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THRIVE: threshold homomorphic encryption based secure and privacy preserving biometric verification system
EURASIP Journal on Advances in Signal Processing volumeÂ 2015, ArticleÂ number:Â 71 (2015)
Abstract
In this paper, we introduce a new biometric verification and template protection system which we call THRIVE. The system includes novel enrollment and authentication protocols based on threshold homomorphic encryption where a private key is shared between a user and a verifier. In the THRIVE system, only encrypted binary biometric templates are stored in a database and verification is performed via homomorphically randomized templates, thus, original templates are never revealed during authentication. Due to the underlying threshold homomorphic encryption scheme, a malicious database owner cannot perform full decryption on encrypted templates of the users in the database. In addition, security of the THRIVE system is enhanced using a twofactor authentication scheme involving userâ€™s private key and biometric data. Using simulationbased techniques, the proposed system is proven secure in the malicious model. The proposed system is suitable for applications where the user does not want to reveal her biometrics to the verifier in plain form, but needs to prove her identity by using biometrics. The system can be used with any biometric modality where a feature extraction method yields a fixed size binary template and a query template is verified when its Hamming distance to the database template is less than a threshold. The overall connection time for the proposed THRIVE system is estimated to be 336 ms on average for 256bit biometric templates on a desktop PC running with quad core 3.2 GHz CPUs at 10 Mbit/s up/down link connection speed. Consequently, the proposed system can be efficiently used in reallife applications.
1 Introduction
In recent times, public and commercial organizations invest on secure electronic authentication (eauthentication) systems to reliably verify identity of individuals. Biometrics is one of the rapidly emerging technologies for eauthentication systems [1]. However, it is almost impossible to discuss biometrics without addressing the associated security and privacy concerns [2, 3]. Biometric data, stored either in a smart card or in a central database, incurs security and privacy risks due to increased number of attacks against identity management systems in recent years [2â€“5]. Security and privacy concerns on biometrics limit their widespread usage in reallife applications.
Biometric systems, which use error correction methods are proposed to cope with the noisy nature of the biometric templates in the literature [6â€“8]. These systems can obtain errorfree biometric templates by using error correction techniques and thus cryptographic primitives (e.g., encryption or cryptographic hash) can successfully be employed free of the aforementioned avalanche effect [8â€“11]. However, their high error correcting capability requirements may render them impractical for reallife applications [12]. Furthermore, side information (e.g., parity bits) is needed for error correction and this may lead to information leakage (i.e., some attacks like error correcting code statistics, and nonrandomness attacks) [13]. Zhou et al. successfully demonstrate that redundancy in an error correction code causes privacy leakage for biometric systems [14, 15].
Although biometric template protection methods are proposed to overcome security and privacy issues [3, 16â€“29], recent research shows that security issues remain to be of a major concern [30â€“36]. In addition to this, there are a number of studies pointing out the privacy leakage of biometric applications [37, 38] as well as biometric template protection methods [14, 15, 39]. In the literature, Zhou et al. propose a framework for security and privacy assessment of the biometric template protection methods [14]. Also, Ignatenko et al. analyze the privacy leakage in terms of the mutual information between the public helper data and biometric features in a biometric template protection method. A tradeoff between maximum secret key rate and privacy leakage is given in the works of Ignatenko et al. [38, 40].
Recently, homomorphic encryption methods are used with biometric feature extraction methods to perform verification via encrypted biometric templates [20, 41â€“43]. However, these methods offer solutions only in the semihonest model where each party is obliged to follow the protocol but can arbitrarily analyze the knowledge that it learns during the execution of the protocol to obtain some additional information. The existing systems have not been designed for the malicious model where each party can arbitrarily deviate from the protocol and may be corrupted. They also do not take into account security and privacy issues of biometric templates stored in the database [20, 43]. The authors state that their security model will be improved in the future work by applying encryption methods also on the biometric templates stored in the database. Moreover, some of these systems are exclusively designed for a single biometric modality or a specific feature extraction method which also limits their application areas [41, 42]. In addition, an adversary can enroll herself on behalf of any user to their systems since they offer no protection against malicious enrollment. Finally, all of these systems suffer from computational complexity.
Biohashing schemes, one of the emerging biometric template protection methods [25â€“29], offer low error rates and fast verification at the authentication stage. However, they are vulnerable to attacks as reported in the literature [33â€“36]. These schemes should be improved to be safely adapted in a wide range of reallife applications. In this study, we present new enrollment and authentication protocols to increase the security and the privacy of the biometric verification. The proposed THRIVE system can work with any biometric feature extraction scheme whose templates are binary or can be binarized (e.g., by using thresholded random projection schemes) and where the verification decision can be based on Hamming distances between templates.
1.1 Problem definition
A simplified view of biometric authentication systems can be described as in Fig. 1. A user claims an identity and provides a biometric which is compared with existing biometric templates stored in a database and if the new biometric matches the one in the database, the user is verified to be a genuine user of the system. The database biometric templates are obtained during a separate enrollment session. There are security and privacy concerns related to different parts of this system as can be seen in Fig. 1. One of the main concerns is the protection of privacy in storage of database biometric templates to make sure that even if the templates are obtained by an adversary, they should not reveal any private information about the users of the system. Other concerns are eavesdropping or tampering with the communication channel between the user and the verifier or overriding the response of the system by adversaries.
A direct solution to biometric template protection might be to store biometric templates on the server side in plain form, and, in this way, it may seem to be possible to realize biometric authentication by simply utilizing wellknown authentication protocols like SSL/TLS. However, as far as security and privacy are concerned, one of the major security issues in this scenario is that any malicious behavior on server side can be very harmful because of storage of biometric data in clear. Furthermore, these standard authentication protocols use conventional cryptographic primitives like Hash, RSA, and AES that cannot be directly used since encryptions cannot be decrypted by the server alone and also biometric data are inherently noisy [9]. More precisely, when a malicious database manager obtains decryption keys, she can perform decryption alone and can access biometric templates of all users. Therefore, biometric templates should be encrypted during the enrollment phase where the private key and encryption are never given to the server and authentication should be still guaranteed. Namely, when a biometric template is encrypted during the enrollment stage, only an approximate comparison between the stored and measured biometric data should be decrypted during the authentication stage. With the conventional cryptographic mechanisms, however, this again can lead to security and privacy issues for biometric templates at the authentication stage [9].
1.2 Our contributions
In this paper, we address adversary attacks in case of an active attacker who wants to gain unauthorized access to the system in the malicious attack model, where dishonest parties can deviate from the protocol and behave arbitrarily. By taking possible attacks into consideration, we propose a new biometric authentication system based on threshold homomorphic encryption. Our aim is to enhance the security of the system and preserve privacy of the users. The contributions of this work can be summarized as follows:

A new biometric authentication system (which we call THRIVE) is proposed in the malicious model. It is a twofactor authentication system (biometric and secret key) and can be used in the applications where the user and the verifier do not trust each other.

THRIVE system can be used with any existing biometric modality whose templates can be transformed into a binary vector and decision can be based on Hamming distances between binary templates.

Even if an adversary gains access to the database and gathers encrypted biometric templates, she can neither authenticate herself by using these templates nor decrypt them due to the (2,2)threshold homomorphic encryption scheme.

Biometric templates are never released even during the authentication phase and only the encrypted binary templates are stored in the database.

The THRIVE system offers a new and advanced biometric template protection method without any helper data.

The THRIVE system ensures security of the communication channel between the user and the verifier since all exchanged messages are randomized and/or encrypted. In addition, usage of nonces^{1} and signature schemes guarantees uniqueness of the communication session.

Since the biometric templates are encrypted, they are irreversible by definition as long as decryption keys, which are shared secrets, are not stolen.

The THRIVE system can generate a number of protected templates from the same biometric data of a user because of randomized encryption. Thus, it ensures diversity. Besides, they are also cancelable, i.e., when they are stolen, they can be reproduced.

