambs ar a ulnrability in rification, thy can b a thrat in idntification bcaus thy can mak th systm idntify many claimants as thm, and los th aailability of th systm. W also not that wols and lambs caus mor srious scurity problms in idntification bcaus FAR incrass (and so dos th numbr of fals accpts causd by th animals) as th numbr of nrolls incrass 26, 27. To clarify th scop of this papr, w introduc a taxonomy which classifis wols as follows: Zro-ffort wols: Ths wols ar thos who (happn to) ha thir own biomtrics similar to many othrs, and imprsonat many nrolls by attmpting a zro-ffort attack 28. That is, thy dirctly input thir own qury sampl as if thy wr attmpting succssful authntication against thmsls. Although Doddington t al. 5 statistically tstd th xistnc of claimants whos oic has high similarity scors against many nrolls, thy fall into this catgory. Thy ar ry powrful in that thy cannot b blockd using anti-spoofing masurs (.g. linss dtction; suprising th authntication procss) 29 bcaus thy do not spoof th systm. Thy can b a thrat n in th modalitis which ar ry difficult to spoof (.g. iris, rtina 26). Spoofing wols: Ths wols ar thos who mak particular ffort (.g. chang thir biomtrics; input an artifact 30) to imprsonat many nrolls, and ar furthr diidd into th following sub-catgoris: Non-adapti spoofing wols: Ths wols ar non-adapti in th sns that thy input th sam qury sampl against all nrolls. For xampl, thy input an artifact which has xtrmly high similarity scors against all tmplats, which is rfrrd to as a unirsal wolf sampl 22. Adapti spoofing wols: Ths wols ar adapti in th sns that thy chang th qury sampl dpnding on th nroll. For xampl, thy chang thir oic dpnding on th nroll to imitat him/hr. W not that this kind of attack is limitd to th modality such as oic whr th attackrs can asily know th qury sampl of ach nroll, and chang thir biomtrics (or crat an artifact) to imitat it. Similarly, our taxonomy classifis lambs as follows: Zro-ffort lambs: Ths lambs ar thos who (happn to) ha thir own biomtrics similar to many othrs,

4 I TRANSACTIONS ON. INFORMATION FORNSICS AND SCURITY, VOL. 9, NO. 2, FBRUARY and nroll thir own tmplat without particular ffort. Thy cannot b blockd using anti-spoofing masurs, and can b a ulnrability (or thrat in idntification) in th modalitis which ar ry difficult to spoof. Spoofing lambs: Ths lambs ar thos who mak particular ffort to nroll th tmplat similar to many othrs. Howr, thy ar non-adapti in th sns that thy cannot chang th tmplat aftr th nrollmnt (w assum a gnral biomtric authntication systm without tmplat updat mchanism 31). Among thm, adapti spoofing wols cannot b blockd using our proposals, as dscribd in dtail in Sction IV, and ar outsid th scop of this papr. B. Scurity masurs for wols and lambs W also dfin scurity masurs for wols and lambs. Som statistical tsts on scors wr usd to dmonstrat th xistnc of th animals dfind by thmsls 5, 8, and som scor-basd masurs to quantify rcognizability of a usr or th xtnt of th biomtric zoo wr studid in 6, 7. Howr, ths tsts and masurs ar not dsignd to dirctly aluat scurity against wols and lambs. To aluat scurity against wols and lambs, w should tak into account fals accpts causd by thm, instad of th statistics on scors. Thus w start with FRR and FAR, th most commonly usd rror rats in rification. To diffrntiat btwn rror rats and rror probabilitis, w also dfin FRP (Fals Rjct Probability) and FAP (Fals Accpt Probability), th probability that a gnuin indiidual is falsly rjctd and an impostor is falsly accptd, rspctily. Lt d {0, 1} b a ariabl which taks 1 or 0 if th final dcision rsult is accpt or rjct, rspctily. Lt furthr W 1 b th nt that a gnuin usr attmpts rification against him/hrslf, and W 0 b th nt that an impostor attmpts rification against somon ls. Thn, FRP and FAP can b writtn as F RP = P (d = 0 W 1 ) (1) F AP = P (d = 1 W 0 ), (2) whr P () is a probability mass function. Sinc thy ar thortical alus, in practic FRR and FAR ar aluatd, instad of FRP and FAP, using a finit numbr of biomtric sampls as follows: F RR = F AR = Th numbr of fals rjcts (3) Th total numbr of gnuin attmpts Th numbr of fals accpts Th total numbr of impostor attmpts. (4) Sinc FRR and FAR ar th arag rror rats takn or all biomtric sampls, thy do not masur th prformanc for a particular usr. Taking this into account, w dfin prformanc masurs for a particular usr. Lt V b a finit st of claimants, b a finit st of nrolls. Lt furthr W, b th nt that attmpts rification against him/hrslf, W, b th nt that V attmpts an impostor attack against somon ls, W, b th nt that somon ls attmpts an impostor Claimant V nroll V Wolf Lamb FAP WAP LAP : Probability that th final dcision rsult is accpt (i.. d = 0) Fig. 1. Thr kinds of fals accpt probabilitis in rification (FAP/WAP/ LAP). WAP and LAP ar th maximum of th claimant-spcific FAP and th nroll-spcific FAP, rspctily. attack against. Thn, w can dfin th following thr rror probabilitis: F RP = P (d = 0 W, ) (5) F AP, = P (d = 1 W, ) (6) F AP, = P (d = 1 W, ). (7) Thy ar th nroll-spcific FRP, th claimant-spcific FAP, and th nroll-spcific FAP, rspctily. Goats caus high F RP, wols caus high F AP,, and lambs caus high F AP,. W also dfin th corrsponding thr rror rats: F RR = F AR, = F AR, = Th numbr of fals rjcts causd by Th total numbr of gnuin attmpts of V (8) Th numbr of fals accpts causd by Th total numbr of impostor attmpts of (9) Th numbr of fals accpts causd by Th total numbr of impostor attmpts of. (10) Un t al. 22 dfind WAP (Wolf Attack Probability) as a scurity masur for wols, which can b writtn as follows: W AP = max V F AP,. (11) That is, WAP is th fals accpt probability causd by th most thratning wolf. Similarly, w dfin LAP (Lamb Accpt Probability), a scurity masur for lambs, as follows: LAP = max F AP,. (12) LAP is th fals accpt probability causd by th most ulnrabl lamb. W furthr dfin WAR (Wolf Attack Rat) and LAR (Lamb Accpt Rat) as th rror rat corrsponding to WAP and LAP, rspctily: W AR = max, V (13) LAR = max,. (14) W us rror probabilitis such as FRP, FAP, WAP and LAP in a thortical analysis, and rror rats such as FRR, FAR, WAR, LAR in an xprimntal aluation. Figur 1 shows th thr kinds of fals accpt probabilitis in rification (FAP/WAP/LAP).

