2013 6th International Congress on Image and Signal Processing (CISP 2013)
A Delaunay Triangle Group Based Fuzzy Vault with Cancellability Wencheng Yang Jiankun Hu*
Song Wang
School of Engineering and Information Technology, University of New South Wales at Canberra Canberra, Australia
[email protected];
[email protected]
Department of Electronic Engineering La Trobe University Victoria, Australia
[email protected]
Abstract—Lack of the capability of cancellability in a biometric cryptosystem makes it impossible to be deployed broadly in real applications. This is because the compromise of a template in one application often means its loss in all other applications. To solve this problem, we equip a minutiae-based fuzzy vault with cancellability by applying a polar transformation to each Delaunay triangle group so as to enhance the security level of the biometric template. The unit of the non-invertible polar transformation is a triangle as a whole instead of a single minutiae point which makes the proposed method to be more stable and insensitive to non-linear distortion. Experimental results on public fingerprint databases FVC2002 DB1 and DB2 demonstrate the valid of the proposed method. Keywords-cancellability; fingerprint; biometrics; template; biocryptography; Denaulay triangle; fuzzy vault; security; polar stransformation.
I.
INTRODUCTION
Biometric cryptography is an emerging technology which combines the advantages of both biometrics and cryptography. A biometric cryptosystem can either release or directly generate a secret key from unique biometric features of an individual, in place of a traditional cryptosystem that relies on passwords and/or tokens. The application of a biometric cryptosystem solves the conflict between the accuracy requirement of a cryptosystem and the uncertainty of biometric features in a biometric system. For instance, a cryptosystem cannot tolerate even one bit of error, while the biometric features tend to be uncertain caused by noisy data, distortion and rotation in the biometric image capture process. Roughly, two categories are separated in biometric cryptosystems [1-6], namely, key binding systems and key generation systems. In a key binding system, a secret key is technically bound with the biometric features, of which the security is provided by secure sketches, e.g., fuzzy commitment, fuzzy vault. A fuzzy commitment sketch, which is the first such encryption method for biometrics, was proposed by Juels et al. [7]. In this method, at the enrollment stage, a secret binary string S is encrypted by the error correction code (ECC) to generate a codeword C=enc(S) which is further bound with a biometric template X to output a helper data H = X ⊕ C , where ⊕ is the XOR operation. At the authentication stage, a biometric query X ' is extracted to retrieve the secret S with the help of H. After
C ' = X ' ⊕ H is computed, S ' is obtained by C ' and the decoding algorithm of ECC, as S ' = dec(C ' ) . S ' would be the
same as S if and only if the difference between X and X ' is smaller than a certain error correction threshold. However, one drawback of this method is that it requires the biometric representation X and X ' to be ordered so that their correspondence can be distinct. This requirement is easy to be achieved on some biometric traits, e.g., Iris and face, but hard to be fulfilled on some other biometrics, e.g., fingerprint, which is represented by a set of unordered minutiae points. To overcome this shortage of a fuzzy commitment scheme, Juels et al.[8], proposed a fuzzy vault scheme which can protect the biometric features that are represented as an unordered set, e.g., fingerprint minutiae set. In a fuzzy vault scheme, a secret key which needs protection is divided and embedded into a polynomial P(x) as its coefficients, where x is the variable from the biometric template feature set M T = {xi }in=1 1 with n1 elements. The genuine point set G = {xi , P ( xi )}in=1 1 that relies on P(⋅) is concealed by a set of chaff points
CF = {( x j , y j ) | x j ≠ xi , ∀i = 1,..., n1 , y j ≠ P( x j )}nj2=1 that do not rely on P(⋅) , where n2 is the number of chaff points. The union of the genuine point set G and the chaff point set CF establish the secure vault V. If without the presence of the genuine biometric user, it is computationally difficult for an adversary to identify the genuine points in the vault V. In this way, the biometric template can be kept safely. At the authentication stage, a genuine user provides his/her biometric query M Q which can be used to identify the genuine points from the vault V that lie on the polynomial P(⋅) . If enough genuine points can be retrieved, the polynomial P(⋅) can be reconstructed and the authentication is successful. After [8], several improved fuzzy vault methods [9-11] [12] are also proposed. In a key generation system, a secret key is directly generated from biometric data. Dodis et al. [13] first provided a framework on how to generate cryptographic keys from biometric data. A refined work can be found in [14] by Dodis et al.. In their work, the fuzzy extractor describes how to recover the template biometric data from a closed feature representation of the same biometric by publishing the sketch
