11_Lilit_Mariam.DVI


Mathematical Problems of Computer Science 39, 88{93, 2013.

I nvestigation of E-Capacity for B iometr ic

I denti¯cation P r otocol with Random P ar ameter ¤

Ma r ia m E . H a r o u t u n ia n a n d L ilit A . Te r -V a r d a n ya n

Institute for Informatics and Automation Problems of NAS RA

e-mail: armar@ipia.sci.am, lilit@sci.am

Abstract

In recent years biometrics is widely used in di®erent tasks in the ¯eld of security.
In this paper we investigated the biometric identi¯cation system from an information-
theoretical point of view. We investigate the exponentially high reliability criterion
in biometric identi¯cation systems. The biometric identi¯cation system with random
parameter is considered, which is more realistic for application. The lower and upper
bounds of identi¯cation E-capacity of the model with random parameter for maximal
and average error probabilities are constructed. When E ! 0 we derive the corre-
sponding bounds of the capacity of the biometric identi¯cation system with random
parameter, which coincide and hence, as a corollary we obtain the identi¯cation capac-
ity for this model.

Keywords: Biometric identi¯cation system, identi¯cation capacity, E-capacity
bounds, error exponents, channel with random parameter.

1 . In t r o d u c t io n

In r e c e n t ye a r s b io m e t r ic s is wid e ly u s e d in d i®e r e n t t a s ks in t h e ¯ e ld o f s e c u r it y. B io m e t r ic s
is b e in g u s e d fo r p h ys ic a l a c c e s s c o n t r o l, c o m p u t e r lo g -in , in t e r n a t io n a l b o r d e r c r o s s in g a n d
ID c a r d s , e -p a s s p o r t s , b a s e d o n b io lo g ic a l fe a t u r e s o f a n y p e r s o n , s u c h a s ¯ n g e r p r in t s , a n e ye
ir is [2 ].

In t h is p a p e r we c o n s id e r t h e p r o t o c o l o f b io m e t r ic id e n t i¯ c a t io n . Th e id e n t i¯ c a t io n
s ys t e m wit h r a n d o m p a r a m e t e r is g ive n in Fig .1 . A t yp ic a l p r o t o c o l fo r id e n t i¯ c a t io n c o n s is t s
o f t wo s t e p s : e n r o llm e n t a n d id e n t i¯ c a t io n . D u r in g t h e e n r o llm e n t , t h e b io m e t r ic d a t a o f
M s u b je c t s a r e c a p t u r e d a n d a n a lyz e d , a ft e r t h a t , fo r e a c h in d ivid u a l, a r e c o r d is a d d e d t o
a d a t a b a s e . A p e r fe c t s ys t e m wo u ld a lwa ys r e c o g n iz e a n in d ivid u a l a n d r e je c t a n im p o s t o r .
To b u ild a n id e a l c h a n n e l is im p o s s ib le b e c a u s e b io m e t r ic d a t a a r e g a t h e r e d fr o m in d ivid u a ls
u n d e r e n vir o n m e n t a l c o n d it io n s a n d t h e c h a n n e ls a r e e xp o s e d t o n o is e . S u c h a s ys t e m is n o t
p e r fe c t ly s e c u r e , it le a d s t o s o m e e r r o r s , a n d it c a n b e in fo r m a t io n -t h e o r e t ic a l s e c u r e u p t o
a c e r t a in le ve l. In t h is p a p e r we in ve s t ig a t e d t h e b io m e t r ic id e n t ī c a t io n s ys t e m fr o m a n
in fo r m a t io n -t h e o r e t ic a l p o in t o f vie w. Th e e n r o llm e n t -d a t a in a d a t a b a s e is a n o is y ve r s io n
o f t h e b io m e t r ic a l d a t a c o r r e s p o n d in g t o t h e in d ivid u a l.

¤Research was supported by Armenian national grant 11-1b255.

8 8



M. Haroutunian and L. Ter-Vardanyan 8 9

Fig. 1. Model of biometric identi¯cation system with random parameter

In t h e id e n t ī c a t io n p h a s e a n u n kn o wn in d ivid u a l is o b s e r ve d a g a in a n d a n o t h e r n o is y
ve r s io n o f t h e b io m e t r ic d a t a is c o m p a r e d t o t h e e n r o llm e n t d a t a in t h e d a t a b a s e . Th e
s ys t e m h a s t o c o m e u p wit h a n e s t im a t e o f t h e in d ivid u a l.

