![](bg5.png)
from the pool. For others take random k points
from the vault.
(c) Find the (k− 1)
st
degree polynomialthat passes
through the selected points. Add the polyno-
mial to the list if it is not already in it.
(d) Check the vault if there are any other points that
lie on the polynomial. Decrement r by the num-
ber of points on this polynomial.
(e) Pick r points from the pool (or the remaining
points if their number is less than r) and eval-
uate these points on the fake polynomial and
place the resulting values in the fuzzy vault.
With the proposed chaff placement method, we al-
low each polynomial intersect with other polynomials
in at least k vault points which increases the maximum
number of polynomials we can embed into the vault.
Note that any two polynomials cannot intersect with
each other in more than k−1 points. As a result of our
experiments in our setting described above, we are
able to hide around 30 fake polynomials in the vault.
Therefore, this method decreases the probability of
finding the secret polynomial using Mihailescu’s at-
tack from 100% to approximately 3.3% after the brute
force attack is applied. Due to the fact that most of the
identification applications allow only limited number
of trials, the proposed method enhance the security
considerably. Moreover, the method does not affect
the false accept or false reject rates since the match-
ing algorithm considers only the x coordinates of the
points and this method changes only the y coordi-
nates.
4.3 Security Analysis
The attacks on the fuzzy vault scheme, mostly assume
the interception of a vault from a database. The basic
attack is the brute force attack over a single vault. The
analysis in this work based on the work of Mihailescu
(Mihailescu, 2007), shows that this attack is not com-
putationally infeasible, therefore fuzzy vault scheme
is insecure without additional security.
If an attacker intercepts a vault, but has no other
information about the locations of the genuine points,
the best method to recover the secret polynomial is
the brute force trials (Clancy et al., 2003)(Mihailescu,
2007). Mihailescu providesa strong brute force attack
in (Mihailescu, 2007), which finds the secret polyno-
mial in less than 8(Ck)(C/n)
k
operations where C is
the number of points in the vault, n is the number of
genuine points in the vault and the degree of the secret
polynomial is k− 1.
In our tests n parameter is on the average 35
and k is constant 10. For C = 300, which gives a
better FRR, breaking the system requires 8 × 300×
10 × (300/35)
10
≈ 2
46
operations. For C = 350,
which gives a worse FRR, the system provides a bet-
ter security; breaking the system requires this time
8× 350 × 10× (350/35)
10
≈ 2
48
operations.
Without the use of the proposed method in section
4.2, the secret polynomial is found with probability
1 after this attack. However, our proposed method
decrease the probability to approximately 0.03 since
the polynomial found as a result of brute force attack
is not guaranteed to be the secret polynomial.
5 TEST RESULTS
We implement polynomial reconstruction phase using
two previously discussed approaches: 1) brute force
method and 2) RS decoding.
For the implementations, we use a database of
180 people where there are two fingerprint images
for each finger, totaling 360 fingerprints. The first
180 fingerprint images are used for enrollment and
the second 180 images are used for verification of the
corresponding fingerprints. Later, all fingerprints are
cross-tested for false accept rates. In the experimental
setting bitmap images of 500× 500 pixels are created
for each fingerprint.
All computations and tests are performed on a
computer with 1.7 GHz Intel Celeron M processor
and 448MB of RAM. The codes are developed in ei-
ther Matlab or C++ (Microsoft Visual Studio) depend-
ing on the nature of the problem.
We basically investigate two issues; firstly, the ef-
fects of the vault and threshold sizes on the perfor-
mance and security of the fuzzy vault, and secondly
time efficiencies of two methods used in the polyno-
mial reconstruction phase of the verification stage.
5.1 Effects of Vault and Threshold Sizes
The false reject rates (FRR) and false accept rates
(FAR) are calculated in four settings where different
values for vault size and minimum distance threshold
are used for our database of fingerprints. We use vault
sizes of 300 and 350 points and minimum distance
thresholds of (t = 15) and (t = 18). The minimum
distance between any two chaff points is taken as 8.
We calculate the FAR and FRR results for both the
brute force and the RS decoding methods.
The FAR rates turn out to be 0% in all settings
after cross testing all fingerprint images with different
fingers in four settings.
Table 3 shows the FRRs for different vault sizes
and threshold values. The results clearly demonstrate
IMPROVED FUZZY VAULT SCHEME FOR FINGERPRINT VERIFICATION
41