Symmetric hash functions for fingerprint minutiae

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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 information
Wish to use similar functions for fingerprint
data
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Obstacles in finding fingerprint hash functions
Fingerprint space
Hash space
f1
h(f1)
h
f2
h(f2)

• 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

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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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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)

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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.
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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.
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Background on complex numbers(1)
Point in 2-dimensional plane can be represented
as a complex number
( is an element satisfying )
Adding complex number to all points
in the complex plane results in a
parallel shift of the plane by vector
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Background on complex numbers (2)
Polar representation of complex numbers
Denote - magnitude of
Then
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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
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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.
Multiplying by r means rotating by angle and
scaling by factor .
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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
.
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Hash functions of minutia points
Consider following functions of minutia positions
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?
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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.
Denote
Thus
And r,t can be calculated given
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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
The difference between two parts of above
equation can serve as a confidence measure for
matching two sets of minutia points.
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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 big changes of symmetric
  functions of higher orders. Thus we limited
  ourselves to the symmetric functions of 1st and
  2nd orders.

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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

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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.

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Fingerprint Matching Algorithm(1)

• Enrollment
• For each triplet generated let (c1,c2,c3) and
  (d1,d2,d3) be the locations and directions of the
  minutia
• Compute hash functions
• h1 (c1 c2 c3)/3
• g1 (d1 d2 d3)/3
• h2 (c12 c22 c32)/3
• g2 (d12 d22 d32)/3
• 3. Store 4 values (h1, h2, g1, and g2)
  corresponding to each triplet in the database.

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Fingerprint Matching Algorithm

• Matching
• Compute hash functions (h1, h2, g1, and g2)
  for all local triplets in the test fingerprint
• For each pair of local hash value sets find the
  distance of match
• 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

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Experimental results
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 tests
and 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.

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Thank you !

• References
•  Davide Maltoni, Dario Maio, Anil K. Jain and
  Salil Prabhakar, Handbook of Fingerprint
  Recognition, Springer-Verlag, New York, 2003
• 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)
• 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.
• 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.

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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.

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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.