*Corresponding author
978-1-4799-2764-7/13/$31.00 ©2013 IEEE
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data which leaks little information about the template. To be more specific, two primitives, secure sketch and fuzzy extractor, are defined. The secure sketch allows recovery of a shared secret given a close approximation thereof, and the strong extractor can extract a uniformly distributed string R from this shared secret in an error-tolerant manner. Some applications of this concept can be found in [6, 15, 16]. All the above mentioned methods are promising strategies to protect the biometric template and enhance the security level of biometric systems. However, in real applications, they also have a limit that all these methods are not cancellable. Given the fact that all the biometric traits are immutable and the sources of biometric traits of a user are limited, e.g., one face, two eyes and ten fingers, when a biometric template is compromised, the user can not revoke it and reissue a new one as easy as passwords or tokens. A loss of a biometric template in one application means its loss in all other applications that utilize the same biometric template. Also fuzzy is subject to cross-matching attacks. Motivated by this, in this paper, we work toward equipping the biometric system with the capability of cancellability. Specifically, we propose a Delaunay triangle group based fuzzy vault and equip it with the capability of cancellability. The main contribution of this work is two-fold. First, a fuzzy vault with cancellability is proposed so as to immensely broaden the scope of its application. Second, the unit of transformation used in our application is each triangle instead of single minutiae which is more stable and insensitive to nonlinear distortion of fingerprint image. The rest of the paper is organized as follows. The proposed Delaunay triangle group based fuzzy vault with cancellability is presented in Section II. In Section III, experimental results are demonstrated and discussed. The conclusion is given in Section IV.
II.
PROPOSED METHODS
A.
Generation of Delaunay triangle groups Delaunay triangle group based local structure is employed in this paper because of some of its nice properties, such as stable local structure under distortion, translation- and rotationinvariant, mentioned by [16-18]. The generation of Delaunay triangle groups is based on Delaunay triangulation net which is composed of a set of minutiae extracted from a fingerprint image. Specifically, assume a set of N minutiae M = {mi }iN=1 are extracted from a fingerprint image, the whole fingerprint image region is divided into several small cells by a Voronoi diagram. Each of the cell satisfies the condition that a minutia mi locates in the center of cell and all the points in the cell surround mi are closer to mi than to any other minutiae in M, as shown in Fig. 1(a). Then the Delaunay triangulation net is generated by connecting the center of each cell and the centers of its neighbor cells, as shown in Fig. 1(b). Totally, there would be (2×N-2-L) Delaunay triangles contained in a Delaunay triangulation net, where L is the number of minutiae locates on the convex hull of the Delaunay triangulation net. In our implementation, each Delaunay triangle group is composed by several Delaunay triangles which share a same vertex in the Delaunay triangulation net. In Fig. 1(c), a Delaunay triangle group TGa , which is centered in minutiae point a and composed of seven triangles abc , acd , ade , aef , afg , agh , and ahb , is given as an example.
Figure 1. (a) The Voronoi diagram (b) The Delaunay triangulation net (c) An example of a Delaunay triangle group TGa Figure 2. Feature data
f abc = (lao , α cax , α bax , β bc ) extracted from triangle
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abc
Biometric feature extraction from each Delaunay triangle group
triangle as the reference point to calculate which sector and level it locates.
For each Delaunay triangle group TGi , we consider its center minutia as the origin of a coordinate system and the center minutia’s orientation acts as the 0 degree axis in the polar space. Take the local structure TGa which centers at minutia a as an example. Several features can be extracted from each of its Delaunay triangles. Below, we define some local features from the Delaunay triangle abc :
Here, the triangle abc in the local structure TGa is considered as an example. We assume that the polar space which originates from vertex a is separated into S=4 sectors and L=3 levels. If the transformation matrixes specified by a user specified transformation key are M s =[5, 7, 10, 15] and M l =[10, 11, 6, 25], the transformation conducted by matrix M s and M l can be expressed as,
B.
