On the complexity of solving quadratic Boolean systems

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) 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. We show that, under precise algebraic assumptions on the input system, 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. (c) 2012 Elsevier Inc. All rights reserved.