The proposed authentication protocol run requires 336 ms and 671 ms on average for 256bit and 512bit biohash vectors, respectively, on a desktop PC with quadcore 3.2 GHz CPUs at 10 Mbit/s up/down link connection speed. Therefore, the THRIVE system is sufficiently efficient to be used in realworld applications.
The paper is structured as follows. Related work is presented in Section 2. Preliminaries are given in Section 3. The proposed biometric authentication system is introduced in Section 4. The security proofs of the proposed protocols are given in Section 5. The complexity analysis of the proposed system is discussed in Section 6. Comprehensive comparison between the proposed system and the existing systems/methods is given in Section 7. Finally, Section 8 concludes the paper.
2 Related work
In this part, we perform literature review on the works which are developed for mitigating security and privacy problems of biometrics and we categorize them as illustrated in Fig. 2. Jain et al. classify biometric template protection schemes into two main categories [3]: 1) Feature transformationbased schemes, and 2) Biometric cryptosystems. We analyze various studies under these two groups below.
2.1 Feature transformationbased schemes
Feature transformationbased schemes use a transformation which is applied on a biometric template. The basic idea is to transform features of a biometric template into another secure domain by using a user defined secret key or password which determines parameters of the transformation.
Biohashing schemes are simple yet powerful biometric template protection methods [25â€“29] that can be classified under saltingbased schemes. It is worth pointing out that biohashing is completely different from cryptographic hashing. Originally proposed to solve the aforementioned security and privacy issues, biohashing schemes fail to be a comprehensive solution. In [33â€“36], the authors claim that biohashes can be reversible under certain conditions and an adversary can estimate the biometric template of a user from her biohash. Consequently, when biohashes are stored in the databases and/or smart cards in their plain form, they can threaten the security of the system as well as the privacy of the users. Moreover, an adversary can use a compromised biohash to exploit the system security by performing malicious authentication. Also, as in the case of a compromised secret key, an adversary can recover the biometric template since these schemes are generally invertible [3].
Noninvertible transformbased schemes utilizing oneway functions have been proposed to protect biometric templates [44â€“46] against inversion attacks. A user secret key, which determines the parameters of noninvertible transformation function, is provided during the authentication stage. Even if an adversary obtains the secret key and/or the transformed biometric template, it is computationally hard to recover the original biometric template. However, these schemes suffer from the tradeoff between discriminability and noninvertibility which limits their recognition performance [3].
2.2 Biometric cryptosystems
The main idea behind biometric cryptosystems (also known as biometric encryption systems) is to use cryptographic techniques to enable template protection. Earlier methods include binding a cryptographic key with a biometric template or generating the cryptographic key directly from the biometric template [47]. Recently, there have been studies which use homomorphic encryption techniques for secure computation of Hamming distances. Thus, the biometric cryptosystems can be classified into three main categories: 1) Key binding schemes, 2) Key generation schemes, and 3) Modern cryptography for biometrics.
Key binding and generation systems use helper data, which is public information, about the biometric template for verification. Although helper data is supposed to leak no critical information about the biometric template, Rathgeb et al. show that helper data is vulnerable to statistical attacks [48]. Furthermore, Ignatenko et al. show how to compute a bound on possible secret rate and privacy leakage rate for helper data schemes [49]. Adler conducts a hillclimbing attack against biometric encryption systems [31]. In addition, Stoianov et al. propose several attacks (i.e., nearest impostors, error correcting code statistics, and nonrandomness attacks) to biometric encryption systems [13].
2.2.1 2.2.1 Key binding systems
In the literature, fuzzy commitment [8] and fuzzy vault schemes [24] are categorized under the key binding schemes. These schemes aim to bind a cryptographic key with a biometric template and it is expected that neither the biometric template nor the random bit string can be recovered if the corresponding userâ€™s biometric data is not known. However, this is not the case in reality as biometric templates are not uniformly random. Furthermore, error correction codes (ECC) used in biometric cryptosystems lead to statistical attacks (i.e., running ECC in a soft decoding or erasure mode and ECC Histogram attack) [13, 50]. Ignatenko et al. show that fuzzy commitment schemes leak information in cryptographic keys and biometric templates which lead to security flaws and privacy concerns [38, 40]. In addition, Zhou et al. argue that fuzzy commitment schemes leak private data. Chang et al. describe a nonrandomness attack against fuzzy vault scheme which causes distinction between the minutiae points and the chaff points [51]. Moreover, Kholmatov et al. describe a correlation attack against fuzzy vault schemes [52].
2.2.2 2.2.2 Key generation systems
Keys are generated from helper data and a given biometric template in key generation schemes [3]. Fuzzy key extraction schemes are classified under the key generation ones and they use helper data [53â€“58]. These schemes can be used as an authentication mechanism where a user is verified via her own biometric template as a key. Although the fuzzy key extraction schemes produce keys from biometric templates, the repeatability and the randomness of the generated keys are two major questions [3]. Boyen et al. describe several vulnerabilities of the fuzzy key extraction schemes from an attacker perspective [59], e.g., improper fuzzy sketch constructions leading to information leakage on the secret, biased codes allowing majority vote attack, and permutation leaks. Moreover, Li et al. argue that when an adversary obtains sketches, they may reveal the identity of the users [60].
2.2.3 2.2.3 Modern cryptography for biometrics
In recent years, a number of papers have been published on systems in which biometrics and homomorphic encryption work together for either authentication or identification purposes. These systems have cryptographic protocols based on secure multiparty computation and most of them especially use superior properties of homomorphic encryption schemes (e.g., allow computation on encrypted data) in order to overcome security and privacy threats to the biometric data.
Kerschbaum et al. [61] propose a protocol to compare fingerprint templates without actually exchanging them by using secure multiparty computation in the semihonest model. At the enrollment stage, the user gives her fingerprint template, minutiae pairs and a PIN to the system. Thus, the verifier knows the fingerprint templates which are collected at the enrollment stage. Although the user does not send her biometric data at the authentication, the verifier already has the userâ€™s enrolled biometric data and this violates the privacy of the user in case of a malicious (or compromised) verifier. A malicious verifier can use these fingerprint templates for malicious authentication. Furthermore, since the fingerprint comparison reveals the matching scores (e.g., Hamming distance [62]), the attacker can launch a hill climbing attack against this system.
Erkin et al. [41] propose a privacy preserving face recognition system for the eigenface recognition algorithm [63]. They present a protocol that performs operations on encrypted images by using the Paillier homomorphic encryption scheme. Later, Sadeghi et al. improve the efficiency of this system [42]. In both studies, the system is limited to using eigenface recognition algorithm with homomorphic encryption when there are better alternative face recognition algorithms. Moreover, they do not use a threshold cryptosystem, which prevents a malicious party from aiming to perform decryption by himself. Storing face images (or corresponding feature vectors) in the database in plain form is the most serious security drawback of this system. An adversary, who has access to the database, may obtain all face images.
Barni et al. [20, 43] propose a privacy preservation system for fingercode templates by using homomorphic encryption in the semihonest model where all parties follow the protocol, but dishonest parties may be curious to violate othersâ€™ privacy. Thus, they do not propose any security and privacy solutions on the biometric templates stored in the database. This issue is mentioned as a future work in their paper. In addition, they do not use threshold encryption, which would protect the system against a malicious party aiming to perform decryption by herself. Another drawback is that the system does not address the malicious enrollment issue. Although they achieve better performance than those in [41, 42] in terms of bandwidth and time complexity, they do not consider the scenarios where the user and the verifier do not trust each other (i.e., the malicious model).
Kulkarni et al. [64] propose a biometric authentication system based on somewhat homomorphic encryption scheme of Boneh et al. [65], which allows an arbitrary number of addition of ciphertexts, but supports only one multiplication operation between the ciphertexts. Although the values stored on the enrollment server are the XORed values of the biometric template vector with the corresponding userâ€™s key, the user first extracts and sends her biometric features to the trusted enrollment server. Again this system uses a trusted enrollment server and fails to provide security and privacy objectives against a malicious database manager. In addition, the system is not efficient since 58 s is required for a successful authentication of a 2048bit binary feature vector.
There are also some works on secure Hamming distance calculation by using cryptographic primitives [64, 66â€“68]. These papers, however, limit the scope of their works only to secure Hamming distance calculation. They do not address biometric authentication as a whole and do not satisfy security and privacy concerns under a malicious model.
Osadchy et al. [66] propose a secure Hamming distance calculation scheme based on Pailler homomorphic encryption for face biometrics. The system is called SCiFI. Although they claim that SCiFI is computationally efficient, it mostly uses precomputation techniques. Its precomputation time includes processing time that must be done locally by each user before using the system each time. They report that SCiFIâ€™s online running time takes 0.31 s for a face vector of size 900 bits; however, its offline computation time takes 213 s. SCiFI provides security against only in the semihonest adversaries.
Rane et al. [67] also propose secure Hamming distance calculation for biometric applications. Nonetheless, their proposed method fails to ensure biometric database security because biometric templates are stored in plain format in the database. Thus, a malicious verifier can threaten a userâ€™s security and privacy. Bringer et al. [68] propose a secure Hamming distance calculation for biometric application, called as SHADE. This technique is based on committed oblivious transfer [69]. However, it cannot guarantee biometric database security since biometric templates are stored in plain form in the database.
3 Preliminaries
3.1 Threshold homomorphic cryptosystem
In this section, we briefly describe underlying cryptographic primitives of the protocols. Given a public key encryption scheme, let \(m \in \mathcal {M}\) denote the message or plaintext space, \(c \in \mathcal {C}\) the ciphertext space, and \(r \in \mathcal {R}\) its randomness. Let c=Enc_{ pk }(m;r) depict an encryption of m under the public key pk where r is a random value. Let sk be the corresponding private key, which allows the holder to retrieve a message from a ciphertext. The decryption procedure is performed with the private key sk as m=Dec_{ sk }(c).
In a (t,n)threshold cryptosystem, the knowledge of a private key is distributed among parties P _{1},â‹¯,P _{ n }. Then, at least t of these parties are required for a successful decryption. On the other hand, there is a public key to perform encryption. More formally, let P _{1},â‹¯,P _{ n } be the participants. We define a (t,n)threshold encryption scheme with three phases as follows:

In the key generation phase, each participant P _{ i } receives a pair of (p k _{ i },s k _{ i }), where p k _{ i } and s k _{ i } are the shares of the public and secret key, respectively. Then, the overall public key pk is constructed by collaboratively combining the shares. Finally pk is broadcasted to allow anyone to encrypt messages in \(\mathcal {M}\). The shares of this public key are also broadcasted to allow all parties to check the correctness of the decryption process.

The encryption phase is performed as in any public key encryption cryptosystem. If \(m\in \mathcal {M}\) is the message, a (secret) random value r from \(\mathcal {R}\) is chosen and c=Enc_{ pk }(m;r) is computed under a public key pk.

In the threshold decryption phase, given that t (or more) participants agree to decrypt a ciphertext c, they follow two steps. First, each participant produces a decryption share by performing \({S_{i}^{j}}=\text {\sf Dec}_{s{k_{i}^{j}}}(c)\), j=1,â‹¯,t. After broadcasting \({S_{i}^{j}}\), they all can apply a reconstruction function \(\mathcal {F}\) on these shares so that they can recover the original message by performing \(m=\mathcal {F}\left ({S_{i}^{1}},\cdots,{S_{i}^{t}}\right)\) where \({P_{i}^{1}},\cdots,{P_{i}^{t}}\) represent the group of t participants willing to recover m.
In case of a (t,n)threshold scheme, the additional requirement is that if less than t parties gather their correct shares of the decryption of a given ciphertext, they will gather no information whatsoever about the plaintext. In the proposed system, we use the (2,2)threshold cryptosystem between the prover (the user) and the verifier where both players must cooperate to decrypt. In this way, we ensure that the verifier cannot decrypt the ciphertexts alone, and the decryption is only performed by both the user and the verifier during the computation of Hamming distance of their corresponding inputs.
A public key encryption scheme is said to be additively homomorphic if given c _{1}=Enc(m _{1};r _{1}) and c _{2}=Enc(m _{2};r _{2}) it follows that c _{1} c _{2}=Enc(m _{1}+m _{2};r _{3}) where \(m_{1}, m_{2} \in \mathcal {M}\) and \(r_{1}, r_{2}, r_{3} \in \mathcal {R}\). That is to say, homomorphic encryption is a form of encryption that allows parties to perform computations on the encrypted values and match the result of operations performed on the plaintexts. Namely, they do not possess the decryption key, and therefore they do not know the plaintexts but can still perform operations under encryption. Conventional cryptosystems do not satisfy the homomorphic property.
There are various versions of threshold homomorphic cryptosystems. The most widely used are ElGamal [70] or Paillier [71] cryptosystems. In our proposal, we will use a threshold version of GoldwasserMicali (GM) encryption scheme (i.e., between a user and a verifier) proposed by Katz and Yung in [72]. GM scheme is XORhomomorphic [73], i.e., given any two bits b _{1},b _{2} in {0,1}, any random values r _{1}, \(r_{2}\in \mathcal {R}\), and any encryptions Enc(b _{1},r _{1}),Enc(b _{2},r _{2}), it is easy to compute Enc(b _{1}âŠ•b _{2},r _{1} r _{2}). Note that our scheme requires encryption of bits instead of bitstrings, and GM encryption is more efficient compared to ElGamal or Paillier in the case of bit encryption.
In the proposed protocol, we use a variant of the threshold decryption protocol which is the socalled private threshold decryption [74]. The requirement of this protocol is that one of the t parties will be the only party who will recover the secret. All tâˆ’1 other parties follow the protocol and broadcast their shares to achieve this requirement. The party who will learn the plaintext proceeds with the decryption process privately, collects all decryption shares from the tâˆ’1 other parties, and privately reconstructs the message. The remaining parties will not get any information about the message.
3.1.1 3.1.1 Threshold XORHomomorphic GoldwasserMicali encryption scheme
We next give a brief explanation of (2,2)GM cryptosystem between two users (in our proposal, between a user and a verifier) using a Trusted Dealer. We note that one can also exclude a trusted dealer using the scheme in [72]:

Key generation:
The trusted dealer first chooses prime numbers p and q (âˆ¥pâˆ¥=âˆ¥qâˆ¥=n) such that N=p q and p â‰¡q â‰¡ 3 mod 4. The dealer next chooses p _{1}, q _{1}, p _{2}, q _{2}âˆˆ_{ R }(0,2^{2n}) such that p _{1}â‰¡q _{1}â‰¡0 mod 4 and p _{2}â‰¡q _{2}â‰¡0 mod 4. He sets p _{0} = pâˆ’p _{1}âˆ’p _{2} and q _{0}=qâˆ’q _{1}âˆ’q _{2} and sends (p _{1},q _{1}) to the first party and (p _{2},q _{2}) to the second party. He finally broadcasts (p _{0},q _{0},N).

Encryption of a bit b âˆˆ{0,1}:
Choose r \(\in _{R}\mathbb {Z}_{N}\) and compute a ciphertext C=(âˆ’1)^{b} r ^{2} mod N.