W ille m s e t a l [1 ,3 ] in ve s t ig a t e d t h e fu n d a m e n t a l p r o p e r t ie s o f b io m e r t ic id e n t i¯ c a t io n
s ys t e m . It h a s b e e n s h o wn t h a t it is im p o s s ib le t o r e lia b ly id e n t ify m o r e p e r s o n s t h a n
c a p a c it y wh ic h is a n in h e r e n t c h a r a c t e r is t ic o f a n y id e n t i¯ c a t io n s ys t e m . Th e y d e r ive d t h e
c a p a c it y C o f t h is m o d e l.

In [4 ] t h e E-c a p a c it y n e w c o n c e p t fo r b io m e t r ic a l id e n t ī c a t io n s ys t e m wa s in t r o d u c e d .
W e in ve s t ig a t e d t h e e xp o n e n t ia lly h ig h r e lia b ilit y c r it e r io n in b io m e t r ic id e n t ī c a t io n s ys -
t e m s . In o t h e r wo r d s , we in t r o d u c e d a n e w p e r fo r m a n c e c o n c e p t o f b io m e t r ic id e n t i¯ c a t io n
E-c a p a c it y, wh ic h t a ke s in t o a c c o u n t a s t r o n g e r r e qu ir e m e n t o n id e n t i¯ c a t io n fa u lt e ve n t s
wit h e xt r e m e ly s m a ll p r o b a b ilit y ( 2 ¡NE in s t e a d o f " ) . In t e r m s o f p r a c t ic a l a p p lic a t io n s a n
e xp o n e n t ia l d e c r e a s e in e r r o r p r o b a b ilit y ( n a m e ly, in u n wa n t e d id e n t i¯ c a t io n fa u lt s ) is m o r e
d e s ir a b le .

In p r a c t ic e t h e in d ivid u a ls a r e o b s e r ve d in va r io u s p la c e s a n d a t va r io u s t im e s . Th a t is
wh y it is m o r e in t e r e s t in g fr o m t h e p r a c t ic a l p o in t o f vie w t o a s s u m e t h a t t h e c o n s id e r e d
m o d e l c h a n n e ls d e p e n d o n a r a n d o m p a r a m e t e r o f t h e b io m e t r ic m o d e l. In t h is p a p e r
we in ve s t ig a t e t h e b io m e t r ic id e n t i¯ c a t io n s ys t e m wit h r a n d o m p a r a m e t e r , wh ic h is m o r e
r e a lis t ic fo r a p p lic a t io n .

Th e c h a n n e l wit h r a n d o m p a r a m e t e r wit h a d d it io n a l in fo r m a t io n o n t h e e n c o d e r wa s
¯ r s t c o n s id e r e d b y S h a n n o n [5 ] a n d s t u d ie d b y Ge lfa n d S . I. a n d P in s ke r M. S . [6 ]. Th e y
fo u n d t h e c a p a c it y o f t h is c h a n n e l fo r t h e a ve r a g e e r r o r p r o b a b ilit y C in a s it u a t io n wh e n
t h e s t a t e s e qu e n c e is kn o wn a t t h e e n c o d e r . A h ls we d e R . F. [7 ] s h o we d t h a t t h e c a p a c it y fo r
t h e m a xim u m e r r o r p r o b a b ilit y C is t h e s a m e . Th e E-c a p a c it y fo r t h e c h a n n e l wit h r a n d o m
p a r a m e t e r , E-c a p a c it y a n d c a p a c it y fo r t h e m u lt ip le -a c c e s s c h a n n e l wit h r a n d o m p a r a m e t e r
wa s in ve s t ig a t e d b y H a r o u t u n ia n E . A ., H a r o u t u n ia n M. E . [8 ,1 0 ].

Th e c h a n n e l c a n b e c o n s id e r e d in fo u r c a s e s , wh e n t h e s t a t e s e qu e n c e is kn o wn o r u n kn o wn
a t t h e e n c o d e r a n d d e c o d e r . P r o c e e d in g fr o m t h e a p p lic a t io n s in t h e id e n t ī c a t io n p r o t o c o l,
t h e s it u a t io n , wh e n t h e s t a t e s e qu e n c e is u n kn o wn a t t h e e n c o d e r a n d d e c o d e r , is p o s s ib le .

In t h is p a p e r t h e lo we r a n d u p p e r b o u n d s o f t h e id e n t ī c a t io n E-c a p a c it y o f t h e m o d e l
wit h r a n d o m p a r a m e t e r fo r m a xim a l a n d a ve r a g e e r r o r p r o b a b ilit ie s a r e c o n s t r u c t e d . W h e n



9 0 Investigation of E-Capacity for Biometric Identi¯cation Protocol with Random Parameter

E ! 0 we d e r ive t h e c o r r e s p o n d in g b o u n d s o f t h e c a p a c it y o f t h e b io m e t r ic id e n t i¯ c a t io n
s ys t e m wit h r a n d o m p a r a m e t e r , wh ic h c o in c id e a n d h e n c e , a s a c o r o lla r y we o b t a in t h e
id e n t ī c a t io n c a p a c it y fo r t h is m o d e l.