5 I TRANSACTIONS ON. INFORMATION FORNSICS AND SCURITY, VOL. 9, NO. 2, FBRUARY In idntification, FAR can b diidd into FAR (nroll FAR) and NFAR (Non-nroll FAR) 21, th fals accpt rat causd by nrolls and non-nrolls, rspctily. Thus, scurity masurs for wols and lambs in idntification can b mor complicatd, and ar not dfind in this papr. A. Biomtric zoo III. RLATD WORK Sinc Doddington t al. 5 introducd a concpt of th biomtric zoo, numrous studis ha bn mad on this issu from sral dirctions. Poh and Kittlr 6 xamind th potntial of som scor-basd masurs as an indx charactrizing rcognizability of a usr. Thy also proposd a masur to quantify th xtnt of th biomtric zoo 7. Yagr and Dunston 8 introducd a nw class of animals considring a rlationship btwn gnuin and impostor scors. Tli t al. 9 instigatd th consistncy of th biomtric zoo across algorithms and datasts. Similarly, Paon and Flynn 10 instigatd th consistncy across algorithms and two iriss for a singl usr. Wang t al. 11 applid th Frog- Boiling attack to kystrok tmplat updat mchanisms to chang th ictim s tmplat littl by littl towards a tmplat of ill-prforming animals such as lambs or goats. Countrmasurs against wols can also b found in som litraturs 12, 13. Inuma t al. 12 proposd a thortical framwork of dcision algorithms in rification using fatur distributions for ach of all human bings to kp WAP lss than a dsird alu. Kojima t al. 13 proposd anothr dcision algorithm in rification using dcision rsults (accpt or rjct) with biomtric sampls othr than th tmplat to dtct wols. Sinc both of thm assum that th claimants do not chang th qury sampl dpnding on th nroll (i.. non-adapti), thy do not ha scurity against adapti spoofing wols. As for lambs, a numbr of scor normalization schms which us nroll-spcific paramtrs or impostor distributions ha bn proposd, and a sury of thm is gin in 14. For xampl, Poh and Kittlr 15 proposd a scor normalization schm which normalizs a scor to a logliklihood ratio using nroll-spcific scor distributions. Anothr xampl is a slcti fusion schm proposd by Ross t al. 16. This schm dtcts lamb tmplats which ha high similarity scors against many othrs and goat tmplats which ha low similarity scors against thmsls, and inoks fusion only for nrolls who ha such wak tmplats. Although this schm rquirs only such nrolls to input multipl biomtrics, it is not intndd to minimiz ANI. B. Squntial fusion To rduc th rror rats whil kping down th numbr of biomtric inputs, w can us squntial fusion schms which combin scors from multipl modalitis and mak a dcision ach tim a claimant inputs his/hr qury sampl. Sinc thy only us scors as information sourcs, thy can b applid to any kind of modality such as fingrprint, fac, and oic. Takahashi t al. 19 proposd a squntial fusion schm in rification which squntially computs a log-liklihood ratio using a gnuin distribution and an impostor distribution which ar common to all usrs (rfrrd to as th LR schm). This schm is basd on SPRT (Squntial Probability Ratio Tst) 32, a statistical hypothsis tst which minimizs th arag numbr of sampls if th sampls ar i.i.d. (indpndnt and idntically distributd) and th rror probabilitis ar sufficintly small. Allano t al. aluatd th LR schm with rspct to th cost which includs th procssing tim and th financial cost of snsors 20. Murakami and Takahashi 21 proposd a squntial fusion schm in idntification using MSPRT (Multi-hypothsis SPRT) 33, a multi-hypothsis rsion of SPRT. Although thy can proid a ry good tradoff btwn th arag rror rats (i.. FRR and FAR) and ANI, thy ar not dsignd to ha scurity against wols and lambs. To sum up, no studis ha r trid to dlop an algorithm which minimizs ANI whil kping WAP and LAP lss than a dsird alu, to th bst of our knowldg. IV. MINIMUM LOG-LIKLIHOOD RATIO BASD SQUNTIAL FUSION Our first proposal is a minimum log-liklihood ratio basd squntial fusion schm (MLR schm), a squntial fusion schm in rification. This schm is a modification of th LR schm 19 to ha scurity against wols and lambs, by computing th minimum of two log-liklihood ratios obtaind using two kinds of usr-spcific impostor distributions. Aftr dscribing its algorithm, w clarify its scurity and optimality from a thortical point of iw. A. Assumptions about scors Bfor dscribing th algorithm of th MLR schm, w mak som assumptions about scors. First of all, w focus on squntial fusion of multipl biomtric traits (.g. fingrprint, fac and iris) or multipl units (.g. indx and middl fingrs) 18, 26. Sinc thy ar highly indpndnt information, a larg impromnt in rcognition accuracy can b xpctd 26. Thn w assum that all scors ar indpndnt. Lt s R b a scor btwn th t-th qury sampl and th corrsponding tmplat (in this papr, w assum that s is continuous; th discussion blow can b asily xtndd to th discrt cas). In addition to th indpndnc of scors, w assum that gnuin scors ar gnratd from a gnuin distribution f which is common to all nrolls: f (s ) = p(s W, ), (15) whr p() is a probability dnsity function. f is traind using gnuin scors from tmplats nrolld in th databas or any othr biomtric sampls which ar collctd in adanc. Strictly spaking, a gnuin distribution is diffrnt from nroll to nroll sinc diffrnt nrolls ha diffrnt dgrs of rcognizability (.g. goats ha low similarity scors against thmsls). Nrthlss, w assum a gnuin distribution common to all nrolls bcaus gnrally ry fw gnuin scors pr nroll (.g. 2 or 3 scors) ar

