Conference Paper
Polly Cracker, Revisited.
DOI: 10.1007/9783642253850_10 In proceeding of: Advances in Cryptology  ASIACRYPT 2011  17th International Conference on the Theory and Application of Cryptology and Information Security, Seoul, South Korea, December 48, 2011. Proceedings
Source: DBLP

Conference Paper: Solving polynomial systems over finite fields: improved analysis of the hybrid approach
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ABSTRACT: The Polynomial System Solving (PoSSo) problem is a fundamental NPHard problem in computer algebra. Among others, PoSSo have applications in area such as coding theory and cryptology. Typically, the security of multivariate publickey 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 socalled 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 tradeoff 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 tradeoff (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(3.313.62 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 
Conference Paper: A fully homomorphic cryptosystem with approximate perfect secrecy
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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 
Conference Paper: Polly cracker, revisited, revisited
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ABSTRACT: In this paper, we consider the Polly Cracker with Noise (PCN) cryptosystem by Albrecht, Farshim, Faugère, and Perret (Asiacrypt 2011), which is a publickey cryptosystem based on the hardness of computing Gröbner bases for noisy random systems of multivariate equations. We examine four settings, covering all possible parameter ranges of PCN with zerodegree noise. In the first setting, the PCN cryptosystem is known to be equivalent to Regev's LWEbased scheme. In the second, it is known to be at most as secure as Regev's scheme. We show that for one other settings it is equivalent to a variants of Regev's with less efficiency and in the last setting it is completely insecure and we give an efficient keyrecovery attack. Unrelated to the attack, we also fix some flaws in the security proofs of PCN.Proceedings of the 15th international conference on Practice and Theory in Public Key Cryptography; 05/2012
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