Decryption:
All parties compute the Jacobi symbol \(J= \left (\frac {C}{N}\right)\). If Jâ‰ 1 then all parties stop because either the encryption algorithm was not run honestly or the ciphertext was corrupted during the transmission. (Note that \(\left (\frac {C}{N}\right)\) is always 1, because \(\left (\frac {C}{N}\right) = \left (\frac {C}{p}\right) \left (\frac {C}{q}\right)= 1\) (i.e., either \(\left (\frac {C}{p}\right)=1\) and \(\left (\frac {C}{q}\right)=1\) or \(\left (\frac {C}{p}\right)=1\) and \(\left (\frac {C}{q}\right)=1)\)). If J=1 then the first party broadcasts \(b_{1} = C^{{(p_{1}q_{1})}/4} \mod N\phantom {\dot {i}\!}\). The second party (who is going to decrypt) will privately compute \(\phantom {\dot {i}\!}b_{0}= C^{{(N p_{0} q_{0} +1)}/4} \mod N\) and \(b_{2} = C^{{(p_{2}q_{2})}/4} \mod N\phantom {\dot {i}\!}\). Finally, the decrypted bit b is computed as b=(1âˆ’b _{0} b _{1} b _{2} mod N)/2.
Note that it is easy to see whether C is a quadratic residue by computing bâ‰¡C ^{(Nâˆ’pâˆ’qâˆ’+1)/4} mod N. The reason is briefly as follows. We first note that by Eulerâ€™s theorem C ^{Ï•(N)}â‰¡1 mod N where Ï•(N)=(pâˆ’1)(qâˆ’1). We also know that C is quadratic residue if and only if C ^{Ï•(N)/2}â‰¡1 mod N. If the Jacobi symbol J \(= \left (\frac {C}{N}\right)\) = 1 then by using \(\left (\frac {C}{p}\right) \left (\frac {C}{q}\right) =\) 1, we have either \(\left (\frac {C}{p}\right)=1\) and \(\left (\frac {C}{q}\right)=1\) or \(\left (\frac {C}{p}\right)=1\) and \(\left (\frac {C}{q}\right)=1\). If \(\left (\frac {C}{p}\right) =\) 1 (resp. âˆ’1) and \(\left (\frac {C}{q}\right) =\) 1 (resp. âˆ’1) then C ^{pâˆ’1/2}â‰¡ 1 mod p (resp. âˆ’1 mod p) and C ^{qâˆ’1/2}â‰¡ 1 mod q (resp. âˆ’1 mod q). Hence, for both cases C ^{(pâˆ’1)(qâˆ’1)/4}â‰¡ 1 mod p and C ^{(pâˆ’1)(qâˆ’1)/4}â‰¡ 1 mod q. By the Chinese Remainder Theorem, we have C ^{(pâˆ’1)(qâˆ’1)/4}â‰¡ 1 mod N. Hence, C is quadratic residue if and only if b=1.
3.2 Biometric verification scheme
Biometric verification schemes perform an automatic verification of a user based on her specific biometric data. They have two main stages: 1) Enrollment stage, and 2) Authentication stage. The user gives her biometric data to the system at the enrollment stage. Then in the authentication stage, she provides her biometric data to the system to prove her identity. Any biometric scheme, which provides binary templates or whose templates can be binarized, can be used with the proposed threshold homomorphic cryptosystem. Most biometric templates are represented as fixed length real vectors and they can be binarized easily using locality sensitive hashing (LSH) techniques and the most natural and widely used distance in the hash space is the Hamming distance. Previous studies have shown that coming up with distance preserving binary hashing is possible [75â€“77]. So, we believe the system we propose can be used in a broad class of verification systems with minor modification in the system to binarize the templates and use Hamming distance for distance calculation, which will result in minimal loss of security properties in the system (such as equal error rate (EER) etc.). In this paper, we use biohashing as an example algorithm for extracting binary biometric templates. Random projection and thresholding used in biohashing is a wellknown type of LSH approach [78]. Although biohashing has its own security and privacy preservation mechanism, we do not rely on these to address the security or privacy concerns.
Biohash is a binary and pseudorandom representation of a biometric template. Biohashing schemes use two inputs: 1) Biometric template, 2) Userâ€™s secret key. A biometric feature vector is transformed into a lower dimension subspace using a pseudorandom set of orthogonal vectors which are generated from the userâ€™s secret key. Then, the result is binarized to produce a pseudorandom bitstring which is called the biohash. In an ideal case, the Hamming distance between the biohashes belonging to the biometric templates of the same user is expected to be relatively small. On the other hand, the distance between the biohashes of different users is expected to be sufficiently high to achieve higher recognition rates.
We adopt the random projection (RP)based biohashing scheme proposed by Ngo et al. [79]. In this scheme, there are three main steps: 1) Feature extraction, 2) Random projection, 3) Quantization. These steps are explained in the following.
3.2.1 3.2.1 Feature extraction
The feature extraction is performed on the biometric data (e.g., face image) which is collected at the enrollment stage, belonging to users, I _{ i,j }âˆˆâ„œ^{mÃ—n} for i=1,â‹¯,n and j=1,â‹¯,L where n and L denote the number of users and the number of training images per user, respectively. The images are lexicographically reordered and the training vectors, x _{ i,j }âˆˆâ„œ^{(mn)Ã—1}, are obtained. Then, Principle Component Analysis (PCA) [63] is applied to these vectors as follows
where Aâˆˆâ„œ^{kÃ—(mn)} is the PCA matrix trained by the images in the training set, w is the mean face vector, and y _{ i,j }âˆˆâ„œ^{kÃ—1} is the vector containing the PCA coefficients belonging to the jth training image of the ith user.
3.2.2 3.2.2 Random projection (RP)
An RP matrix, Râˆˆâ„œ^{â„“Ã—k}, is generated to reduce the dimension of the PCA coefficient vectors. The RP matrix elements are independent and identically distributed (i.i.d) and generated from a Gauss distribution with zero mean and unit variance by using a Random Number Generator (RNG) with a seed derived from the userâ€™s secret key. The GramSchmidt (GS) procedure is applied to obtain an orthonormal projection matrix R _{ GS }âˆˆâ„œ^{â„“Ã—k} to have more distinct projections. Finally, the PCA coefficients are projected onto a lower â„“dimensional subspace using
where z _{ i,j }âˆˆâ„œ^{â„“Ã—1} is an intermediate biohash vector belonging to the jth training image of the ith user.
3.2.3 3.2.3 Quantization
The elements of the intermediate biohash vector z _{ i,j } are binarized with respect to a threshold as follows
where Î» _{ i,j }âˆˆ {0,1}^{â„“} denotes biohash vector of the jth training image of the ith user and Î² denotes the mean value of the intermediate biohash vector z _{ i,j }.
A biohash vector, \(\textit {\textbf {B}}_{enroll_{i}}\) for the ith user, which can be any vector among Î» _{ i,j } vectors in a realworld application, is stored in the database during the enrollment stage, which is accessed for verification purpose later during the authentication phase. For simulation purposes, we take into account all possible biohashes for a user by computing Î» _{ i,j }. The user is authenticated when the Hamming distance between \(\textit {\textbf {B}}_{enroll_{i}}\) and \(\textit {\textbf {B}}_{auth_{i}}\) is below a threshold Î¼, where \(\textit {\textbf {B}}_{auth_{i}}\) is the biohash vector measured during the authentication stage as follows
where \(\textit {B}_{enroll_{i}}^{k}\) denotes the kth bit of \(\textit {\textbf {B}}_{enroll_{i}}\), \(\textit {B}_{auth_{i}}^{k}\) denotes the kth bit of \(\textit {\textbf {B}}_{auth_{i}}\), and âŠ• denotes the binary XOR (exclusive OR) operator. Consequently, the verifier decides whether the prover is an authorized user depending on the threshold, Î¼.
4 The proposed biometric authentication system
stop In this section, we introduce the THRIVE system which has two participants: User (U _{ i }) and Verifier (V). The user has control of the biometric sensor, the feature extractor, and the biohash generator whereas the verifier has control of the database and the matcher as can be observed in Fig. 3 in the enrollment stage and Fig. 4 in the authentication stage. We assume that there is a trusted third party (TTP) which initially sets up the system public/private keys.
The TTP distributes the keys in the proposed system. There are publicprivate key pairs \(\left ({pk}_{i}, \left (s{k_{i}^{1}}, s{k_{i}^{2}}\right)\right)\), which are shared between the user and the verifier. Here, \(s{k_{i}^{1}}\) is the private key share of the ith user, U _{ i }, and \(s{k_{i}^{2}}\) is the private key share of the verifier. p k _{ i } is the public key of the ith user, U _{ i }, and both the user and the verifier know it. When an enrollment biometric template is encrypted by p k _{ i }, this can solely be decrypted using the private key shares of the user \(\left (s{k_{i}^{1}}\right)\) and the verifier \(\left (s{k_{i}^{2}}\right)\) collaboratively since the proposed system is based on the (2,2)threshold homomorphic cryptosystem. In addition, there is another publicprivate key pair \(\left ({pk}_{U_{i}}, {sk}_{U_{i}}\right)\), which belongs to the ith user, U _{ i }, where \({pk}_{U_{i}}\) is the public key and \({sk}_{U_{i}}\) is its associated private key to perform the signature operation. The verifier also knows the public key \({pk}_{U_{i}}\).
4.1 Enrollment stage
The proposed enrollment protocol is illustrated in Fig. 5 and its steps are introduced as follows:

1.
Step 1: The ith user, U _{ i }, computes her biohash, \(\textit {\textbf {B}}_{enroll_{i}}\) = \({\textit {B}}^{1}_{enroll_{i}}\cdots {\textit {B}}^{n}_{enroll_{i}}\) where \({\textit {B}}^{j}_{enroll_{i}}\in \{0,1\}\), j=1,â‹¯,n. Next, the user encrypts her biohash, \({C^{j}_{i}} = \)Enc\(_{pk_{i}}\left ({\textit {B}}^{j}_{enroll_{i}}\right)\) for j=1,â‹¯,n, by using the public key p k _{ i }. Then, the user signs her encrypted biohash, Sign\(_{sk_{U_{i}}}\left (<{C^{j}_{i}}:j=1,\cdots,n>\right)\), and sends it to the verifier.