6 I TRANSACTIONS ON. INFORMATION FORNSICS AND SCURITY, VOL. 9, NO. 2, FBRUARY aailabl as training sampls. For xampl, Poh t al. 34 showd that thr wr low corrlation btwn th standard diations of th nroll-spcific gnuin distributions in th training st and thos in th aluation st. This indicats th difficulty of rliably stimating th nroll-spcific gnuin distributions. Furthrmor, if ach nroll prsnts only on biomtric sampl during nrollmnt, thr ar no gnuin scors which can b obtaind from th sampl. n in such a cas, w can train f using gnuin scors obtaind from othr biomtric sampls which ar collctd in adanc. W furthr dfin th following two impostor distributions: g (s ) = p(s W, ) (16) g (s ) = p(s W, ). (17) That is, g is a claimant-spcific impostor distribution, and g is an nroll-spcific impostor distribution. Suppos V attmpts rification against. Thn, g is traind using scors btwn a qury sampl of V and biomtric sampls othr than th tmplat of. W rfr to th biomtric sampls usd for stimating g as dummytmplats. For xampl, w can us all or part of tmplats of othr nrolls in th databas as dummy-tmplats. W can also us any othr biomtric sampls which ar collctd sparatly from th tmplats (.g. biomtric sampls collctd for prformanc aluation). Similarly, g is traind using scors btwn a tmplat of and dummy-tmplats. Not that f and g ar traind bfor authntication (.g. right aftr nrollmnt), whil g is traind aftr V inputs th t-th qury sampl. As will b dscribd in dtail in Sction IV-B, f, g and g ar usd for stimating th following two kinds of log-liklihood ratios: Z = log f (s ) g (s ) = log f (s ) Z g (18) (s ). (19) Thus, it is also possibl to dirctly train Z and Z, instad of training f, g, and g (in this cas, Z is traind bfor authntication, and Z is traind aftr V inputs th t-th qury sampl). For xampl, w can us logistic rgrssion 35 which modls Z and Z as follows: whr w 1, w 0 Z = w 1 s + w 0 (20) = w 1 s + w 0, (21) Z, w 1, and w 0 ar rgrssion cofficints. Th ffctinss of th logistic rgrssion modl is shown in som litraturs on biomtrics 21, 36, 37. W also show th alidity of this modl in our xprimnts in Sction VI. B. Algorithm W now dscrib th algorithm of th MLR schm. Suppos th claimant V attmpts rification against th nroll. Lt r = (r 1,, r N ) b a squnc of scors btwn th t-th qury sampl of V and N s Claimant Qury sampl ( t) r1 r N Tmplat Dummy-tmplats Probability dnsity ( t) r1 g g r N f s Scor Z tot A 0 accpt rjct 1 2 3(=T) Fig. 2. Oriw of th MLR schm in th cas whr th scor distributions f, g, and g ar traind. f and g ar traind in adanc, whil g is traind using r = (r 1,, r N ). Aftr computing s, Z tot is updatd and compard to A. Hr, two xampls ar gin: on rsults in accptanc at th scond input; th othr rsults in rjction (T = 3). dummy-tmplats. r is just usd to train g (or Z ). Lt furthr s t = (s (1),, s ) b a squnc of scors btwn th qury sampls of V and th tmplats of, and H i (i = 0, 1) b th following hypothss: H 1 : Th claimant is a gnuin usr. H 0 : Th claimant is an impostor. Thn, sinc w assum that all scors ar indpndnt, th logliklihood ratio aftr obtaining s t can b writtn as follows: log l(h 1 s t ) l(h 0 s t ) = t τ=1 log l(h 1 s (τ) ) l(h 0 s (τ) ), (22) whr l(h 1 s (τ) ) is a liklihood of th hypothsis H 1 gin a scor s (τ) and l(h 0 s (τ) ) is a liklihood of H 0 gin s (τ). Th MLR schm stimats th two kinds of log-liklihood ratios using th two kinds of impostor distributions g (τ) and g (τ) (s (18) and (19)), and adopts th minimum alu of thm (i.. th on which causs lss fals accpts) as log l(h 1 s (τ) )/l(h 0 s (τ) ). That is, th MLR schm computs Z tot aftr obtaining s t as follows: t Z (τ) whr and Z (τ) Z tot = τ=1 Z (τ) min = min { Z (τ) min, (23) Z (τ) }, Z (τ). (24) Thn, it compars Z tot to a rification thrshold A, and maks th following dcision: If Z tot is gratr than or qual to A, accpt th hypothsis H 1 (i.. accpt th claimant); Othrwis if th numbr of inputs t rachs th maximum alu T (i.. th numbr of modalitis), accpt th hypothsis H 0 (i.. rjct th claimant); Othrwis, rquir anothr biomtric input. To sum up, th algorithm of th MLR schm is as follows: Th MLR algorithm: 1) t 1, Z (0) tot 0; 2) Comput r = (r 3) Comput s ; 4) Z log f (s ) { 5) Z min min 1,, r, g (s ) Z Z, Z N ), and train g log f (s ) } ; g (s ), Z tot Z (t 1) tot t (or Z ); + Z min ; 6) If Z tot A, accpt th claimant; Othrwis if t = T, rjct th claimant; Othrwis, t t + 1 and go to 2). Figur 2 shows th oriw of th MLR schm.

7 I TRANSACTIONS ON. INFORMATION FORNSICS AND SCURITY, VOL. 9, NO. 2, FBRUARY TABL I NOTATIONS USD IN DSCRIBING THORTICAL PROPRTIS OF TH MLR SCHM. Symbol Dscription δ a squntial fusion algorithm δ 0 th MLR algorithm V a st of claimants a st of nrolls F RP FRP (Fals Rjct Probability) of (s (5)) F AP, FAP (Fals Accpt Probability) of V (s (6)) F AP, FAP (Fals Accpt Probability) of (s (7)) W AP WAP (Wolf Attack Probability) (s (11)) LAP LAP (Lamb Accpt Probability) (s (12)) ANI ANI (th arag numbr of inputs) of f a gnuin distribution common to all nrolls (s (15)) g an impostor distribution of V (s (16)) g an impostor distribution of (s (17)) Z /Z two kinds of log-liklihood ratios (s (18) and (19)) A a rification thrshold of th MLR schm T th maximum numbr of inputs (th numbr of modalitis) α a rquird WAP and LAP β a rquird F RP (α, β) a st of squntial fusion algorithms whos F AP, and F RP do not xcd α and β, rspctily (s (25)) C. Thortical proprtis W now show thortical proprtis of th MLR schm. W first brifly xplain th outlin of thm. Lt δ b a squntial fusion algorithm, and W AP (δ) b WAP of δ. W apply th sam rul to othr prformanc masurs such as LAP, FRP, and FAP. Lt furthr δ 0 b th MLR algorithm. In this papr, w pro th following thr proprtis of th MLR schm: Scurity against wols and lambs: Th MLR schm can kp th fals accpt probability causd by any claimant and any nroll, xcpt for adapti spoofing wols, lss than a dsird alu: W AP (δ 0 ) α and LAP (δ 0 ) α, whr α is a rquird WAP and LAP (Thorm 1). Optimality with rgard to FRP: Th MLR schm can minimiz, for any nroll, th fals rjct probability among all squntial fusion schms with th sam fals accpt probability: F RP (δ 0 ) F RP (δ) for any δ such that F AP, (δ) = F AP (δ 0 ) (Thorm 2). Asymptotic optimality with rgard to ANI: Th MLR schm can minimiz, for any nroll, ANI among all squntial fusion schms in th asymptotic stting whr both th fals accpt probability and th fals rjct probability (F AP, and F RP ) ar sufficintly small (Thorm 3). Th first proprty guarants th scurity of th MLR schm in trms of fals accpts, whil th scond on guarants th optimality with rgard to fals rjcts. W can significantly rduc both fals accpts and fals rjcts by stting th maximum numbr of inputs (i.. th numbr of modalitis) T ry larg. Thn, th third proprty guarants that ANI of this schm can achi almost th minimum alu. In th rst of this subsction, w formally dscrib ths thortical proprtis. Tabl I shows th notations usd thr. 1) Scurity against wols and lambs: Th MLR schm modls both th claimant-spcific impostor distribution g and th nroll-spcific impostor distribution g, and adopts th on which causs lss fals accpts (i.. th minimum alu of Z and Z ). By this mans, it achis scurity against any claimant and any nroll xcpt for adapti spoofing wols, if Z and Z ar prfctly stimatd: Thorm 1. If (i) th log-liklihood ratios Z and Z (1 t T ) ar prfctly stimatd in th MLR algorithm δ 0, thn w ha F AP, (δ 0 ) α and F AP, (δ 0 ) α for any V and, and hnc w ha W AP (δ 0 ) α and LAP (δ 0 ) α (xcpt in th cas of adapti spoofing wols), whr α = A and A is a rification thrshold. Th proof is gin in Appndix A. It should b notd that this thorm holds irrspcti of th input ordr (i.. irrspcti of which modality to start with). Th optimality of th MLR schm (Lmma 1 and Thorm 3) also holds irrspcti of th input ordr. Howr, w assum that th input ordr is fixd in proing th optimality of th MLR schm with rgard to FRP (Thorm 2). 