Ts = So + M s
- lao , the length between the vertices a and o which is the incircle center of abc .; - α cax , the angle between the 0 degree axis (X axis) and edge ac in the counter clock-wise direction; - α bax , the angle between the 0 degree axis (X axis) and edge ab in the counter clock-wise direction; - βbc , the orientation difference between the
= [1, 2, 3, 4] + [5, 7,10,15] = [mod(6, 4), mod(9, 4), mod(13, 4), mod(19, 4)] = [2,1,1, 3] and Tl = Lo + M l = [1, 2, 3] + [10,11, 6, 25] = [mod(11, 3), mod(13, 3), mod(28, 3)] = [2,1,1]
orientation θb of vertex b and the orientation θ c of vertex c; Therefore, a set of feature data, e.g., f abc = (lao , α cax , α bax , β bc ) as shown in Fig. 2, can be extracted from each triangle of a Delaunay triangle group. C. Add cancellability to the Delaunay triangle group based features The concept of cancellable biometric was first proposed by Ratha et al [19] and three different transformations, Cartesian transformation, functional transformation and polar transformation, were included in their work to generate a transformed version of the original features. In this work, a polar transformation similar as that in [19] is adopted to achieve non-invertible transformation. However, the difference compared with existing works such as [20, 21] [22-26] is that, instead of each individual minutia, the polar transformation unit is each triangle of a Delaunay triangle group.
(1)
Tl = Lo + M l
(2) where So and Lo are the original sector and level of a triangle, while Ts and Tl are the new sector and level of a triangle after transformation, respectively, using the incircle center of the
(4)
In this way, the triangle abc originally locates in the So =1 sector and Lo =2 level according to the location of its incircle center o, would be transformed to location of So =2 sector and Lo =1 level after transformation as shown in Fig. 3. This transformation is a many-to-one mapping, e.g., the triangles that originally locate in (S=2, L=2) and (S=2 L=3) are both mapped to (S=1, L=1).
To be specific, a polar coordinate space which originates from the center minutia of a Delaunay triangle group is separated into S sectors and L levels. The polar transformation process is equal to change the sectors and levels of each triangle and the transformation process is conducted by a user specified transformation key which composed of two matrix M s and M l . If the template is compromised by an adversary, a new template could be reissued by simply changing the transformation key. By using these two matrixes, the transformation can be expressed as,
Ts = S o + M s
(3)
Figure 3. The original triangle
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abc and transformed triangle a1b1c1
After non-invertible transformation, feature data f a1b1c1 = (la1o1 , α c1a1 x , α b1a1 x , β b1c1 ) which are defined in Section IIB, are extracted from the transformed triangle a1b1c1 . The feature data f a1b1c1 would be surely different to f abc that are extracted from the original triangle abc . To tolerate the variation caused by image non-linear distortion, the quantization technique is applied to each feature and each triangle is denoted by four quantized values. For instance, the transformed triangle would be expressed a1b1c1 by qf a1b1c1 = ( qla1o1 , qα c1a1 x , qα b1a1 x , q β b1c1 ) . The condition of two triangles is considered as a matching pair is that all their four corresponding elements should be the same. Similarly, each Delaunay triangle group can be expressed by a (NTG×4) matrix, where NTG is the number of triangles in a Delaunay triangle group, e.g., TGa can be expressed by a 7×4 matrix as,
1) Encoding stage: Given a secret k which needs protection, it is divided into (ndegr+1) partitions which act as the coefficients of a Polynomial P(⋅) of degree of ndegr, e.g.,
