fingerprint hash

8/6/2019 Fingerprint Hash

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Symmetric hash functions for

fingerprint minutiae

S. Tulyakov, F. Farooq and V. Govindaraju

Center for Unified Biometrics and Sensors

SUNY at Buffalo, New York, USA


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Securing password information

It is impossible to learn the original password given stored

hash value of it.


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Securing fingerprint informationWish to use similar functions for fingerprint data:


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Obstacles in finding fingerprint hashfunctions

Since match algorithm will work with the values of hash functions, similar fingerprints should have similar hash values

rotation and translation of original image should not have big

impact on hash values

partial fingerprints should be matched

Fingerprint space Hash space

hf1

f2h(f1)

h(f2)


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Existing Approaches Davida, Frankel, Matt (1998)

- use error correcting codes, features should be

ordered

Biometric encryption (Soutar et al., 1998)- use filters for Fourier transform of fingerprint image

- translation is accounted for, but not rotation

(example of such

filters for face

verification)


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Minutia points of the fingerprint

Minutia points - points where

ridge structure changes: end

of the ridge and branching ofthe ridge.

The positions of minutia

points uniquely identifies the

fingerprint.


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Assumptions on minutiae setsextracted from the same finger

Assume that two fingerprints originating from one finger

differ by scale and rotation. Thus the set of minutia points

of one fingerprint image can be obtained from the set of

minutia points of another fingerprint image by scaling and

rotating.


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Background on complex

numb

ers(1)

ibyax )()(

Point in 2-dimensional plane can be represented as a complex

number: ( is an element satisfying )yixyx pn),( i 12 !i

Adding complex number to all points in the

complex plane results in a parallel shift of the plane by vector

bia yix

),( ba

yix

ibyax

biayix

)()(

)()(

!


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numb

ers (2)Polar representation of complex numbers:

!! iyx

y

yx

xyxyixz

2222

22

N

||z

yixz!

y

x

Denote - magnitude of22 yxz ! z

Then )sin(cos UU iziz

y

z

xzz !

!


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numb

ers (3)Multiplying all points in the complex plane by some complex

number results in a rotation around origin

by angle and scaling by factor

z

)sin(cos NN irr !N || r

))sin()(cos(||||

))sincoscos(sinsinsincos(cos||||

)sinsinsincoscossincos(cos||||

)sin(cos||)sin(cos||

2

NUNU

UNUNUNUN

UNUNUNUN

UUNN

!

!

!

v!v

izr

izr

iiizr

izirzr


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Transformation of minutiae setIf we represent minutia points as points on a complex plane,

then scaling and rotation can be expressed by function:

where is the complex number determining

rotation and scaling, and is the complex number

determining translation of minutia point.

trzzf !)(

N||z

|||| rz v

z

rzt

rz

t

)sin(cos|| NN irr !

Multiplying by r means

rotating by angle and

scaling by factor .

N

|| r

rt


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Transformation function

If is a set of minutia points of first

fingerprint and is a set of minutia points of

second fingerprint (same finger), then we assume that there

is a transformation such that

for any .

trzzf !)(

),,,( 21 nccc -),,,( 21 nccc ddd -

trccfc iii !!d )(

ni -,1!

trzzf !)(


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Hash functions of minutia points

Consider following functions of minutia positions:

m

n

mm

nm

nn

nn

cccccch

cccccch

cccccch

!

!

!

--

/

--

--

2121

22

2

2

1212

21211

),,,(

),,,(

),,,(

The values of these symmetric functions do not depend on

the order of minutia points.


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Hash functions of transformed minutiae

What happens with hash functions if minutia point set is transformed?

ntcccrhntcccr

trct

rct

rc

cccccch

nn

n

nn

!!

!

ddd!ddd

),,,()()()()(

),,,(

21121

21

21211

--

-

--

2

211212

2

2

21

22

2

2

1

2

22

2

2

1

22

2

2

1212

),,,(2),,,(

)(2)(

)()()(

),,,(

ntcccrthccchr

ntcccrtcccr

trctrctrc

cccccch

nn

nn

n

nn

!

!

!

ddd!ddd

--

--

-

--


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Finding transformation parameters from hash

function values

Thus can be expressed as a linear

combinations of with coefficients

depending on transformation parameters r and t.

),,,( 21 ni ccch ddd -ijccch

nj e),,,,( 21 -

Denote:

),,,( 21 nii ccchh ddd!d -

),,,( 21 nii ccchh -!

ntrhh !d 112

12

2

2 2 ntrthhrh !d

Thus

And r,t can be calculated given 2121 ,,, hhhhdd


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Verifying fingerprint match using hash functions

When and are found we can use higher order hash

functions to check if fingerprints match.