2) Optimality with rgard to FRP: W thn pro th optimality of th MLR schm with rgard to FRP. Hr w assum that th input ordr is fixd as mntiond abo. Thn, th following thorm holds: (1 t T ) ar prfctly stimatd in th MLR algorithm δ 0, thn w ha F RP (δ 0 ) F RP (δ) for any squntial fusion algorithm δ such that F AP, (δ) = F AP, (δ 0 ). Thorm 2. If (i) th log-liklihood ratios Z and Z Th proof is gin in Appndix B. This thorm mans that th MLR schm can achi, for any, th minimum F RP among all squntial fusion schms with th sam F AP,. In othr words, th MLR schm can proid an optimal trad-off btwn F RP, and F AP. 3) Lowr bound for ANI: W finally pro th optimality of th MLR schm with rgard to ANI in th asymptotic stting whr th idntification rror probabilitis (both th fals accpt probability and th fals rjct probability) ar sufficintly small. To pro this optimality, w first show th lowr bound for ANI of a squntial fusion algorithm δ. Lt (α, β) th following st of squntial fusion algorithms: (α, β) = {δ : F AP, (δ) α, F RP (δ) β}, (25) whr β is a rquird F RP. Th MLR algorithm δ 0 satisfis this rquirmnt by stting th thrshold A = log α 1 (so that F AP, α; s Thorm 1), and th numbr of modalitis T sufficintly larg (so that F RP (δ 0 ) β). Lt ANI b ANI of. Thn, th following lmma gis th lowr bound for ANI of any δ (α, β): Lmma 1. If (ii) th KL (Kullback-Liblr) dirgnc btwn f and g taks a alu D indpndntly of th modality, thn inf ANI log α 1 (δ) as max(α, β) 0, (26) δ (α,β) D whr inf δ (α,β) X is th infimum of X or δ (α, β). Th proof is gin in Appndix C. Hr w xplain th maning of this lmma. Th KL dirgnc D(f g )

8 I TRANSACTIONS ON. INFORMATION FORNSICS AND SCURITY, VOL. 9, NO. 2, FBRUARY btwn f and g is writtn as follows 23: D(f g ) = f (s ) log f (s ) g (s ) ds, (27) and has a maning of a distanc masur btwn f and g. If this taks th alu D indpndntly of th modality (i.. if th condition (ii) holds), thn ANI of any squntial fusion schm is lowr boundd by th right sid of (26) in th asymptotic cas whr th idntification rror probabilitis ar sufficintly small (i.. max(α, β) 0). 4) Asymptotic optimality with rgard to ANI: From Lmma 1, to pro that th MLR schm is optimal with rgard to ANI, it suffics to show that ANI of th MLR algorithm δ 0 can achi th right sid of (26). W pro that this is indd th cas: Thorm 3. If (i) th log-liklihood ratios Z and Z (1 t T ) ar prfctly stimatd in th MLR algorithm δ 0, and (ii) th KL dirgnc btwn f and g taks a alu D indpndntly of th modality, thn ANI (δ 0 ) log α 1 D as max(α, β) 0, (28) whr X Y as r 0 mans that lim r 0 (X/Y ) = Th proof is gin in Appndix D. This thorm mans that th MLR schm achis, for any, th minimum ANI (i.. th right sid of (26)) among all squntial fusion schms in th asymptotic cas whr th idntification rror probabilitis ar sufficintly small (i.. max(α, β) 0). D. Limitations In Sction IV-C, w clarifid th thortical proprtis of scurity and optimality of th MLR schm. Howr, this schm has som limitations. Firstly, w xcludd adapti spoofing wols in Thorm 1. Suppos thr is an adapti spoofing wolf who can prfctly imitat th oic of othrs. If this wolf attmpts an impostor attack against ach nroll by prsnting th oic which has high similarity scor only against, thr is no way to block such an attack and consquntly WAP rachs 100%. W nd to adopt a modality which is ry difficult to spoof (.g. iris and rtina), to prnt such an attack. Scondly, th condition (i), which is common to Thorm 1, 2, and 3, dos not hold in gnral. That is, it is gnrally impossibl to prfctly stimat th log-liklihood ratios Z (= log f /g ) and Z (= log f /g ). In our xprimnts in Sction VI, w us th logistic rgrssion modl which dirctly stimats log f /g and log f /g as a linar function of a scor (s (20) and (21)), and show that it works ry wll though not prfct. Thirdly, Lmma 1 and Thorm 3 only guarant th optimality of th MLR schm with rgard to ANI of in th asymptotic stting whr F AP, and F RP ar sufficintly small (i.. A and T ar sufficintly larg). It rmains unsttld whthr it is optimal in th cas whr F AP, and F RP ar not small. Last but not last, th condition (ii) in Lmma 1 and Thorm 3 also dos not hold in gnral. As mntiond abo, th KL dirgnc is a distanc masur btwn two probability distributions, and som studis proposd to us th KL dirgnc btwn th gnuin distribution and th impostor distribution as a mtric of idntification prformanc 38, 39. Thus, it is natural to considr that th KL dirgnc D(f g ) diffrs from on modality to anothr, spcially in th cas of multipl biomtric traits (.g. fingrprint, fac and iris) 18. In Sction V, w propos an input ordr dcision schm using th KL dirgnc as an impromnt of th MLR schm in th cas whr (ii) dos not hold. A. Algorithm V. INPUT ORDR DCISION BASD ON TH KULLBACK-LIBLR DIVRGNC Our scond proposal is an input ordr dcision schm basd on th KL dirgnc, which furthr rducs ANI of th MLR schm in th cas whr th KL dirgnc diffrs from on modality to anothr (i.. whn th condition (ii) in Lmma 1 and Thorm 3 dos not hold). This schm dcids, for ach nroll, th input th KL dirgnc btwn f and g. Aftr nrolls his/hr tmplats, this schm dcids th input ordr (th ordr of modalitis) as follows: 1) Comput th KL dirgnc D for ach modality (1 t T ); 2) Sort th modalitis in dscnding ordr of D. Thn, whn th claimant claims his/hr idntity as, this schm rquirs or rcommnds him/hr to squntially input a qury sampl according to th abo ordr. Altrnatily, it can prsnt th abo ordr to right aftr nrolls ordr using th KL dirgnc. Lt D his/hr tmplats, bcaus f and g can b stimatd right aftr th nrollmnt (as dscribd in Sction IV-A), and so can th KL dirgnc D. Suppos attmpts rification against him/hrslf by squntially inputting his/hr qury sampl according to th abo ordr. Th KL dirgnc D can b writtn, using (19) and (27), as follows: D That is, D = D(f g ) (29) = f (s ) log f (s ) g (s ) ds (30) =. (31) Z can b rgardd as th xpctation of Z attmpts rification against him/hrslf, g to g, and hnc D. Sinc is qual can b rgardd as th xpctation ). Thus, th xpctation of Z tot is of Z min (= Z = Z maximizd at any numbr of biomtric inputs t, by sorting th input ordr in dscnding ordr of D. As a rsult, it can b xpctd that Z tot xcds A with th smallr numbr of biomtric inputs, and ANI is furthr rducd. Not that th abo input ordr can b ithr a rquirmnt or a rcommndation to furthr rduc ANI. In th lattr cas,

