Polly Cracker, Revisited.
ABSTRACT We initiate the formal treatment of cryptographic constructions ("Polly Cracker") based on the hardness of computing remainders modulo an ideal over multivariate polynomial rings. We start by formalising the relation between the ideal remainder problem and the problem of computing a Gröbner basis. We show both positive and negative results. On the negative side, we define a symmetric Polly Cracker encryption scheme and prove that this scheme only achieves bounded CPA security. Furthermore, we show that a large class of algebraic transformations cannot convert this scheme to a fully secure Polly-Cracker-style scheme. On the positive side, we formalise noisy variants of the ideal membership, ideal remainder, and Gröbner basis problems. These problems can be seen as natural generalisations of the LWE problem and the approximate GCD problem over polynomial rings. We then show that noisy encoding of messages results in a fully IND-CPA-secure somewhat homomorphic encryption scheme. Our results provide a new family of somewhat homomorphic encryption schemes based on new, but natural, hard problems. Our results also imply that Regev's LWE-based public-key encryption scheme is (somewhat) multiplicatively homomorphic for appropriate choices of parameters.
Full-textDOI: · Available from: Ludovic Perret, Jun 19, 2015
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ABSTRACT: The Polynomial System Solving (PoSSo) problem is a fundamental NP-Hard problem in computer algebra. Among others, PoSSo have applications in area such as coding theory and cryptology. Typically, the security of multivariate public-key schemes (MPKC) such as the UOV cryptosystem of Kipnis, Shamir and Patarin is directly related to the hardness of PoSSo over finite fields. The goal of this paper is to further understand the influence of finite fields on the hardness of PoSSo. To this end, we consider the so-called hybrid approach. This is a polynomial system solving method dedicated to finite fields proposed by Bettale, Faugère and Perret (Journal of Mathematical Cryptography, 2009). The idea is to combine exhaustive search with Gröbner bases. The efficiency of the hybrid approach is related to the choice of a trade-off between the two methods. We propose here an improved complexity analysis dedicated to quadratic systems. Whilst the principle of the hybrid approach is simple, its careful analysis leads to rather surprising and somehow unexpected results. We prove that the optimal trade-off (i.e. number of variables to be fixed) allowing to minimize the complexity is achieved by fixing a number of variables proportional to the number of variables of the system considered, denoted n. Under some natural algebraic assumption, we show that the asymptotic complexity of the hybrid approach is 2(3.31-3.62 log2(q)-1)n, where q is the size of the field (under the condition in particular that log(q) ≪ n). This is to date, the best complexity for solving PoSSo over finite fields (when q > 2). We have been able to quantify the gain provided by the hybrid approach compared to a direct Gröbner basis method. For quadratic systems, we show (assuming a natural algebraic assumption) that this gain is exponential in the number of variables. Asymptotically, the gain is 21.49n when both n and q grow to infinity and log(q) ≪ n.Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation; 07/2012
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ABSTRACT: This work presents a study of the complexity of the Blum–Kalai–Wasserman (BKW) algorithm when applied to the Learning with Errors (LWE) problem, by providing refined estimates for the data and computational effort requirements for solving concrete instances of the LWE problem. We apply this refined analysis to suggested parameters for various LWE-based cryptographic schemes from the literature and compare with alternative approaches based on lattice reduction. As a result, we provide new upper bounds for the concrete hardness of these LWE-based schemes. Rather surprisingly, it appears that BKW algorithm outperforms known estimates for lattice reduction algorithms starting in dimension $n \approx 250$ n ≈ 250 when LWE is reduced to SIS. However, this assumes access to an unbounded number of LWE samples.Designs Codes and Cryptography 02/2015; 74(2). DOI:10.1007/s10623-013-9864-x · 0.73 Impact Factor
Conference Paper: A fully homomorphic cryptosystem with approximate perfect secrecy[Show abstract] [Hide abstract]
ABSTRACT: We propose a new fully homomorphic cryptosystem called Symmetric Polly Cracker (SymPC) and we prove its security in the information theoretical settings. Namely, we prove that SymPC approaches perfect secrecy in bounded CPA model as its security parameter grows (which we call approximate perfect secrecy). In our construction, we use a Gröbner basis to generate a polynomial factor ring of ciphertexts and use the underlying field as the plaintext space. The Gröbner basis equips the ciphertext factor ring with a multiplicative structure that is easily algorithmized, thus providing an environment for a fully homomorphic cryptosystem.Proceedings of the 13th international conference on Topics in Cryptology; 02/2013