For example, if extracted minutia set is identical to the stored in

the database, then for the hash function of third order we should

get:3

1

2

2

2

3

3

3 33 nthrtthrhrh !d

The difference between two parts of above equation can serve

as a confidence measure for matching two sets of minutia

points.

r t


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Practical considerations for matching

localized hash values

Since direction of the minutia (direction of the ridge where

minutia is located) is also important in fingerprint matching, we

consider unit direction vectors and same hash functions of that

vectors (associating direction vector with complex number)

The small changes in locations of minutia points result in bigchanges of symmetric functions of higher orders. Thus we limited

ourselves to the symmetric functions of 1st and 2nd orders.

i

n

ii

ni ddddddg ! -- 2121 ),,,(


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Matching Localized SubsetsSince it is rare that two fingerprint images contain exactly sameminutia points, we consider subsets of minutia points.

To ensure privacy we must have less symmetric functions than

points in the minutia subset.

Consider two subsets of 3 minutiae points & 2 functions

The distfunction provides a goodness of match between thesubsets


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Goodness ofMatch

For all local subsets find how many subsets are matched and

whether values of and are similar.For each minutiae point, find the 3 nearest neighbors and form 3

triplets that always include the initial minutia.

r t

|2|

||

|2|

||),,,,,(

2

12

2

24

113

2

12

2

22

111

ntrtggrg

ntrgg

ntrthhrh

ntrhhtrgghhdist iiii

d

d

d

d!dd

E

E

E

E


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Fingerprint Matching Algorithm(1)

Enrollment:1. For each triplet generated let (c1,c2,c3) and (d1,d2,d3) be

the locations and directions of the minutia

2. Compute hash functionsh1 = (c1 + c2 + c3)/3

g1 = (d1 + d2 + d3)/3h2 = (c1

2 + c22 + c3

2)/3

g2 = (d12

+ d22

+ d32

)/3

3. Store 4 values (h1, h2, g1, and g2) corresponding to eachtriplet in the database.


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Fingerprint Matching Algorithm Matching:

1. Compute hash functions (h1, h2, g1, and g2) for alllocal triplets in the test fingerprint

2. For each pair of local hash value sets find the distanceof match

3. Note that tcan be derived from the match between hand h and establishing the pivot. For a given t,search several quantized r

|2|

||

|2|

||),,,,,(

212

224

113

2

12

2

22

111

ntrtggrg

ntrgg

ntrthhrh

ntrhhtrgghhdist iiii

d

d

d

d!dd

E

E

E

E


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Experimental results

0

2 0

4 0

6 0

8 0

10 0

0 2 4 6 8 10 12

F a l se P o si t i v e

P la in M at c h ing Co. 1 Co. 2 Co. 3

Co.3: 3 pts and 2 hash fns

ERR = 3%

Original no-hash matching

ERR = 1.7%

Co.2: 3 pts and 1 hash fn

Co.1: 2 points and 1 hash fn

tested on FVC2002 set,

with 2800 genuine testsand 4950 impostor tests


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Algorithm limitations Different local minutia sets can have same hash value sets. Thus

the expected performance of the algorithm is lower than the

performance of the matching algorithm using all available

fingerprint information. Usually there are less matching hash values than matching

minutiae. This means bigger difficulty in producing good match

score, and setting match thresholds.

1c

2c

3c

1cd 2cd

3cd

1h1hd3 matching minutiae

can result in only one

matching hash pair


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Thank you !References

1. Davide Maltoni, Dario Maio,Anil K. Jain and Salil

Prabhakar, Handbook of FingerprintRecognition, Springer-

Verlag, New York, 20032. Colin Soutar, Danny Roberge,Alex Stoianov, Rene Gilroy

and B.V.K. Vijaya Kumar, Biometric Encryption, in ICSA

Guide to Cryptography, R.Nichols, ed. (McGraw-Hill, 1999)

3. G.I. Davida, Y. Frankel, and B.J. Matt. On enabling secure

applications through offline biometric identification. InIEEESymposium on Privacy andSecurity, 1998.

4. Tsai-Yang Jea, Viraj S. Chavan, Venu Govindaraju and John

K. Schneider, Security and matching of partial fingerprint

recognition systems, In SPIEDefense andSecurity

Symposium, 2004.


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Security

If the number of stored hash functions is less thanthe number of minutia points, it is not possible to find thepositions of minutia points from local hash values.

Using system of hash equations is difficult, since it is notknown which minutia correspond to particular hash value.

ih

jc

x

ox x

x

ox

o

x

x

x o

x

x

oxx

x

o

x

x

(a) (b)

(c)


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