9 I TRANSACTIONS ON. INFORMATION FORNSICS AND SCURITY, VOL. 9, NO. 2, FBRUARY th systm allows othr input ordrs, which can b particularly hlpful in th cas whr cannot us som modalitis at th rification tim (du to injury, for xampl). Thorm 1 in Sction IV-C1 also guarants th scurity against wols and lambs, irrspcti of th input ordr. B. stimation of th KL dirgnc W also xplain how to stimat th KL dirgnc D in our input ordr dcision schm. Hr it is important to not that w only ha to comput th ordr rlation btwn th KL dirgncs. In Sction IV-A, w dscribd that it is difficult to corrctly stimat an nroll-spcific gnuin distribution from a small numbr of training sampls. In th sam way, it is difficult to corrctly stimat th KL dirgnc D using a small numbr of gnuin scors from. Howr, sinc w us not th KL dirgncs thmsls but th ordr rlation btwn thm to dcid th input ordr, a littl stimation rror of th KL dirgncs dos not mattr. Furthrmor, Poh t al. 34 showd that th arag of th nroll-spcific gnuin distribution can b mor rliably stimatd than th standard diation. Taking ths mattrs into account, w propos to stimat D by taking th arag of Z = log f (s )/g (s ) or gnuin scors from (rcall that D is th xpctation of Z as shown in (31)), in th cas whr mor than on biomtric sampl can b obtaind from. By doing so, w can xpct that th ordr rlation btwn th KL dirgncs of can b rliably stimatd, whil considring th rcognizability of. In our xprimnts in Sction VI, w dmonstrat th ffctinss of this stimation mthod. A. xprimntal stup VI. XPRIMNTAL VALUATION As dscribd in Sction II-A, zro-ffort wols and lambs cannot b blockd using anti-spoofing masurs such as linss dtction. Thus, it is spcially important to dmonstrat th scurity of our schms against ths animals, using datasts whos subjcts ha no intnt to spoof th systm. To this nd, w aluatd our two schms using th NIST BSSR1 St3 datast 25 and th CASIA-FingrprintV5 datast 24. W usd ths datasts bcaus thy ar frly aailabl and rlatily larg-scal (th formr contains fac scors from 3000 subjcts, and th lattr contains imags of 8 diffrnt fingrs from 500 subjcts, whil many othr datasts contain fwr subjcts 40, 41, 42, 43). W assumd that fac and fingrprint biomtrics ar mutually indpndnt, and cratd 500 irtual subjcts who ha on fac and 8 fingrprints (T = 9), in a similar way to 44, 45. W dscrib this in dtail blow. 1) Datasts: Th NIST BSSR1 St3 datast contains fac scors from 3000 subjcts, ach of whom contributd two qury sampls and on tmplat ( scors in total; thr ar no scors btwn tmplats). Although thr ar scors from two algorithms ( C and G ) in this datast, w usd thos from th algorithm G bcaus som subjcts ha inappropriat scors (th alus -1 ) in C. Th CASIA-FingrprintV5 contains fingrprint imags (lft and right thumb / indx / middl / ring fingr) from 500 subjcts, ach of whom contributd 5 sampls pr fingr. W assumd, for ach fingr, th first sampl as a tmplat, th rmaining four sampls as qury sampls, and computd scors using SourcAFIS Vrsion , a frly aailabl fingrprint matchr ( scors in total, including scors btwn tmplats). W thn randomly slctd 500 subjcts from 3000 subjcts in th fac datast, and cratd 500 irtual subjcts who ha on fac and 8 fingrprints (T = 9). Hr, w trid 100 ways to randomly slct 500 subjcts from 3000 subjcts, and carrid out, for ach st of irtual subjcts, th following xprimnt. 2) xprimnt: From 500 irtual subjcts, w randomly slctd 300 subjcts for aluation (i.. claimants or nrolls), and usd th tmplats of th rmaining 200 subjcts as dummy-tmplats. W thn carrid out an xprimnt whr ach of 300 claimants attmpts rification against ach of 300 nrolls by squntially inputting th last qury sampl of ach modality (w usd th rmaining on fac qury sampl and thr fingrprint qury sampls to stimat th KL dirgnc, as dscribd in dtail in Sction VI-A4). Th numbr of gnuin attmpts was 300, whil th numbr of impostor attmpts was (= ). W aluatd FAR, WAR, LAR, FRR, and ANI in th abo xprimnt, and aragd thm or th 100 sts of irtual subjcts to obtain stabl prformanc. 3) aluatd schms: For comparison, w aluatd th following squntial fusion schms: LR: Th LR schm 19. As dscribd in Sction III-B, it computs a log-liklihood ratio log f /g, whr g is an impostor distribution common to all usrs. W usd th logistic rgrssion modl which stimats log f /g = w 1 s + w 0, whr w 1 and w 0 ar rgrssion cofficints (w dscrib th training mthod in Sction VI-A4). W randomly dcidd th input ordr for ach. MLR: Th MLR schm. As dscribd in Sction IV-B, it computs th minimum of log f /g and log f /g. W usd th logistic rgrssion modl which stimats log f /g = w 1 s + w 0 and log f /g = w 1 s + w 0 (s (20) and (21)). Th input ordr is th sam as that of th LR schm. MLRKL: Th MLR schm using th input ordr dcision schm basd on th KL dirgnc. 4) Training of th rgrssion cofficints and th KL dirgncs: W traind ach of th rgrssion cofficints (w 1, w 0, w 1, w 0, w 1, and w 0 ) using 200 gnuin scors and 200 impostor scors. First, w traind w 1 and w 0 in th LR schm using 200 gnuin scors and 200 impostor scors btwn 200 dummy-tmplats and 200 corrsponding first qury sampls (w randomly slctd on impostor qury sampl for ach dummy-tmplat). Thn, w traind w 1 and w 0 in th MLR schm using th abo 200 gnuin scors and 200 impostor scors btwn dummy-tmplats and th tmplat of (hr w substitutd th first qury sampls for dummy-tmplats in th NIST BSSR1 St3