2.
Step 2: The verifier V verifies Sign\(_{sk_{U_{i}}}\left (<{C^{j}_{i}}:j=1,\cdots,n>\right)\) by using \({pk}_{U_{i}}\) and stores the signature and encrypted biohash in the database. This data will be used for verification at the authentication stage.
Note that both the user and the verifier have to cooperate to decrypt a ciphertext due to the (2,2)threshold homomorphic cryptosystem. Furthermore, the signature ensures that the data stored in the database is generated by an authorized user.
Lemma 1.
Biohashes are not revealed at the enrollment stage.
Proof.
(Sketch) At the enrollment stage, the ith user U _{ i } first encrypts her biohash and then signs it. After these computations, U _{ i } sends her encrypted and signed biohash Sign\(_{sk_{U_{i}}}\left (<{C^{j}_{i}}:j=1,\cdots,n>\right)\) to the verifier. Since the userâ€™s biohash is not sent in plain form, biohashes are not revealed to the verifier at the enrollment stage.
Lemma 2.
An adversary cannot register as a user at the enrollment stage.
Proof.
(Sketch) At the enrollment stage, the ith user U _{ i } encrypts her biohash by using the public key p k _{ i } and then signs her encrypted biohash by using her private key \({sk}_{U_{i}}\). Thus, U _{ i } sends encrypted and signed biohash Sign\(\phantom {\dot {i}\!}_{sk_{U_{i}}}\left (<{C^{j}_{i}}:j=1,\cdots,n>\right)\) to the verifier. Since the verifier verifies the signature of the user, an adversary cannot register himself as a user without knowing the user private key \({sk}_{U_{i}}\) used to compute Sign\(_{sk_{U_{i}}}\left (<{C^{j}_{i}}:j=1,\cdots,n>\right)\).
4.2 Authentication stage
In this stage, U _{ i } tries to prove herself to the verifier by executing the proposed authentication protocol shown in Fig. 6. Similar to the enrollment case, the biometric sensor must be authorized by the system before the authentication protocol is carried out. Steps of the protocol are given as follows:

1.
Step 1: U _{ i } wants to verify her identity by using her biohash and sends a connection request to the verifier. Then, U _{ i } computes her biohash \(\textit {\textbf {B}}_{auth_{i}}\) = \({\textit {B}}^{1}_{auth_{i}}\cdots {\textit {B}}^{n}_{auth_{i}}\) where \({\textit {B}}^{j}_{auth_{i}}\in \{0,1\}\), j=1,â‹¯,n. Note that the user cannot produce exactly the same biometric template at each attempt and this results in different biohashes computed by the same user. Therefore, \(\textit {\textbf {B}}_{enroll_{i}}\) and \(\textit {\textbf {B}}_{auth_{i}}\) are different biohashes although they are generated by the same user at different sessions (e.g., enrollment and authentication stages). First, U _{ i } chooses a random vector \(\textit {\textbf {r}}_{i}^{j}\in _{R}\{0,1\}\) for j=1,â‹¯,n. She computes \({R^{j}_{i}} = {r^{j}_{i}}\oplus {\textit {B}}^{j}_{auth_{i}}\) for j=1,â‹¯,n. Then, U _{ i } generates a nonce, nonce\(_{U_{i}}\), which is uniquely defined and contains information about user id, session id, and timestamp. Finally, the user sends \(<{R^{j}_{i}}:j=1,\cdots,n>\), nonce\(_{U_{i}}\) to the verifier.

2.
Step 2: The verifier retrieves Sign\(_{sk_{U_{i}}}\left (<{C^{j}_{i}}:j=1,\cdots,n>\right)\) from the database where \({C^{j}_{i}}\) = \(\text {\sf Enc}_{pk_{i}}\left (\textit {B}^{j}_{enroll_{i}}\right)\) for j=1,â‹¯,n. Then, it generates a nonce nonce\(_{V_{i}}\) which contains information about the verifier, session id, and timestamp. Finally, it sends Sign\(_{sk_{U_{i}}}\left (<{C^{j}_{i}}:j=1,\cdots,n>\right)\), nonce\(_{V_{i}}\) to the user.

3.
Step 3: The user verifies Sign\(_{sk_{U_{i}}}\left (<{C^{j}_{i}}:j=1,\cdots,n>\right)\) by using public key \({pk}_{U_{i}}\). She computes \(C^{\prime j}_{i}\) = Enc\(_{pk_{i}}\left ({r^{j}_{i}}\right)\cdot {C^{j}_{i}}\) = Enc\(_{pk_{i}}\left ({r^{j}_{i}}\oplus \textit {B}^{j}_{enroll_{i}}\right)\) for j=1,â‹¯,n. Then, she performs partial decryption over \(C^{\prime j}_{i}\), i.e., \(T^{1,j}_{i}\) = Dec\(_{s{k_{i}^{1}}}\left (C^{\prime j}_{i}\right)\) = \(\left (C^{\prime j}_{i}\right)^{\left ({p_{i}^{1}}{q_{i}^{1}}\right)/4}\mod {N_{i}}\), for j=1,â‹¯,n using her private key share \(s{k_{i}^{1}}\). Finally, she sends Sign\(_{sk_{U_{i}}}\left (<\text {\sf Enc}_{pk_{i}}\right.\left ({r^{j}_{i}}\right), T^{1,j}_{i}:j=1,\cdots,n>\), nonce\(_{U_{i}}\), nonce\(\left.{\!~\!}_{V_{i}}\right)\) to the verifier.