P ( x ) = kdegr x degr + … k1 x + k0 . After a set of feature data {TGiT }iN=T1 are extracted from a template image fpT , where NT are the number of local structures in fpT , a hash function
hash(·) is applied to each TGiT to extract a 32-bit length
TGa = {qf a1b1c1 , qf a1c1d1 , qf a1d1e1 , qf a1e1 f1 , qf a1 f1 g1 , qf a1 g1h1 , qf a1h1b1 }
binary string SGiT = hash(TGiT ) which is further input into
⎧(la1o1 , α c1a1 x , α b1a1 x , β b1c1 ) ⎫ ⎪ ⎪ ⎪(la1o2 , α d1a1 x , α c1a1 x , β c1d1 ) ⎪ ⎪ ⎪ ⎪(la1o3 , α e1a1 x , α d1a1 x , β d1e1 ) ⎪ ⎪ ⎪ = ⎨(la1o4 , α f1a1 x , α e1a1 x , β e1 f1 ) ⎬ ⎪ ⎪ ⎪(la1o5 , α g1a1 x , α f1a1 x , β f1 g1 ) ⎪ ⎪ (l , α ,α , β )⎪ ⎪ a1o6 h1a1 x g1a1 x g1h1 ⎪ ⎪(la o , α b a x , α h a x , β h b ) ⎪ 1 1 1 1 1 1 ⎩ 17 ⎭
P(⋅) as a variable and output a value P( SGiT ) . In this way, a
(5)
from a query image, is denoted as TG Q = {qf1Q ,..., qf NQLT } , where NLT and NLQ are the number of triangles in TG T and TG Q , respectively. Assume the number of matched triangles between TG T and TG Q are NLB, then the similarity between them is calculated by, N × N LB S L = LB N LT × N LQ
genuine point set GP = {TGiT , P ( SGiT )}iN=T1 is generated. To protect this genuine point set GP, a chaff point set CP which contains a number of N cf chaff points is added to generate a N +N
D. Similarity measurement between two Delaunay triangle groups The similarity between two Delaunay triangle groups is calculated based on how many triangles from each local structure are matched. Assume TG T , a Delaunay triangle group from a template image, is denoted as TG T = {qf1T ,..., qf NTLT } and TG Q , a Delaunay triangle group
(6)
If S L is equal to or larger than a pre-defined threshold t L , then TG T and TG Q are considered as a matching pair.
The Delaunay triangle group based fuzzy vault In a fuzzy vault scheme, two stages, encoding stage and decoding stage, are included. In the encoding stage, the biometric features extracted from a template image are technique bound with a user-specified secret k by using a
E.
polynomial P(⋅) and the genuine points are hidden in vault which contains both genuine points and chaff points. In the decoding stage, a set of points are extracted from a query image and compared with the points in the vault to acquire a number of points for polynomial reconstruction. If enough genuine points could be found, then the secret k can be retrieved, vice verse.
secure vault GP ∪ CP = {(TGiV , P ( SGiV ))}i =T1 cf , which is considered as a key lock set. During the chaff point adding process, we make sure that the similarity calculated by (6) between the abscissas of any two points in GP ∪ CP is larger than 0.25. Finally, hash(k) which is the hash value of the secret key k and the secure vault GP ∪ CP are stored in the database for secret key retrieve process, while the feature set {TGiT }iN=T1 is destroyed. N
2) Decoding stage: After a feature set {TG Qj } j =Q1 extracted from a query image fp Q , each local structure feature data
TG Qj would be compared with the abscissa of each point in the secure vault GP ∪ CP . If the matching score S L between
TG Qj and any element TGiV from GP ∪ CP is equal to or larger than a pre-defined threshold t L , then (TGiV , P( SGiV )) , the corresponding vault entry of TGiV , would be added to a key unlock set UP . If NUP , the number of elements in the unlock set UP, is smaller than (ndegr+1), it means no enough points can be used to reconstruct the Polynomial P(⋅) and the authentication is considered to be failure. If NUP is equal or greater than (ndegr+1), (ndegr+1) elements of UP would be utilized to reconstruct the Polynomial P(⋅) . All the coefficients of Polynomial P(⋅) is concatenated to generate a secret key k ' . If hash( k ' ) = hash(k), the authentication is successful, otherwise, the next (ndegr+1) elements from UP are tested. If all the elements are tried and there is still no hash( k ' ) = hash(k), the authentication is failure.
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III.