10 I TRANSACTIONS ON. INFORMATION FORNSICS AND SCURITY, VOL. 9, NO. 2, FBRUARY WAR/LAR% 0.01 MLR(WAR) MLRKL(WAR) MLR(LAR) MLRKL(LAR) Rquird WAR/LAR% Fig. 3. Rlationship btwn WAR/LAR and a rquird WAR/LAR (i.. α = A ) in our proposals. WAR/LAR is lss than α in th gray ara. Log-liklihood Ratio Scor Log-liklihood Ratio Scor Fig. 4. Logarithm of th ratio btwn th frquncy distribution of gnuin scors and that of impostor scors (lft: fac, right: lft thumb). Th width of th intral is 0.5 and 1.0, rspctily. W do not show both sid dgs whr th numbr of gnuin scors or impostor scors is lss than 10. bcaus thr wr no scors btwn tmplats). Similarly, w traind w 1 and w 0 using th abo 200 gnuin scors and 200 impostor scors btwn th qury sampl of V and dummy-tmplats. As a training mthod, w adoptd th Nwton-Raphson mthod 35 which approximats th maximum liklihood stimats sinc thr wr a numbr of (hundrds of) training sampls. As for th KL dirgnc D, w usd a small numbr of gnuin scors from as dscribd in Sction V-B: thr scors for a fingrprint and on scor for a fac. First, w computd, for ach fingrprint, thr gnuin scors btwn th rmaining thr qury sampls (dscribd in Sction VI-A2) and th tmplat. Thn, w stimatd th KL by taking th arag of th stimatd logliklihood ratios log f /g (= w 1 s + w 0 ) or th thr scors. In th sam way, w computd, for ach fac, on gnuin scor btwn th rmaining on qury sampl and dirgnc D th tmplat, and stimatd th KL dirgnc by computing th corrsponding log-liklihood ratio log f /g. B. xprimntal rsults Figur 3 shows th rlationship btwn WAR/LAR and a rquird WAR/LAR (i.. α = A ) in our proposals. It was found that th MLR schm kpt WAR and LAR lss than th rquird alu, irrspcti of th input ordr, as dscribd in Thorm 1 in Sction IV-C1. W considr this is bcaus th logistic rgrssion paramtrs wr corrctly stimatd. W also show in Figur 4 th logarithm of th ratio btwn th frquncy distribution of gnuin scors and Fig. 5. Fig. 6. FAR% FAR/WAR% FAR/LAR% LR MLR MLRKL FRR% Trad-off btwn FRR and FAR in th thr schms LR(WAR/LAR) MLR(WAR/LAR) MLRKL(WAR/LAR) LR(FAR) MLR(FAR) MLRKL(FAR) Arag Numbr of Inputs Arag Numbr of Inputs Trad-off btwn FAR/WAR/LAR and ANI in th thr schms. that of impostor scors (lft: fac, right: lft thumb), obtaind using all th scors btwn qury sampls and tmplats (lft: scors, right: scors). Thr is a clos-to-linar rlationship btwn th log-liklihood ratio and th scor, spcially in th cas of lft thumbs. W also confirmd that a similar tndncy was obtaind for th othr typs of fingrs. Figur 5 shows th trad-off btwn FRR and FAR in th thr schms. It was found that th MLR schm outprformd th LR schm. W considr this is bcaus th MLR schm can proid an optimal trad-off btwn F RP and F AP for any if th input ordr is th sam (s Thorm 2 in Sction IV-C2). It was also found from Figur 5 that th trad-off btwn FRR and FAR was furthr improd by changing th input ordr using our input ordr dcision schm. W considr this is bcaus ANI was significantly improd by using this schm, as shown in th following. Figur 6 shows th trad-off btwn FAR/WAR/LAR and ANI in th thr schms. It was found that th trad-off of th MLR schm was bttr than that of th LR schm and was significantly improd by using th input ordr dcision

11 I TRANSACTIONS ON. INFORMATION FORNSICS AND SCURITY, VOL. 9, NO. 2, FBRUARY TABL II PROPORTION OF ACH MODALITY FOR ACH RANK DCIDD BY TH INPUT ORDR DCISION SCHM %. 1st 2nd 3rd 4th 5th 6th 7th 8th 9th Fac Lft thumb Lft indx Lft middl Lft ring Right thumb Right indx Right middl Right ring schm. W xamind, for ach nroll and ach modality, th stimatd alu of th KL dirgnc, and confirmd that th alu diffrd from on modality to anothr (i.. th condition (ii) did not hold). Thus, w considr th rason th input ordr dcision schm workd ry wll is that th ordr rlation of th KL dirgncs of was rliably stimatd using gnuin scors from, and th xpctation of th liklihood ratio Z tot was maximizd using th ordr rlation, as dscribd in Sction V. For xampl, whn th maximum of WAR and LAR was fixd to 0.01%, ANI of th LR schm, th MLR schm, and th MLR schm using th input ordr dcision schm wr 1.94, 1.72, and 1.19, rspctily. To mor thoroughly instigat th ffctinss of th input ordr dcision schm, w finally xamind, for ach rank (1st, 2nd,, 9th) dcidd by th input ordr dcision schm, th proportion of ach modality. Tabl II shows th rsults. It can b sn that a fac was rankd ry low in most cass, whil th 1st modality was a fingrprint in most cass. This indicats that th KL dirgnc significantly diffrs from on biomtric trait (.g. fac, fingrprint, iris) to anothr. Thus, in th cas of multipl biomtric traits, th optimal input ordr can b common to most nrolls, and finding th ordr can b a triial problm. For xampl, if w dcid th input ordr common to all nrolls by sorting biomtric traits in ascnding ordr of R, th ordr may b optimal for most nrolls. On th othr hand, it can b sn from Tabl II that th typ of th most discriminati fingr diffrd from on nroll to anothr, which indicats finding th optimal input ordr in th cas of multipl units (.g. multipl fingrprints, multipl fingr-ins) is not triial. n in such a cas, th input ordr dcision schm can proid an optimal input ordr for ach nroll, and significantly rduc ANI as shown in Figur 6. Thus, w can conclud that this schm is highly ffcti not for multipl biomtric traits but for multipl units. VII. CONCLUSION In this papr, w first introducd a taxonomy of wols and lambs, dfind scurity masurs for th animals, and proposd th MLR schm as a countrmasur against th animals. W prod its scurity against th animals xcpt for adapti spoofing wols and optimality with rgard to FRP undr th condition (i) (Thorm 1 and 2). W also prod its asymptotic optimality with rgard to ANI undr th condition (i) and (ii) (Lmma 1 and Thorm 3). W finally proposd th input ordr dcision schm basd on th KL dirgnc to furthr rduc ANI in th cas whr (ii) dos not hold. Th xprimntal rsults dmonstratd th scurity of our schms against zro-ffort wols and lambs who cannot b blockd using anti-spoofing masurs such as linss dtction. On th othr hand, our schms ar not scur against adapti spoofing wols as discussd in Sction IV-D, whil anti-spoofing masurs can block spoofing wols and lambs to som xtnt. Thus, it is dsirabl to us