Beating brute force for systems of polynomial equations over finite fields

We consider the problem of solving systems of multivariate polynomial equations of degree k over a finite field. For every integer k ≥ 2 and finite field F_q where q = p^d for a prime p, we give, to the best of our knowledge, the first algorithms that achieve an exponential speedup over the brute force O(qn) time algorithm in the worst case. We present two algorithms, a randomized algorithm with running time qn+o(n) · q^n+O(k) time if q ≤ 2^4ekd, and q^n+O(n) · ( logq dek ) -dn otherwise, where e = 2.718. . . is Napier's constant, and a deterministic algorithm for counting solutions with running time q^n+O(n) · q^n+O(kq6/7d ). For the important special case of quadratic equations in F2, our randomized algorithm has running time O(20.8765n). For systems over F2 we also consider the case where the input polynomials do not have bounded degree, but instead can be efficiently represented as a circuit, i.e., a sum of products of sums of variables. For this case we present a deterministic algorithm running in time 2^n-δn for δ= 1/O(log(s/n)) for instances with s product gates in total and n variables. Our algorithms adapt several techniques recently developed via the polynomial method from circuit complexity. The algorithm for systems of polynomials also introduces a new degree reduction method that takes an instance of the problem and outputs a subexponential-sized set of instances, in such a way that feasibility is preserved and every polynomial among the output instances has degree O(log(s/n)).