4.
Step 4: V verifies the signature Sign\(_{sk_{U_{i}}}\left (<\text {\sf Enc}_{pk_{i}}\right. \left ({r^{j}_{i}}\right), T^{1,j}_{i}:j=1,\cdots,n>\), nonce\(_{U_{i}}\), nonce\(\left.{\!~\!}_{V_{i}}\right)\) by using the public key \({pk}_{U_{i}}\). Then, it computes \({C^{\prime \prime }}_{i}^{j} = \)Enc\(_{pk_{i}}\left ({r_{i}^{j}}\right) \cdot {C^{j}_{i}}\) (this is done to assure correctness of the result and will prevent a malicious user computing different values than expected.). Next, the verifier performs the full decryption by computing \(T^{2,j}_{i}= \)Dec\(_{s{k_{i}^{2}}}\left ({C^{\prime \prime }}_{i}^{j}\right)\) = \({\left (C^{\prime \prime j}_{i}\right)}^{\left ({p_{i}^{2}}{q_{i}^{2}}\right)/4}\mod {N_{i}}\) and \(T^{3,j}_{i}\) = \({\left ({C^{\prime \prime }}^{j}_{i}\right)}^{\left (N{p_{i}^{0}}{q_{i}^{0}}+1\right)/4}\mod {N_{i}}\). Finally, the verifier computes the decrypted jth bits \({T^{j}_{i}}\) = \(\left (1T^{1,j}_{i}T^{2,j}_{i}T^{3,j}_{i} \mod N_{i}\right)/2\) and the Hamming distance between \({R^{j}_{i}}\) and \({T^{j}_{i}}\) is calculated as follows
$$ \sum^{n}_{j=1}{R^{j}_{i}} \oplus {T^{j}_{i}} \leq \mu, $$((5))where Î¼ is the distance threshold. Therefore, the verifier decides whether the user is authentic with respect to the predefined distance threshold. Note that the Hamming distance between \({r^{j}_{i}}\oplus \textit {\textbf {B}}^{j}_{enroll_{i}}\) and \({r^{j}_{i}}\oplus \textit {\textbf {B}}^{j}_{auth_{i}}\) is equal to the Hamming distance between \(\textit {\textbf {B}}^{j}_{enroll_{i}}\) and \(\textit {\textbf {B}}^{j}_{auth_{i}}\).
Finally, the verifier sends its decision (either Accept or Reject) to the user. However, the user may get dummy output if there is an error or an attack (i.e., override response attack) in the communication channel. The proposed system can easily be updated to cope with such an attack, for instance, by allowing the verifier to sign its decision including the nonces generated during the authentication session (i.e., either Sign(Accept, nonce\(_{U_{i}}\), nonce\(_{V_{i}}\)) or Sign (Reject, nonce\(_{U_{i}}\), nonce\(_{V_{i}}\)) and then sends it to the user. This way, authenticity, integrity, and origin of the data can easily be verified. Signing the nonces (nonce\(_{U_{i}}\) and nonce\(_{V_{i}}\)) also makes the communication unique and avoids replay attacks.
Lemma 3.
Biohashes are not revealed at the authentication stage.
Proof.
Authentication is performed in a randomized domain. In other words, the authentication is determined by comparing \({R^{j}_{i}}\) and \({T^{j}_{i}}\). An adversary can only obtain R \(^{j}_{i}\) and T \(^{j}_{i}\) which are revealed at the authentication stage. Recall that these are randomized biohashes. Thus, from the adversaryâ€™s perspective, there are three unknowns \(\left (\textit {r}_{i}^{j}\right.\), \(\textit {B}^{j}_{enroll_{i}}\) and \(\left.\textit {B}^{j}_{auth_{i}}\right)\) and two equations which are shown in the below.
where \({r_{i}^{j}}\) is the random bit generated by the U _{ i } for the jth bit. Since this is a linear equation system with fewer equations than unknowns, it has many solutions. Consequently, it is impossible for the adversary to obtain a honest userâ€™s biohash by using \({T_{i}^{j}}\) and \({r_{i}^{j}}\) which are revealed at the authentication stage. As a result, the proposed biometric authentication system ensures security and privacy.
5 Security proof of the proposed authentication protocol
In this section, we prove that the proposed authentication protocol shown in Fig. 6 is secure against malicious users and the verifier. More concretely, the primary goal of privacy preservation is to protect usersâ€™ biometric templates during protocol executions. In a secure computation framework, parties have their own private input and are willing to evaluate a desired functionality f on their inputs without revealing any information except the outputs. This was originally formalized by Goldreich et al. in [80]. Intuitively, the following two scenarios should be absolutely indistinguishable: 1) securely computing f by realizing a protocol, and 2) privately sending their private inputs to a trusted third party, who then computes f and privately returns the outputs to each party. This formalization of secure computation is referred to as the simulationbased approach. The idea of the standard simulationbased privacy definition is that, given a welldefined privacyleakage, a polynomialtime simulator (i.e., an adversary) can generate a transcript that is indistinguishable from the output of the real protocol. If such an efficient simulator exists, then an adversary cannot learn any additional information beyond the defined leakage. The simulator must perform its task without knowing the private information of the party who proves her identity [80].
In this proof, we show that given a party is corrupted (either user or verifier), there exists a simulator that can produce a view which is statistically indistinguishable from the view of that party interacting with the other honest party. Assuming that one party is corrupted, we build an efficient simulator that has access to the public input and private secret shares of the secret key of the corrupted party. Besides, the simulator knows the public output. We want to point out that the simulator already knows the shares of the secret key of the corrupted party before the simulation is run. Since the threshold cryptosystem is set up before the protocol starts, we assume that the simulator extracts this information when the distributed key generation is run.
It is worth mentioning that the proposed authentication protocol gives computational privacy to both the user and the verifier due to the semantic security of the underlying cryptosystem. Furthermore, it is shown that the proposed authentication protocol is simulatable for both parties and these simulations produce views which are statistically indistinguishable from the views in the real protocol executions.
Theorem 1.
The proposed authentication protocol shown in Fig. 6 is secure in the presence of static malicious adversaries.
Proof.
We show that given a party is corrupted, there exists a simulator that can produce a view to the adversary that is statistically indistinguishable from the view in the real protocol execution based on its private decryption share as well as public information.
Case 1  User U _{ i } is corrupted. In this case, we prove the security for the case where U _{ i } is corrupted. The simulator has the private key share of the user \(s{k_{i}^{1}}\), the userâ€™s private key \({sk}_{U_{i}}\), and the userâ€™s biohash \(B^{j}_{auth_{i}}\) apart from the userâ€™s public information (i.e., \({pk}_{U_{i}}\) and p k _{ i }) as described in the proposed authentication protocol. The simulator constructs a view for the user which is statistically close to the one the user observes when interacting with the honest verifier by using this information. The simulator proceeds as follows:

1.
The simulator first obtains \(<{R^{j}_{i}}:j =1,\cdots, n>, \textit {nonce}_{U_{i}}\). As in the second round of the real protocol, the simulator needs to output the signature of the encrypted biohash of the user. To do so, the simulator computes \(\tilde {C}^{j}_{i} = \text {\sf Enc}_{pk_{i}}\left (B^{j}_{auth_{i}}\right)\) for j=1,â‹¯,n by using the userâ€™s public key p k _{ i }, and then computes Sign\(_{sk_{U_{i}}}\left (\left \langle \tilde {C}^{j}_{i}:j=1,\cdots,n\right)\right \rangle \). The simulator also generates a nonce called \(\tilde {\text {nonce}}_{V_{i}}\). The values Sign\(_{sk_{U_{i}}}\left (\left \langle \tilde {C}^{j}_{i}:j=1,\cdots,n\right)\right \rangle \) and \(\tilde {\text {nonce}}_{V_{i}}\) are the simulated outputs. Note that the simulator uses \(\tilde {\textbf {B}}_{\textit {auth}}\) instead of B _{ enroll } since it is the only available biohash to him.

2.
The simulator obtains Sign\(_{sk_{U_{i}}}\left (<\text {\sf Enc}_{pk_{i}}\left ({r_{i}^{j}}\right),\right.\left.T^{1,j}_{i}:j =1, \cdots, n>, \textit {nonce}_{U_{i}}, \tilde {\text {nonce}}_{V_{i}}\right)\) as in the second round of the protocol. The simulator next verifies the signature Sign\(_{sk_{U_{i}}}\left (<\text {\sf Enc}_{pk_{i}}\left ({r_{i}^{j}}\right),\right.\left.T^{1,j}_{i}:j =1, \cdots, n>, \textit {nonce}_{U_{i}}, \tilde {\text {nonce}}_{V_{i}}\right)\) that U _{ i } would run. Next, it computes \(\tilde {C}^{\prime \prime j}_{i} = \)Enc\(_{pk_{i}}\left ({r_{i}^{j}}\right)\cdot \tilde {C}_{i}^{j}\).
Given \(\tilde {C}^{\prime \prime j}_{i}\), its plaintext and the share of private key \(s{k_{i}^{2}}\) of the user U _{ i } the decryption shares \(T_{i}^{2,j}\) can be simulated as follows: The simulator computes \(\tilde {b}_{0} = \left [\tilde {C}^{\prime \prime j}_{i}\right ]^{{(N  p_{0}  q_{0} +1)}/4} \mod N\) from the public information and computes \(\tilde {b}_{1} = \left [\tilde {C}^{\prime \prime j}_{i}\right ]^{{(p_{1}q_{1})}/4} \mod N\) since it knows \(s{k_{i}^{1}}\) (i.e., p _{1},q _{1}). Let \(\tilde {b}\) denote the plaintext of \(\tilde {C}^{\prime \prime j}_{i}\). Then, the simulator can compute \(\tilde {b}_{2} \mod N \equiv (1 \tilde {2b})/(\tilde {b}_{0} \tilde {b_{1}}) \mod N\) (which is \(T_{i}^{2,j}\) in the real protocol). Note that in the real setting this is not possible since the plaintext inside the ciphertext is unknown \(\tilde {C}^{\prime \prime j}_{i}\).
Similarly, \(T_{i}^{3,j}\) can also be simulated since \({p_{i}^{0}}\) and \({q_{i}^{0}}\) are known by the simulator. The simulator finally computes \(\sum ^{k}_{j=1} {R_{i}^{j}} \oplus {T_{i}^{j}}\).
Each step of the proposed authentication protocol for the simulator is simulated and this completes the simulation for the malicious user. The transcript is consistent and statistically indistinguishable from the userâ€™s view when interacting with the honest verifier.
Case 2  The verifier V is corrupted. We now prove the security for the case where the verifier is corrupted. The simulator has the private key share of the verifier \(\left (s{k_{i}^{2}}\right)\) apart from the verifierâ€™s public information (i.e., \({pk}_{U_{i}}\), p k _{ i }, and Sign\(_{sk_{U_{i}}}\left (<{C_{i}^{j}}:j=1,\cdots,n>\right)\)) as described in the proposed authentication protocol. The simulator constructs a view for the verifier which is statistically close to the one when interacting with the honest user by using this information. The simulator proceeds as follows:

1.
Note that the simulator already knows \(<{C_{i}^{j}}:j=1,\cdots,n>\) because of the knowledge of Sign\(_{sk_{U_{i}}}\left (<{C_{i}^{j}}:j=1,\cdots,n>\right)\). The simulator chooses a random bit \(\tilde {r}_{i}^{j}\) and arbitrary \(\tilde {B}^{j}_{auth_{i}} \in _{R} \{0,1\}\) and computes \(\tilde {R}_{i}^{j} = \tilde {r}_{i}^{j} \oplus \tilde {B}^{j}_{auth_{i}}\). Recall that the simulator must perform its task without knowing the private information of the honest user in this case. Thus, although it does not have real \({r_{i}^{j}}\) and \(B^{j}_{auth_{i}}\), it can successfully execute the simulated conversation since \(\tilde {r}_{i}^{j}\) is uniformly random.

2.
The simulator obtains Sign\(_{sk_{U_{i}}}\left (<{C_{i}^{j}}:j=1,\cdots,n>\right)\). The simulator verifies the signature Sign\(_{sk_{U_{i}}}\left (<{C_{i}^{j}}:j=1,\cdots,n>\right)\) (by using the userâ€™s public key \({pk}_{U_{i}}\)) that V would run. Next, it next computes \(\tilde {C}^{\prime j}_{i} = \text {\sf Enc}_{pk_{i}}(\tilde {r}_{i}^{j}) \cdot {C_{i}^{j}}\).

3.
Given \(\tilde {C}^{\prime j}_{i}\), its plaintext (which is \(\tilde {r}_{i}^{j} \oplus B^{j}_{enroll_{i}}\)) and the share of private key \(s{k_{i}^{2}}\) of the verifier V the decryption share \(\tilde {T}_{i}^{1,j}\) can also be simulated as follows: The simulator computes \(\tilde {b}_{0}= \left [\tilde {C}^{\prime j}_{i}\right ]^{{(N p_{0} q_{0} +1)}/4} \mod N\) from the public information and computes \(\tilde {b}_{2} = \left [\tilde {C}^{\prime j}_{i}\right ]^{{(p_{2}q_{2})}/4} \mod N\) since it knows \(s{k_{i}^{2}}\) (i.e., p _{2},q _{2}). Let \(\tilde {b}\) denote the plaintext of \(\tilde {C}^{\prime j}_{i}\). Then, the simulator can compute \(\tilde {b}_{1} \mod N \equiv (1 \tilde {2b})/(\tilde {b}_{0} \tilde {b_{2}}) \mod N\) (which is \(T_{i}^{1,j}\) in the real protocol). Note that in the real setting, this is not possible since the plaintext inside the ciphertext is unknown \(\tilde {C}^{\prime \prime j}_{i}\).

4.
Finally, the simulator needs to simulate the signature
\(\text {\sf Sign}_{sk_{U_{i}}}\left (<\text {\sf Enc}_{pk_{i}}\left ({r^{j}_{i}}\right),T^{1,j}_{i}:j =1, \cdots, n>,\right. \left.\textit {nonce}_{U_{i}}, \textit {nonce}_{V_{i}}{\vphantom {{r^{j}_{i}}}}\right)\). However, this is not possible since the simulator does not know \({sk}_{U_{i}}\). In order to successfully simulate this final step, we need to provide additional information to the simulator. For example, performing a following modification over the key generation phase in the real protocol, the simulation will be possible:

During the key generation in the real protocol, the private key \({sk}_{U_{i}}\) is distributed to the user U _{ i } and V in a threshold fashion. For example, distributed RSA setting can be used for the signature algorithm (note that for an RSA setting (e,n) denotes the public key and (p,q,d) denotes the private key where n=p q and e dâ‰¡1 mod (pâˆ’1)(qâˆ’1)). Namely, the private key d can be divided into d _{1} and d _{2} such that the ciphertext c can be decrypted together with U _{ i } and V as m \(\phantom {\dot {i}\!}\equiv {c^{d}}\equiv {c^{d_{1}+d_{2}}} \mod n\).

In order to simulate, instead of signing procedures, U _{ i } will compute an encryption, compute its partial decryption and finally will send to the verifier. This will assure that the user indeed used its private decryption key over the encrypted value. Next, the verifier will also compute its partial decryption and will compute the decrypted value privately.
Thus, with this modified version the decryption share can be simulated in a similar way as described at the third step of the simulation.