our schms in conjunction with anti-spoofing masurs to ha total scurity against wols and lambs. It should b also notd that wols and lambs caus mor srious scurity problms in idntification, as dscribd in Sction II-A. As futur work, w plan to dfin scurity masurs for wols and lambs in idntification, and xtnd our proposals to th idntification scnario. ACKNOWLDGMNT Portions of th rsarch in this papr us th CASIA- FingrprintV5 collctd by th Chins Acadmy of Scincs Institut of Automation (CASIA). APPNDIX A PROOF OF THORM 1 W pro that if th condition (i) holds, w ha F AP, (δ 0 ) A for any V. W can pro that w ha F AP, (δ 0 ) A for any in a similar way by intrchanging V and. Suppos th claimant V attmpts an impostor attack against ach nroll ( ) in th st. Lt Z tot = t τ=1 Z(τ), and S t b a st of scor squncs s t = (s (1),, s ) such that Z tot, (1), Z (t 1) tot < A and Z tot A. That is, S t is a st of st such that th attack of V rsults in succss at th t-th input, in th cas whr only g is usd as an impostor distribution modl. Not that S t is indpndnt of bcaus w assum that V is non-adapti and g (and hnc Z tot) is indpndnt of. Sinc th MLR algorithm δ 0 modls both g and g, and adopts th on which causs lss fals accpts (i.. th minimum alu of Z and Z ), it is not always tru that th attack rsults in succss (i.. Z tot A) in such a scor squnc. Thus, F AP, in (6) can b boundd as follows: F AP, (δ 0 ) t=1 S t p(s t W, )ds t (32)

12 I TRANSACTIONS ON. INFORMATION FORNSICS AND SCURITY, VOL. 9, NO. 2, FBRUARY If (i) holds, Z can b writtn, using (15), (16) and (18), as Z = log p(s W, ) p(s W, ). (33) Thus, for any s t S t, w can dri th following inquality: p(s t t W, ) τ=1 p(s t = p(s(τ) W, ) W, ) t (34) τ=1 p(s(τ) W, ) t = xp (35) = xp Using (32) and (37), w ha T F AP, (δ 0 ) A τ=1 Z (τ) Z tot (36) A. (37) t=1 S t p(s t W, )ds t (38) A. (39) Th last inquality follows from th fact that th probability that ithr of Z tot, (1) (T 1), Z tot or Z (T ) tot xcds (or rachs) A is lss than or qual to 1. APPNDIX B PROOF OF THORM 2 Suppos attmpts rification against him/hrslf. Lt C t 0 and Ct b a st of scor squncs s t such that th rification attmpt rsults in rjct (i.. d = 0) at th t-th input in δ 0 and δ, rspctily. Thn, w ha F RP (δ) F RP (δ 0 ) (40) = t=1 C t p(st W, )ds t p(s t W, )ds t. (41) t=1 C0 t = t=1 C t C p(s t W, )ds t 0 t t=1 C0 t C t p(st W, )ds t. (42) In (42), w xcludd th common st C t C0 t from both trms. If (i) holds, Z can b writtn, using (15), (17) and (19), as follows: Z = log p(s W, ) p(s W, ). (43) Furthrmor, sinc attmpts rification against him/hrslf, g is qual to g, and hnc Z min = Z = Z. Thus, for any s t C0 t, w can dri th following inquality: p(s t W, ) p(s t W, ) = = xp t τ=1 p(s(τ) W, ) t τ=1 p(s(τ) W, ) t = xp τ=1 Z tot Z (τ) (44) (45) (46) < A. (47) Not that this inquality holds for not only th MLR algorithm δ 0 but any squntial fusion algorithm δ, as long as w fix th input ordr. Conrsly, for any s t C 0 t (i.. any scor squnc which rsults in accptanc at th t-th input in th MLR algorithm δ 0 ), w ha p(s t W, ) p(s t W, ) = xp Z tot A. (48) This inquality also holds for any squntial fusion algorithm δ. By (42), (47), and (48), if F AP, (δ) = F AP, (δ 0 ), thn F RP (δ) F RP (δ 0 ) (49) T A t=1 C t C p(s t W, )ds t 0 t t=1 C0 t C t p(st W, )ds t (50) T = A t=1 C t p(st W, )ds t p(s t W, )ds t (51) t=1 C t 0 = A (1 F AP, (δ)) (1 F AP, (δ 0 )) (52) = A F AP, (δ 0 ) F AP, (δ) (53) = 0. (54) In (50), w usd th fact that (47) and (48) hold for any δ. In (51), w addd th common st C t C0 t to both trms. APPNDIX C PROOF OF LMMA 1 Suppos attmpts rification against him/hrslf in any squntial fusion algorithm δ (α, β). If th condition (ii) holds, (27) can b also writtn as follows: D = D(f g ) (55) = f (s ) log f (s ) g (s ) ds (56) =. (57) Z Hr w introduc a random ariabl t which rprsnts th numbr of biomtric inputs rquird to trminat th rification procss. Thn, ANI (δ) can b xprssd as ANI (δ) = t. (58) Sinc Z (1 t t ) is indpndntly distributd with man D (s (57)), Z (1) + + Z (t ) is dcomposd as follows, Z (1) + + Z (t ) = D t. (59) This quation is known as Wald s idntity (or Wald s quation)

13 I TRANSACTIONS ON. INFORMATION FORNSICS AND SCURITY, VOL. 9, NO. 2, FBRUARY , 48. Using (58) and (59), w ha Z (1) + Z (t ) ANI (δ) = (60) D t τ=1 log f (τ) (s (τ) )/g (τ) (s (τ) ) = (61) D log t τ=1 g(τ) (s (τ) )/f (τ) (s (τ) ) = (62) D t log τ=1 g(τ) (s (τ) )/f (τ) (s (τ) ). (63) D Th last inquality follows from Jnsn s inquality 23. Lt C t b a st of scor squncs s t such that th rification attmpt rsults in rjct (i.. d = 0) at th t-th input in δ (in th sam way as th proof of Thorm 2). As max(α, β) 0, th probability that th gnuin attmpts of rsult in accptanc gos to 1. Thn, th xpctation in (63) is takn or th st C1 C T and can b writtn, using (7) and (17), as follows: t τ=1 g(τ) (s (τ) )/f (τ) (s (τ) ) (64) t C f (τ) (s (τ) ) g t (τ) (s (τ) )/f (τ) (s (τ) )ds t τ=1 = t=1 t=1 (65) C t p(st W, )ds t (66) = F AP, (δ) (67) α. (68) It follows from (63) and (68) that inf ANI log α 1 (δ) as max(α, β) 0. (69) δ (α,β) D APPNDIX D PROOF OF THORM 3 Suppos attmpts rification against him/hrslf in th MLR algorithm δ 0. Lt t b a random ariabl which rprsnts th numbr of biomtric inputs rquird to trminat th rification procss (in th sam way as th proof of Lmma 1). Thn, if (ii) holds, ANI (δ 0 ) can b xprssd, in th sam way as (60), as Z (1) + Z (t ) ANI (δ 0 ) =. (70) D Sinc attmpts rification against him/hrslf, if (i) holds, thn g is qual to g, and hnc Z min = Z Z = =. Thus, ANI (δ 0 ) in (70) can b furthr writtn as follows: Z (1) min + ) Z(t min ANI (δ 0 ) = (71) D Z (t ) tot D. (72) As max(α, β) 0, th probability that th gnuin attmpts of rsult in accptanc gos to 1. At th sam tim, sinc A (= log α 1 ) gos to infinity whil th xpctd gain of th minimum log-liklihood ratio Z min (= Z ) at ach input is fixd to D (s (57)), th xcss of Z (t ) tot or th thrshold A bcoms ngligibl compard to A. Thus, w ha Z (t ) tot A (73) = log α 1 (74) as max(α, β) 0. It follows from (72) and (74) that ANI (δ 0 ) log α 1 D as max(α, β) 0. (75) RFRNCS 1 T. Murakami, K. Takahashi, and K. Matsuura, Towards optimal countrmasurs against wols and lambs in biomtrics, in Procdings of th I Fifth Intrnational