Details:
Title  On the complexity of solving quadratic Boolean systems  Author(s)  Magali Bardet, JeanCharles Faugère, Bruno Salvy, PierreJean Spaenlehauer  Type  Article in Journal  Abstract  A fundamental problem in computer science is that of finding all the common zeros of m quadratic polynomials in n unknowns over F 2 . The cryptanalysis of several modern ciphers reduces to this problem. Up to now, the best complexity bound was reached by an exhaustive search in 4 log 2 n 2 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. We show that, under precise algebraic assumptions on the input system, the deterministic variant of our algorithm has complexity bounded by O ( 2 0.841 n ) when m = n , while a probabilistic variant of the Las Vegas type has expected complexity O ( 2 0.792 n ) . 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.  Keywords  Boolean quadratic system, Gröbner bases, Complexity, Semiregularity, Multivariate cryptography  ISSN  0885064X 
URL 
http://www.sciencedirect.com/science/article/pii/S0885064X12000611 
Language  English  Journal  Journal of Complexity  Volume  29  Number  1  Pages  53  75  Year  2013  Edition  0  Translation 
No  Refereed 
No 