Consequently, each step of the proposed authentication protocol for the simulator is simulated and this completes the simulation for the malicious verifier. The transcript is consistent and statistically indistinguishable from the verifierâ€™s view when interacting with the honest user.
â–¡
6 Complexity analysis of the proposed system
In this section, we discuss the complexity of the THRIVE enrollment and authentication protocols. The complexity of the THRIVE enrollment and authentication protocols are examined in terms of protocol steps for the round complexity, the number of cryptographic operations for the computational complexity and the number of messages exchanged by the two parties for the communication complexity. Without loss of generality, we will provide complexity of the THRIVE protocols using the (2,2)threshold homomorphic GM cryptosystem as an instance [73]. In the protocol, we use (2,2)threshold XORhomomorphic GM cryptosystem for confidentiality (i.e., encryption and decryption) while for signature generation and verification a conventional cryptosystem such as RSA (using the key pair \(({pk}_{U_{i}},{sk}_{U_{i}})\)) is employed.
The round complexity of the enrollment protocol is only one. For the computational complexity, the enrollment protocol requires n XORhomomorphic encryptions, and one conventional signature generation for a user, but one signature verification for the server. For the communication complexity, the user sends a conventional signature and n ciphertexts (i.e., \({C_{i}^{j}}\) for j=1,â‹¯,n).
In the authentication protocol, there are only four rounds. For the computational complexity of the authentication protocol, the user generates one conventional signature and verifies another, computes n XORhomomorphic encryptions and n XORhomomorphic decryptions, and performs n modular multiplications over homomorphic ciphertexts (i.e., \({Enc}_{pk_{i}}\left ({r_{i}^{j}}\right)\cdot {C_{i}^{j}}\) for j=1,â‹¯,n). The verifier verifies one conventional signature, computes n+2 modular multiplications, 2n decryptions, and performs n Jacobi computations to check \({Enc}_{pk_{i}}\left ({r_{i}^{j}}\right)\) for j=1,â‹¯,n. In total, there are n XORhomomorphic encryptions, 3n XORhomomorphic decryptions, two signature verifications, one signature generation, n Jacobi computations, and 2n+2 modular multiplications during the entire authentication protocol.
For the communication complexity of the authentication protocol, the user sends 2n homomorphic ciphertexts, one conventional signature and one nonce value. The verifier sends one conventional signature, n homomorphic ciphertexts, and one nonce to the user. In total, 3n homomorphic ciphertexts, two conventional signatures, and two nonce values are exchanged.
In the following, we provide timing estimates for the entire protocol for 80bit security level, on a desktop computer, which has Intel processor with various clock speeds (2.4 and 3.2 GHz). On the computing platform independent from the clock speed, one modular multiplication using Montgomery arithmetic takes about 2000 clock cycles. Furthermore, one encryption operation in XORhomomorphic GM cryptosystem takes only a single modular multiplication. On the other hand, one decryption in XORhomomorphic GM cryptosystem requires one modular exponentiation operation which takes about 3.2 million clock cycles. In addition, one signature generation and verification operation in conventional cryptosystem such as RSA are equivalent to one modular exponentiation operation.
The bandwidth usage of the proposed protocol for various lengths of biohashes of the user are given in Table 1. The required bandwidth for the proposed protocol increases with the increasing length of the biohash. Bandwidth usage also affects the overall connection time.
The computation times for the user and the verifier at 2.4 GHz for the proposed protocol with different biohash lengths are given in Table 2. Naturally, it is expected that the required computation times of the proposed protocol increase as lengths of the biohashes increase.
In our timing estimates, we assume sequential (i.e., singlethreaded) implementations of the user and server sides of the protocol that run on a single core. On the other hand, one advantage of the proposed system from the time complexity point of view is that majority of the expensive operations (i.e., mainly modular exponentiations) can be performed in parallel. Therefore, custom ASIC [81] or GPU implementations [82] can accelerate protocol considerably. For instance, the custom modular exponentiation circuit for RSA in [81] reports 0.89 ms execution time on a circuit of 153,000 equivalent gate counts. With specialized hardware that incorporates many custom modular exponentiation modules, overall execution time can be reduced significantly. Similarly, a GPU implementation is reported [82] to reach the peak throughtput of 34,981 RSA1024 decryptions per second with 2.6 ms latency. Apparently, a parallel GPU implementation of modular exponentiation will trivialize the computation complexity of the proposed technique. Finally, we expect an acceleration proportional to the number of cores if the protocol is implemented on a multicore CPU platform.
We compare the communication complexity of the proposed system with the existing systems in the literature assuming that all systems run on a computer platform with 2.4 GHz clock speed. Erkin et al.â€™s system [41] requires 56.25 ms, Barni et al.â€™s system [20, 43] requires 50 ms and Sadeghi et al.â€™s system [42] requires 25 ms for authentication at the server side for single user with 112bit binary feature vector [20]. On the other hand, our solution requires 449 ms for the same authentication setup. Although existing solutions seem faster than the proposed system, they propose their solutions in the semihonest model whereas our solution is secure under malicious adversary model. The similar timing estimations are also computed for 3.2 GHz as shown in Table 3.
We also compare the communication complexity of the proposed system with those of the existing systems at 3.2 GHz in the literature. Kulkarni et al.â€™s system [64] requires 58 s at the server side, 10 ms at the user side, and 400 Kbit bandwidth usage for authentication of single user with 2048bit binary feature vector at 3.2 GHz. Our proposed system requires 6146 ms at the server side, 2051 ms at the user side, and 6296 Kbit bandwidth usage for the same authentication setup. Thus, it is faster than Kulkarni et al.â€™s system. In addition, our proposes system offers other advantages such as that it is secure under malicious adversary model, that the biometric is protected via both a template protection method (e.g., biohash) and cryptographic primitives (e.g., threshold homomorphic encryption). On the other hand, Kulkarni et al.â€™s system offers solution for semihonest model which also means that it is insecure for the malicious model. The comprehensive comparison between the proposed system and existing systems are shown in Fig. 7.
7 Comparison between the proposed system and the existing methods/systems
In this section, we compare the proposed THRIVE system with currently available biometric template protection solutions in the literature. The comprehensive comparison between the proposed system and existing solutions are shown in Fig. 7. This comparison is performed by checking whether they use or satisfy the following properties or not. These properties are given in the below but more information can be found in Sections 2 and 3.

Irreversibility: It is computationally hard to reconstruct the original biometric template by using the protected biometric template (e.g., biohash, encrypted biometric data).

Renewability: It is possible to generate different protected templates from the same biometric data of a user.

Diversity: Different protected templates generated from the same biometric data of a user do not allow crossmatching or information leakage.

Helper data: It is auxiliary data/side information, which is needed for running biometric system successfully (e.g., data for alignment, parity bits of error correction codes). Without use of helper data such biometric systems do not work properly. Furthermore, an attacker may deduce further information that can threat privacy of a user and security of a system from helper data.

Homomorphic encryption: It is an encryption method, which allows computations to be carried out on ciphertext and obtaining the same results with the operations performed on the plaintext.

Threshold homomorphic encryption: It is a special type of homomorphic encryption, which distributes the knowledge of a private key among n parties and at least t of the parties are required for successful decryption. This method can also ensure security of biometric data stored in the database. Even if biometric templates are encrypted, a database manager can easily decrypt them as soon as he knows the private key. However, threshold homomorphic encryption requires cooperation between parties for a successful decryption.

Semihonest attack model: In this adversary model, all parties follow the protocol but dishonest party may be curious to violate others privacy by keeping a record of all its intermediate computations and messages. This is weak attack model in comparison with malicious attack model. Semihonest adversaries are also called honestbutcurious or passive attackers.

Malicious attack model: In this adversary model, attacker makes arbitrary feasible deviation(s) from the protocol specification and does not have to follow instructions of the protocol. A malicious attacker can enter the protocol with arbitrary input, which may not be a true input, and this makes the malicious model harder to deal with. Malicious adversaries are also called active attackers.
It is clearly seen in Fig. 7 that the proposed THRIVE system offers superior security and privacy preservation solutions under the malicious attack model.
8 Conclusion
In this work, we propose a novel biometric authentication system. The aim of the THRIVE system is to increase security against adversary attacks defined in [83] and preserve the privacy of users. The proposed system can be used with any biometric feature extraction method which can produce binary templates or whose templates can be binarized by postprocessing. The biohashing is chosen as an example binary biometric template generation system since it offers satisfactory performance and fast authentication. The comparison is performed in a randomized domain at the authentication stage and the binary templates (e.g., biohashes) are never released. In addition, only encrypted binary templates are stored in the database. Since we use the (2,2)threshold cryptosystem, the verifier cannot decrypt the data stored in the database by itself. Namely, the user and the verifier both has to cooperate to decrypt the encrypted binary templates. The THRIVE system can be used in applications where the user is not willing to reveal her biometrics to the verifier although she needs to proof her physical presence by using biometrics. It is also suitable for applications where the user and the verifier do not necessarily trust each other. The THRIVE system appears to be sufficiently efficient compared to the existing scheme and can be used in reallife applications. A common drawback of all existing schemes (including the THRIVE system), which consider the existence of malicious verifiers, is to utilize expensive asymmetric encryptions. Providing more efficient constructions (i.e., less communication, storage and computational overheads) while ensuring user privacy in the presence of malicious verifiers is still an open problem.
Biometric authentication systems work with some error rates which can be represented by using equal error rate (EER), false acceptance rate (FAR), and false rejection rate (FRR). These error rates may occur due to the nature of biometric data and can vary with respect to the various factors. The proposed THRIVE system can work with any biometric authentication scheme whose outputs can be binarized. The main goal of the THRIVE system is to increase security of authentication process and enhance privacy of users. Error rates (e.g., EER, FAR, FRR) of the THRIVE system related with authentication process is dependent on chosen biometric modality, underlying feature extraction method, and biometric authentication scheme (e.g., biohashing is given as an example in this paper).
9 Endnote
^{1} A nonce is an arbitrary number used only once in a cyptosystem.
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Acknowledgements
This work has been supported by the BEAT project 7^{th} Framework Research Programme of the European Union (EU), grant agreement number: 284989. The authors would like to thank the EU for the financial support and the partners within the consortium for a fruitful collaboration. For more information about the BEAT consortium, please visit http://www.beateu.org.
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Karabat, C., Kiraz, M.S., Erdogan, H. et al. THRIVE: threshold homomorphic encryption based secure and privacy preserving biometric verification system. EURASIP J. Adv. Signal Process. 2015, 71 (2015). https://doi.org/10.1186/s1363401502555
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DOI: https://doi.org/10.1186/s1363401502555