EXPERIMENTAL RESULTS
Two fingerprint databases, FVC2002 DB1 and DB2, are chosen to evaluate the proposed method. In each database, there are 800 gray-level fingerprint images from 100 different fingers and each finger provides eight images. VeriFinger 6.0 from Neurotechnology [27] is utilized to extract minutiae from each image. In the experimental, three indices (FAR, FRR and EER) are employed to evaluate the performance of the proposed methods. False Accept Rate (FAR) is calculated by the number of successful imposter tests divide the total number of imposter tests. False Reject Rate (FRR) is calculated by the number of unsuccessful genuine tests divide the total number of genuine tests. Equal Error Rate(EER) is the error rate when both FAR and FRR are equal. In the experiments, the 1st image of each finger is set as the template image and compared with the 2nd image which is considered as a query image from the other fingers in the database to calculate FAR. And we choose the 1st image from each finger as the template and the 2nd image from the same finger as the query in the database to calculate FRR. In this way, totally 9900 impostor matching tests and 100 genuine matching tests are conducted on each database. Here, we give a brief introduction to the parameters set in the experiments. Different parameters are test and only the parameters that achieve the best results in our experiments are listed here. We set the quantization step sizes as qsl =20 pixels, qsα = 5π 36 and qsβ = π 9 ; the similarity threshold t L is set to be 0.15; S=8 and L=3 are set for the cancellable transformation. In Fig. 4, the performances of the proposed fuzzy vault method, before and after cancellable transformation, are demonstrated.
respectively. One direct reason of this performance decrease is that the many-to-one mapping transformation enables some of local structures from template and query images that match before transformation to be non-matching after transformation. The security of a fuzzy vault [9] can be computed by (7) as, ⎛ ⎛ NT ⎞ ⎞ ⎜ ⎜ ⎟ ⎟ ndegr ⎠ ⎟ ⎜ H ∞ (TG | GP ∪ CP ) = −log ⎜ ⎝ N + N cf ⎞ ⎟ ⎜ ⎛⎜ T ⎟⎟ ⎜ ndegr ⎠ ⎟ ⎝⎝ ⎠
(7)
The security depends on three parameters, NT , N cf and ndegr from (7). In our application, the average NT is 40 and N cf is set to be 400 which is ten times as NT . In Table I and II, the performances of the proposed fuzzy vault method in terms of FRR, FAR and security under different polynomial degree ndegr are listed. There is balance between the system performance and the security level which can be adjusted according to the real applications. It is worth noting that the security of the original biometric template cannot be simply evaluated by the computation complexity which is indicated by bit because even if an adversary compromises the fuzzy vault, what he/she has obtained is just the transformed template and the original biometric template is still safe. TABLE I.
PERFORMANCE UNDER DIFFERENT POLYNOMIAL DEGREE ON DATABASE 2002DB1
ndegr, degree of P(·) FRR(%) FAR(%) Security (bits) 3 4 5 6 7
TABLE II.
19 23 27 34 42
0.38 0.09 0.03 0 0
14 18 21 25 29
PERFORMANCE UNDER DIFFERENT POLYNOMIAL DEGREE ON DATABASE 2002DB2
ndegr, degree of P(·) FRR(%) FAR(%) Security (bits) 3 4 5 6 7
8 12 15 17 23
2.25 0.79 0.29 0.09 0.02
14 18 21 25 29
Figure 4. Performance of the proposed fuzzy vault method before and
after transformation in terms of EER over databases FVC2002 DB1 and DB2
It can be seen from Fig. 4 that the performance after transformation is slightly worse than that before transformation. Specifically, the EERs on database 2002DB1 and 2002DB2 before transformation are 5.54% and 3.4%, respectively, while, after transformation, they become to be 8.46% and 5.7%,
IV.
CONCLUSION
A fuzzy vault scheme as a classical secure sketch can provide reasonable security to a biometric template. However, lack of cancellability makes it infeasible to be broadly deployed in real applications since the compromise of a template in one application means its loss in all other
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applications that utilize the same biometric template. To solve this problem, in this paper, we equip the fuzzy vault with the capability of cancellability by using a polar transformation. The unit of transformation is each triangle in a Delaunay triangle group instead of a single minutia which makes the system more stable and insensitive to the non-linear distortion existing in fingerprint images. Experiments on public databases show that the performance of the proposed fuzzy vault method decreased slightly after the non-invertible transformation compared to using the original template features. However, security of the biometric template is enhanced because even if an adversary compromises the fuzzy vault, the original biometric template is still safe. A new biometric template can be released by just changing the transformation key.
[11]
[12] [13] [14] [15]
ACKNOWLEDGMENT
[16]
This work was supported in part by ARC grants LP110100602, LP100200538, LP100100404, and LP120100595.
[17]
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