2 . N o t a t io n s a n d D e ¯ n it io n s

L e t X ; Y; Z; S b e ¯ n it e s e t s a n d W b e a fa m ily o f d is c r e t e m e m o r yle s s c h a n n e ls Ws : X ! Y,
wit h a n in p u t a lp h a b e t X a n d a n o u t p u t a lp h a b e t Y. Th e s is t h e c h a n n e l s t a t e , va r yin g
in d e p e n d e n t ly a t e a c h m o m e n t o f t h e c h a n n e l a c t io n wit h t h e s a m e kn o wn P D Q( s) o n S.
Th e r e a r e M in d ivid u a ls a n d e a c h in d ivid u a l h a s a n in d e x m = f 1 ; 2 ; ¢ ¢ ¢ ; Mg. A b io m e t r ic
d a t a s e qu e n c e x( m ) = fx1; x2; ¢ ¢ ¢ ; xN g, wh e r e xn 2 X ; n = 1 ; N c o r r e s p o n d s t o e a c h
in d ivid u a l m . A ll t h e s e s e qu e n c e s a r e s u p o s e d t o b e g e n e r a t e d a t r a n d o m wit h a g ive n
p r o b a b ilit y d is t r ib u t io n

P N ( x ) =
NY

n=1

P ( xn ) ; x 2 X N :

E nr ollment phase. L e t u s h a ve t h e s t a t io n a r y a n d d is c r e t e m e m o r yle s s c h a n n e l
W ( yjx; s) wit h r a n d o m p a r a m e t e r . In t h is p h a s e a ll b io m e t r ic d a t a s e qu e n c e s x ( m) a r e
o b s e r ve d via t h is c h a n n e l. Th e s t a t e o f t h e c h a n n e l is c h a n g e d b y t h e fo llo win g p r o b a b ilit y
d is t r ib u t io n Q( s) , it m e a n s

W N ( yjx; s) =
NY

n=1

W ( ynjxn; sn ) ; QN ( s) =
NY

n=1

Q ( sn ) x 2 X N ; y 2 YN ; s 2 SN :

Th e r e s u lt in g y ( m ) e n r o llm e n t o u t p u t s e qu e n c e s fo r a ll m = f 1 ; 2 ; ¢ ¢ ¢ ; Mg a r e s t o r e d in
t h e d a t a b a s e ( d e ¯ n e it a s YDB ) .

I denti¯cation phase. In t h e id e n t ī c a t io n p h a s e t h e b io m e t r ic d a t a s e qu e n c e o f a n
u n kn o wn in d ivid u a l is o b s e r ve d via t h e s a m e m e m o r yle s s c h a n n e l W ( zjx; s) wit h r a n d o m
p a r a m e t e r .

W N ( zjx; s) =
NY

n=1

W ( ynjxn; sn ) ; z 2 ZN ; x 2 X N ; s 2 SN :

Th e r e s u lt in g id e n t i¯ c a t io n o u t p u t s e qu e n c e z is c o m p a r e d t o t h e s e qu e n c e s y ( m ) , m =
1 ; 2 ; ¢ ¢ ¢ ; M, fr o m t h e d a t a b a s e a n d t h e id e n t i¯ c a t io n fu n c t io n

gN : ZN ! f 0 ; 1 ; 2 ; ¢ ¢ ¢ ; Mg

p r o d u c e s t h e in d e x o f t h e u n kn o wn in d ivid u a l m0 = gN ( z) ; h e r e 0 s t a n d s fo r t h e c a s e , wh e n
t h e u n kn o wn in d ivid u a l h a s n o t b e e n o b s e r ve d b y t h e e n r o llm e n t p h a s e . If t h e s t a t e s e qu e n c e
is u n kn o wn a t t h e e n c o d e r a n d d e c o d e r , le t u s d e n o t e

W ¤ ( yjx ) =
X

s2S
Q( s) W ( yjx; s) ; W ¤ ( zjx) =

X

s2S
Q( s) W ( zjx; s) :

A n d
P ¤ = fP ¤ ( y ) =

X

x

W ¤ ( yjx ) P ( x) ; x 2 X; y 2 Y g;

W ¤ ( zjy ) =
P

x W
¤ ( yjx ) W ¤ ( zjx ) P ( x )

P ¤ ( y )
:



M. Haroutunian and L. Ter-Vardanyan 9 1

Th e c h a n n e l W ¤ : Y ! Z is m e m o r yle s s :

W N ( zjy ) =
NY

n=1

W ( znjyn ) ; z 2 ZN ; y 2 YN :

D e n o t e b y R t h e fo llo win g r a t e

R =
1

N
lo g 2 M:

Th e e r r o r p r o b a b ilit y o f t h e id e n t i¯ c a t io n o f t h e p e r s o n m is

e ( N; m ) = W N ( ZN ng¡1N ( m) jy ( m) ) ;

wh e r e
g¡1N ( m) = fz : gN ( z ) = mg:

W e c o n s id e r t h e maximal a n d t h e aver age er r or pr obabilities

e( N ) = m a x
m2M

e( m ) ;

e ( N ) =
1

M

X

m2M
e( m ) :

Th e E-c a p a c it y fu n c t io n fo r t h e g ive n E > 0 is d e ¯ n e d a s

C ( E; P ¤; W ¤ ) = lim
N!1

1

N
lo g M ( E; P ¤; W ¤; N ) ;

wh e r e
M ( E; P ¤; W ¤; N ) = s u p

gN
fM : eQ ( N ) · e xp ( ¡N E ) g:

W e d e n o t e b y C ( E; P ¤; W ¤ ) t h e E-c a p a c it y fo r t h e a ve r a g e e r r o r p r o b a b ilit y.
W e s h a ll u s e t h e fo llo win g P D in t h e fo r m u la t io n o f r e s u lt s :

P = fP ( y ) ; y 2 Yg;

V = fV ( zjy ) ; z 2 Z; y 2 Yg:
Fo r in fo r m a t io n -t h e o r e t ic qu a n t it ie s , s u c h a s e n t r o p y HP ( Y ) , m u t u a l in fo r m a t io n

IP;V ( Z ^ Y ) , t h e d ive r g e n c e D ( V jjWjP ) a n d fo r t h e n o t io n o f t h e t yp e we r e fe r t o [9 ]{ [1 5 ].

3 . Fo r m u la t io n o f t h e R e s u lt

To d e ¯ n e t h e lo we r b o u n d ( random coding bound ) o f t h e id e n t ī c a t io n o f E-c a p a c it y fo r t h e
c h a n n e l wit h r a n d o m p a r a m e t e r le t u s d e n o t e :

Rr ( E; P
¤; W ¤ )

4
=

4
= m in

P;V :D(P ±V jjP ¤±W ¤)·E

¯̄
¯̄IP;V ( Z ^ Y ) + D ( P ± V jjP ¤ ± W ¤ ) ¡ E

¯̄
¯̄
+

: ( 1 )

Fo r t h e fo r m u la t io n o f t h e u p p e r b o u n d ( sphere packing bound ) o f t h e in d e n t i¯ c a t io n o f E-
c a p a c it y le t u s in t r o d u c e t h e fo llo win g fu n c t io n :

Rsp ( E; P
¤; W ¤ )

4
= m in

P;V :D(P ±V jjP ¤±W ¤)·E
IP;V ( Z ^ Y ) : ( 2 )



9 2 Investigation of E-Capacity for Biometric Identi¯cation Protocol with Random Parameter

T heor em. F or the biometric identi¯cation system with random parameter for the given
P ¤; W ¤ and for all E > 0

Rr ( E; P
¤; W ¤ ) · C ( E; P ¤; W ¤ ) · C ( E; P ¤; W ¤ ) · Rsp ( E; P ¤; W ¤ ) :

Th e p r o o f o f t h e t h e o r e m is s im ila r t o t h e p r o o f e xp o s e d in [4 ].
Cor ollar y. W hen E ! 0 we derive the lower and upper bounds capacity of the channel

with random parameter, which coincide and hence, we obtain the capacity

C = IP ¤;W ¤ ( Z ^ Y ) :

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M. Haroutunian and L. Ter-Vardanyan 9 3

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Submitted 22.11.2012, accepted 14.02.2013.