Confrnc on Biomtrics: Thory, Applications and Systms (BTAS 2012), 2012, pp J. Daugman, Probing th uniqunss and randomnss of iriscods: Rsults from 200 bilion iris pair comparisons, Procdings of th I, ol. 94, no. 11, pp , M. A. Turk and A. P. Ptland, Fac rcognition using ignfacs, in Procdings of 1991 I Confrnc on Computr Vision and Pattrn Rcognition (CVPR 91), 1991, pp A. Jain, L. Hong, and R. Boll, On-lin fingrprint rification, I Transactions on Pattrn Analysis and Machin Intllignc, ol. 19, no. 4, pp , G. Doddington, W. Liggtt, A. Martin, M. Przybocki, and D. Rynolds, Shp, goats, lambs and wols: A statistical analysis of spakr prformanc in th NIST 1998 spakr rcognition aluation, in Procdings of th Fifth Intrnational Confrnc on Spokn Languag Procssing (ICSLP 1998), 1998, pp N. Poh and J. Kittlr, A mthodology for sparating shp from goats for controlld nrollmnt and multimodal fusion, Procdings of Biomtrics Symposium 2008 (BSYM 08), , A biomtric mnagri indx for charactrising tmplat/modlspcific ariation, Procdings of th Third Intrnational Confrnc on Adancs in Biomtrics (ICB 09), pp , N. Yagr and T. Dunston, Th biomtric mnagri, I Transactions on Pattrn Analysis and Machin Intllignc, ol. 32, no. 2, M. N. Tli, J. R. Bridg, P. J. Phillips, and G. H. Gins, Biomtric zoos: Thory and xprimntal idnc, in Procdings of th I/IAPR Intrnational Joint Confrnc on Biomtrics (IJCB), 2011, pp J. Paon and P. J. Flynn, On th consistncy of th biomtric mnagri for iriss and iris matchrs, in Procdings of th 2011 I Intrnational Workshop on Information Fornsics and Scurity (WIFS 11), 2011, pp Z. Wang, A. Srwadda, K. S. Balagani, and V. V. Phoha, Transforming animals in a cybr-bhaioral biomtric mnagri with frog-boiling attacks, in Procdings of th I Fifth Intrnational Confrnc on Biomtrics: Thory, Applications and Systms (BTAS 2012), 2012, pp M. Inuma, A. Otsuka, and H. Imai, Thortical framwork for constructing matching algorithms in biomtric authntication systms, in Procdings of th 3rd Intrnational Confrnc on Biomtrics (ICB 2009), 2009, pp Y. Kojima, R. Shigtomi, M. Inuma, A. Otsuka, and H. Imai, A matching algorithm scur against th wolf attack in biomtric authntication systms, in Procdings of th Joint COST 2101 & 2102 Intrnational Confrnc on Biomtric ID Managmnt and Multimodal Communication (BioID MultiComm 09), 2009, pp N. Poh, Usr-spcific scor normalization and fusion for biomtric prson rcognition, in Adancd Topics in Biomtrics. World Scintific Publishing Company, 2010, ch. 16, pp

14 I TRANSACTIONS ON. INFORMATION FORNSICS AND SCURITY, VOL. 9, NO. 2, FBRUARY N. Poh and J. Kittlr, On th us of log-liklihood ratio basd modlspcific scor normalisation in biomtric authntication, in Procdings of th Intrnational Confrnc on Biomtrics (ICB 07), 2007, pp A. Ross, A. Rattani, and M. Tistarlli, xploiting th Doddington zoo ffct in biomtric fusion, in Procdings of th 3rd I Intrnational Confrnc on Biomtrics: Thory, Applications, and Systms (BTAS 09), 2009, pp I. Arc, Th wakst link risitd, I Scurity & Priacy, ol. 1, no. 2, pp , A. Ross and A. K. Jain, Multimodal biomtrics: An oriw, in Procdings of th 12th uropan Signal Pocssing Confrnc (USIPCO), 2004, pp K. Takahashi, M. Mimura, Y. Isob, and Y. Sto, A scur and usrfrindly multimodal biomtric systm, in Procdings of SPI, ol. 5404, 2004, pp L. Allano, B. Dorizzi, and S. Garcia-Salictti, Tuning cost prformanc in multi-biomtric systms: A nol and consistnt iw of fusion stratgis basd on th squntial probability ratio tst (SPRT), Pattrn Rcognition Lttrs, ol. 31, no. 9, pp , T. Murakami and K. Takahashi, Accuracy impromnt with high conninc in biomtric idntification using multihypothsis squntial probability ratio tst, in Procdings of th First I Intrnational Workshop on Information Fornsics and Scurity (WIFS 09), 2009, pp M. Un, A. Otsuka, and H. Imai, Wolf attack probability : A thortical scurity masur in biomtric authntication systms, IIC Transactions on Information and Systms, ol. 91, no. 5, pp , T. M. Cor and J. A. Thomas, lmnts of Information Thory, Scond dition. Wily-Intrscinc, CASIA-FingrprintV5, 25 NIST Biomtric Scors St - Rlas 1 (BSSR1), itl/iad/ig/biomtricscors.cfm. 26 A. K. Jain, A. Ross, and S. Prabhakar, An introduction to biomtric rcognition, I Transactions on Circuits and Systms for Vido Tchnology, ol. 14, no. 1, pp. 4 20, R. M. Boll, J. H. 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Wald, Som gnralizations of th thory of cumulati sums of random ariabls, Th Annals of Mathmatical Statistics, ol. 16, D. Blackwll, On an quation of Wald, Th Annals of Mathmatical Statistics, ol. 17, pp , Takao Murakami is a rsarchr of th Yokohama Laboratory, Hitachi, Ltd. H rcid th BS dgr and th MS dgr from th Unirsity of Tokyo in 2004 and 2006, rspctily. H joind Hitachi, Ltd. in Sinc thn, h has workd on rsarch and dlopmnt of biomtric authntication systms. H is now a doctoral candidat in Institut of Industrial Scinc, th Unirsity of Tokyo. H rcid th Yamashita SIG Rsarch Award from th Information Procssing Socity of Japan (IPSJ) in H is a mmbr of IIC, IPSJ and I. Knta Takahashi is a rsarchr of th Yokohama Laboratory, Hitachi, Ltd. H rcid th BS dgr, th MS dgr, and Ph.D dgr from th Unirsity of Tokyo in 1998, 2000, and 2012, rspctily. H joind Hitachi, Ltd. in Sinc thn, h has workd on rsarch and dlopmnt of biomtric authntication systms. H rcid th bst papr award from th Information Procssing Socity of Japan (IPSJ) in H is a mmbr of IIC, IPSJ and I. Kanta Matsuura rcid B.ng., M.ng., and Ph.D. dgrs from th Unirsity of Tokyo, Japan, in 1992, 1994, and 1997, rspctily. H is currntly an associat profssor at Institut of Industrial Scinc, th Unirsity of Tokyo. His rsarch intrsts includ cryptography, ntwork scurity, and scurity managmnt. In 2008, h won Distinguishd- Sric Award from th IIC Communications Socity. H srd as an associat ditor of IIC Transactions on Communications btwn 2005 and H has bn sring as th ditor-in-chif of JSSM (Japan Socity of Scurity Managmnt) Journal sinc 2008, and a mmbr of ditorial Board of Dsigns, Cods and Cryptography sinc H was a program co-chair of IWSC2008, and a gnral co-chair of IWSC2011. H is a mmbr of IACR and IPSJ. H is a snior mmbr of ACM, I, and IIC. H is a mmbr of th Board-of-Dirctors of JSSM.