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An Efficient and Provably Secure Public Key Encryption Scheme Based on Coding Theory Rongxing Lu1, Xiaodong Lin2, Xiaohui Liang1 and Xuemin (Sherman) Shen1∗

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An Efficient and Provably Secure Public Key Encryption Scheme Based on Coding Theory Rongxing Lu1, Xiaodong Lin2, Xiaohui Liang1 and Xuemin (Sherman) Shen1∗ SECURITY AND COMMUNICATION NETWORKS Security Comm. Networks (2010) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/sec.274 RESEARCH ARTICLE An efficient and provably secure public key encryption scheme based on coding theory Rongxing Lu1, Xiaodong Lin2, Xiaohui Liang1 and Xuemin (Sherman) Shen1∗ 1 Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada 2 Faculty of Business and Information Technology, University of Ontario Institute of Technology, Oshawa, Ontario, L1H 7K4, Canada ABSTRACT Although coding-based public key encryption schemes such as McEliece and Niederreiter cryptosystems have been well studied, it is not a trivial task to design an efficient coding-based cryptosystem with semantic security against adaptive chosen ciphertext attacks (IND-CCA2). To tackle this challenging issue, in this paper, we first propose an efficient IND- CCA2-secure public key encryption scheme based on coding theory. We then use the provable security technique to formally prove the security of the proposed scheme is tightly related to the syndrome decoding (SD) problem in the random oracle model. Compared with the previously reported schemes, the proposed scheme is merited with simple construction and fast encryption speed. Copyright © 2010 John Wiley & Sons, Ltd. KEYWORDS coding-based cryptography; public key encryption; semantic security; chosen-ciphertext attacks; syndrom decoding problem *Correspondence Xuemin (Sherman) Shen, Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada. E-mail: [email protected] 1. INTRODUCTION size, the security level also grows much faster. Niederreiter cryptosystem [5] is a dual encryption scheme proposed in Post-quantum cryptography (PQC), as a popular cryptog- 1986, which is not only ten times faster than the McEliece raphy terminology aiming at providing ‘Post-Quantum’ cryptosytem in terms of encryption speed, but also equiv- alternative to the currently existing number theory cryp- alent to the McEliece cryptosystem in terms of security. tography [1--3], has obtained great attention in recent Following these two seminal works, over the past years, years. In essence, PQC overlaps many existing cryptogra- many efforts have been put in coding-based cryptography phy branches including coding-based cryptography [4,5], [8--14]. For example, Stern has proposed a coding-based lattice-based cryptography, hash-based cryptography, and zero knowledge identification scheme in 1993 [14]; Cour- multivariate-quadratic-equations cryptography [6]. How- tois, Finiasz, and Sendrier have presented the first practical ever, driven largely by the possible invention of a large coding-based signature scheme in 2001 [10]. More recently, quantum computer in the near future, PQC becomes a how to reduce the public key size and how to secure the new buzz word in cryptography communities and these parameter choice in coding-based cryptography are also non-number theory cryptography branches, especially the deeply explored [15--19]. coding-based cryptography, have brought renewed atten- The semantic security (a.k.a indistinguishability) against tion. adaptive chosen ciphertext attacks (IND-CCA2) is the In coding-based cryptography, there are two well- strongest known notion of security for the public key known public key encryption schemes, namely McEliece encryption schemes. However, in coding-based cryptog- and Niederreiter cryptosystems [4,5]. McEliece cryptosys- raphy, ‘IND-CCA2’ has not been widely discussed. To tem was first proposed in 1978 [4], which represents the best of our knowledge, only a few papers have the first public key encryption scheme based on linear touched this research issue [20--22]. Because McEliece error-correcting codes. Compared with the classical RSA cryptosystem has some special architecture, some gen- cryptosystem [7], the McEliece cryptosystem has two eral IND-CCA2 conversions [23,24], though they achieve advantages: (i) the speeds of both encryption and decryption IND-CCA2 versions of McEliece cryptosystem, may incur algorithms are faster; and (ii) with the increase of the key some redundancy. Therefore, Kobta and Imai have pro- Copyright © 2010 John Wiley & Sons, Ltd. Public key encryption scheme based on coding theory R. Lu et al. posed two specific conversions to reduce the redundancy parameters params in a polynomial time of κ; we write R [20]. Recently, Nojima et al. [21] have studied the seman- params ←− Setup(κ). tic for the McEliece cryptosystem without random oracles. • The randomized key generation algorithm Kgen takes However, they only achieve the semantic security against the