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áÉáñïÇ ï³ñµ»ñ ËݹÇñÝ»ñáõÙ: ²ß˳ï³Ýùáõ٠ѻﳽáïí»É ¿ Ï»Ýë³ã³÷³Ï³Ý
ÝáõÛݳϳݳóÙ³Ý Ñ³Ù³Ï³ñ·Á‘ ÇÝýáñÙ³óÇáÝ-ï»ë³Ï³Ý ï»ë³ÝÏÛáõÝÇó: Ø»Ýù Ñ»ï³-
½áïáõÙ »Ýù óáõóã³ÛÇÝ µ³ñÓñ Ñáõë³ÉÇáõÃÛ³Ý ã³÷³ÝÇßÁ‘ Ï»Ýë³ã³÷³Ï³Ý ÝáõÛݳϳ-
ݳóÙ³Ý Ñ³Ù³Ï³ñ·»ñáõÙ: ¸Çï³ñÏí³Í ¿ å³ï³Ñ³Ï³Ý å³ñ³Ù»ïñáí Ï»Ýë³ã³÷³Ï³Ý
ÝáõÛݳϳݳóÙ³Ý Ñ³Ù³Ï³ñ·Á, áñÁ ÏÇñ³éáõÃÛáõÝÝ»ñÇ ï»ë³Ï»ïÇó ³í»ÉÇ Çñ³ï»ë³Ï³Ý
¿: γéáõóí³Í »Ý E- áõݳÏáõÃÛ³Ý í»ñÇÝ ¨ ëïáñÇÝ ·Ý³Ñ³ï³Ï³ÝÝ»ñÁ ³é³í»É³·áõÛÝ
¨ ÙÇçÇÝ ë˳ÉÇ Ñ³í³Ý³Ï³ÝáõÃÛáõÝÝ»ñÇ ¹»åùáõÙ å³ï³Ñ³Ï³Ý å³ñ³Ù»ïñáí Ùá¹»ÉÇ
ѳٳñ: ºñµ E ! 0 , ëï³ÝáõÙ »Ýù å³ï³Ñ³Ï³Ý å³ñ³Ù»ïñáí Ï»Ýë³ã³÷³Ï³Ý
ÝáõÛݳϳݳóÙ³Ý Ñ³Ù³Ï³ñ·Ç áõݳÏáõÃÛ³Ý Ñ³Ù³å³ï³ëË³Ý ·Ý³Ñ³ï³Ï³ÝÝ»ñÁ,
áñáÝù ѳÙÁÝÏÝáõÙ »Ý ¨, áñå»ë ѻ勉Ýù, ëï³ÝáõÙ »Ýù ³Û¹ Ùá¹»ÉÇ ÝáõÛݳϳݳóÙ³Ý
áõݳÏáõÃÛáõÝÁ:

Èññëåäîâàíèå E-ïðîïóñêíîé ñïîñîáíîñòè äëÿ ïðîòîêîëà
áèîìåòðè÷åñêîé èäåíòèôèêàöèè ñî ñëó÷àéíûì ïàðàìåòðîì

Ì. Àðóòþíÿí è Ë. Òåð-Âàðäàíÿí

Àííîòàöèÿ

Çà ïîñëåäíèå ãîäû áèîìåòðèêà øèðîêî èñïîëüçóåòñÿ â ðàçëè÷íûõ çàäà÷àõ â
ñôåðå áåçîïàñíîñòè.  äàííîé ðàáîòå ìû èññëåäîâàëè ñèñòåìó áèîìåòðè÷åñêîé
èäåíòèôèêàöèè ñ èíôîðìàöèîííî-òåîðåòè÷åñêîé òî÷êè çðåíèÿ. Ìû èññëåäóåì
êðèòåðèé ýêñïîíåíöèàëüíî âûñîêîé íàäåæíîñòè â áèîìåòðè÷åñêèõ ñèñòåìàõ
èäåíòèôèêàöèè. Ðàññìîòðåíà ñèñòåìà áèîìåòðè÷åñêîé èäåíòèôèêàöèè ñî
ñëó÷àéíûì ïàðàìåòðîì, êîòîðàÿ áîëåå ðåàëèñòè÷íà ñ òî÷êè çðåíèÿ ïðèëîæåíèé.
Ïîñòðîåíû íèæíÿÿ è âåðõíÿÿ ãðàíèöû äëÿ E-ïðîïóñêíîé ñïîñîáíîñòè ìîäåëè
ñî ñëó÷àéíûì ïàðàìåòðîì äëÿ ìàêñèìàëüíîé è ñðåäíåé âåðîÿòíîñòè îøèáêè.
Êîãäà E ! 0 ìû ïîëó÷àåì ñîîòâåòñòâóþùèå îöåíêè ïðîïóñêíîé ñïîñîáíîñòè
äëÿ ñèñòåìû áèîìåòðè÷åñêîé èäåíòèôèêàöèè ñî ñëó÷àéíûì ïàðàìåòðîì,
êîòîðûå ñîâïàäàþò è, êàê ñëåäñòâèå, ìû ïîëó÷àåì ïðîïóñêíóþ ñïîñîáíîñòü
èäåíòèôèêàöèè äëÿ ýòîé ìîäåëè.