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On the Complexity of Solving quadratic boolean systems
, 2012
"... A fundamental problem in computer science is to find all the common zeroes of m quadratic polynomials in n unknowns over F2. The cryptanalysis of several modern ciphers reduces to this problem. Up to now, the best complexity bound was reached by an exhaustive search in 4log 2 n2 n operations. We giv ..."
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A fundamental problem in computer science is to find all the common zeroes of m quadratic polynomials in n unknowns over F2. The cryptanalysis of several modern ciphers reduces to this problem. Up to now, the best complexity bound was reached by an exhaustive search in 4log 2 n2 n operations. We give an algorithm that reduces the problem to a combination of exhaustive search and sparse linear algebra. This algorithm has several variants depending on the method used for the linear algebra step. Under precise algebraic assumptions on the input system, we show that the deterministic variant of our algorithm has complexity bounded by O(2 0.841n) when m = n, while a probabilistic variant of the Las Vegas type has expected complexity O(2 0.792n). Experiments on random systems show that the algebraic assumptions are satisfied with probability very close to 1. We also give a rough estimate for the actual threshold between our method and exhaustive search, which is as low as 200, and thus very relevant for cryptographic applications.
Solving polynomial systems over finite fields: Improved analysis of the hybrid approach
 In: ISSAC’12: Proceedings of the 2012 International Symposium on Symbolic and Algebraic Computation
, 2012
"... 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 ..."
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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