system public parameters params as input, and the chosen plaintext attacks and the tight reductions are returns a pair (pk, sk) consisting of a public key and a also questionable, especially for the Niederreiter cryptosys- corresponding private key in a polynomial time of κ, tem. In Reference [22], Dowsley et al. have also discussed R we write (pk, sk) ←− Kgen(params). the CCA2 secure public key encryption scheme based on • The randomized encryption algorithm Enc takes a pub- the McEliece assumption in the standard model, but their lic key pk, a random number r, and a plaintext M as scheme needs some special constructions. Therefore, how input, and returns a ciphertext C in a polynomial time to design an efficient and IND-CCA2 secure coding-based of κ; we write C ←− Enc(pk, r,M). cryptosystem with/without random oracles is still worth of • The deterministic decryption algorithm Dec takes the investigation. private key sk and a ciphertext C as input, and returns In this paper, we propose an efficient IND-CCA2 secure the corresponding plaintext M or a special symbol ⊥ public key encryption scheme based on coding theory. Con- indicating that the ciphertext was invalid in a poly- cretely, we design our scheme based on the syndrome nomial time of κ; we write x ← Dec(sk, C), where decoding (SD) problem, and use the provable security tech- x∈{M, ⊥}. nique to get a tight reduction in the random oracle model [25]. Compared with Niederreiter cryptosystem, only two All algorithms should satisfy the standard consistency additional hash operations are required in the proposed constraint of public key encryption, i.e., for any message scheme. Thus, our scheme achieves fast encryption speed. M, Dec(sk, C = Enc(pk, r,M)) = M. The remainder of this paper is organized as follows. In Section 2, we formalize the definition of public key encryp- tion and the corresponding security model. In Section 3, we review the coding theory and the complexity assumption, 2.3. Security model the base of our proposed scheme. In Section 4, we present our efficient public key encryption scheme based on coding We recall the standard notion of security of public key theory, following by its formal security proof and parameter encryption schemes in terms of indistinguishability [26]. selection in Section 5 and Section 6, respectively. Finally, Concretely, we consider the security notion for a public key we draw our conclusions in Section 7. encryption scheme is indistinguishable against the adaptive chosen ciphertext attacks, call it the ‘IND-CCA2’ security model for brevity. 2. DEFINITION AND SECURITY Definition 1. (IND-CCA2) Let k and t be integers MODEL and ε a real number in [0, 1], and PKE a secure pub- lic key encryption scheme with the security parameter k. 2.1. Notation Let A be an IND-CCA2 adversary, which is allowed to access the decryption oracle OD (and some random ora- OH ,OH , ···, Let N ={1, 2, 3,...} denote the set of natural numbers, cles 1 2 in the random oracle model), against and k∈N be a security parameter. An event is said to be the indistinguishability of PKE. We consider the following random experiment: negligible if it happens with probability less than the inverse − n Expind cca2 k of any polynomial in k.Ifn∈N, then 0 denotes the string of Experiment PKE,A ( ) n zeros. If x, y are strings, then |x| denotes the length of x, n R [x]n denotes the n least significant bits of x,[x] denotes the params ←− Setup(k) n most significant bits of x, and x ⊕ y denotes the bit XOR if |x|=|y|, while if S is a finite set, then |S| is its cardinality, R R and s ←− S indicates the process of selecting s uniformly (pk, sk) ←− Kgen(params) and at random in S.IfA is a randomized algorithm, then y ←− A(x1,x2, ···) denotes the processing of A on inputs OD ,OH ,OH ,··· M ,M , ←− A ( 1 2 ) params, pk x1,x2, ···, and letting y denote its output. ( 0 1 state) ( ) R 2.2. Definition b ←− { 0, 1},Cb ←− Enc(pk, r,Mb) In general, a public key encryption scheme PKE = OD(,OH ,OH ,···) (Setup, Kgen, Enc, Dec) consists of four algorithms: b ←− A 1 2 (params, pk, Cb, state) • The randomized setup algorithm Setup takes a security if b = b then return b∗ ← 1 else b∗ ← 0 parameter κ as input, and returns the system public return b∗ Security Comm. Networks (2010) © 2010 John Wiley & Sons, Ltd. DOI: 10.1002/sec R. Lu et al. Public key encryption scheme based on coding theory We define the success probability of A via pute a word x∈Fn such that hw(x) ≤ t and HxT = s. Note 2 − − that, to ensure the hardness of SD problem, the parameters Succind cca2(k) = 2Pr Expind cca2(k) −1 PKE,A PKE,A should be carefully chosen [27,29,30]. = b = b − 2Pr 1 Definition 2. (SD Assumption) Let C be an [n, k, d]- binary linear code defined by a (n−k) × n binary matrix H k, t, ε d t ≤ d−1 PKE is said to be ( )-IND-CCA2 secure, if no adver- with the minimal distance , and 2 . An adversary A t Succind−cca2 k ≥ ε s∈Fn−k sary running in time has a success PKE,A ( ) .
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