symmetric hash functions for fingerprint minutiae s. tulyakov, f. farooq and v. govindaraju center...
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Symmetric hash functions for fingerprint minutiae
S. Tulyakov, F. Farooq and V. GovindarajuCenter for Unified Biometrics and Sensors
SUNY at Buffalo, New York, USA
Securing password information
It is impossible to learn the original password given stored hash value of it.
Securing fingerprint information
Wish to use similar functions for fingerprint data:
Obstacles in finding fingerprint hash functions
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
f2
h(f1)h(f2)
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)
Existing Approaches
•Ratha, Connell, Bolle (2001)- polynomial transform; need alignment.
• Juels, Sudan (2002)- map points to the values of error correcting codes , introduce variation by adding some other points.
Follow-ups:• Clancy et al, 2003• Uludag, Jain (2004)
Minutia points of the fingerprint
Minutia points - points where ridge structure changes: end of the ridge and branching of the ridge.
The positions of minutia points uniquely identifies the fingerprint.
Assumptions on minutiae sets extracted 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.
Background on complex numbers(1)
ibyax )()(
Point in 2-dimensional plane can be represented as a complex number: ( is an element satisfying )yixyx ),( 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
)()(
)()(
Background on complex numbers (2)
Polar representation of complex numbers:
i
yx
y
yx
xyxyixz
2222
22
|| z
yixz
y
x
Denote - magnitude of 22|| yxz z
Then )sin(cos||||||
|| iziz
y
z
xzz
Background on complex numbers (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|| irr
|| r
))sin()(cos(||||
))sincoscos(sinsinsincos(cos||||
)sinsinsincoscossincos(cos||||
)sin(cos||)sin(cos||2
izr
izr
iiizr
izirzr
Transformation of minutiae set
If 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 )(
|| z
|||| rz
z
rztrz t
)sin(cos|| irr
Multiplying by r means rotating by angle and scaling by factor .
|| r
rt
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
trccfc iii )(ni ,1
trzzf )(
Hash functions of minutia points
Consider following functions of minutia positions:
mn
mmnm
nn
nn
cccccch
cccccch
cccccch
2121
222
21212
21211
),,,(
),,,(
),,,(
The values of these symmetric functions do not depend on the order of minutia points.
Hash functions of transformed minutiae
What happens with hash functions if minutia point set is transformed?
ntcccrhntcccr
trctrctrc
cccccch
nn
n
nn
),,,()(
)()()(
),,,(
21121
21
21211
2211212
2
221
222
21
2
222
21
222
21212
),,,(2),,,(
)(2)(
)()()(
),,,(
ntcccrthccchr
ntcccrtcccr
trctrctrc
cccccch
nn
nn
n
nn
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 ijccch nj ),,,,( 21
Denote:
),,,( 21 nii ccchh ),,,( 21 nii ccchh
ntrhh 112
122
2 2 ntrthhrh Thus
And r,t can be calculated given 2121 ,,, hhhh
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:
31
22
23
33 33 nthrtthrhrh
The difference between two parts of above equation can serve as a confidence measure for matching two sets of minutia points.
r t
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 big changes of symmetric functions of higher orders. Thus we limited ourselves to the symmetric functions of 1st and 2nd orders.
in
iini ddddddg 2121 ),,,(
Matching Localized Subsets•Since it is rare that two fingerprint images contain exactly same minutia 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 dist function provides a “goodness” of match between the subsets
Goodness of Match
• 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|
||),,,,,(
212
224
113
212
222
111
ntrtggrg
ntrgg
ntrthhrh
ntrhhtrgghhdist iiii
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 functions h1 = (c1 + c2 + c3)/3
g1 = (d1 + d2 + d3)/3h2 = (c1
2 + c22 + c3
2)/3
g2 = (d12 + d2
2 + d32)/3
3. Store 4 values (h1, h2, g1, and g2) corresponding to each triplet in the database.
Fingerprint Matching Algorithm• Matching:
1. Compute hash functions (h1’, h2’, g1’, and g2’) for all “local” triplets in the test fingerprint
2. For each pair of local hash value sets find the distance of match
3. Note that t can be derived from the match between h and h’ and establishing the pivot. For a given t, search several quantized r
|2|
||
|2|
||),,,,,(
212
224
113
212
222
111
ntrtggrg
ntrgg
ntrthhrh
ntrhhtrgghhdist iiii
Experimental results
0
20
40
60
80
100
0 2 4 6 8 10 12
False Positive
Plain Matching 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 fnCo.1: 2 points and 1 hash fn
tested on FVC2002 set, with 2800 genuine tests and 4950 impostor tests
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.
1c2c
3c
1c 2c
3c
1h1h
3 matching minutiae can result in only one matching hash pair
Thank you ! References1. Davide Maltoni, Dario Maio, Anil K. Jain and Salil
Prabhakar, Handbook of Fingerprint Recognition, Springer-Verlag, New York, 2003
2. 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”. In IEEE Symposium on Privacy and Security, 1998.
4. Tsai-Yang Jea, Viraj S. Chavan, Venu Govindaraju and John K. Schneider, “Security and matching of partial fingerprint recognition systems”, In SPIE Defense and Security Symposium, 2004.
Security
• If the number of stored hash functions is less than the number of minutia points, it is not possible to find the positions of minutia points from local hash values.
• Using system of hash equations is difficult, since it is not known which minutia correspond to particular hash value.
ih
jc
xox x
xo
xo
x
x
x ox
x
oxx
x
ox
x
(a) (b)
(c)
Cancelable Fingerprint Templates
• If fingerprint database is compromised, the different set of symmetric hash functions should be chosen. It can be any function set, constituting a basis in the set of symmetric polynomial functions of order less than • Also, different set of hash functions can be chosen for each individual, resulting in cancelable fingerprint templates.
m
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