Beating Brute Force for Systems of Polynomial Equations over Finite Fields

Abstract

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(q n ) time algorithm in the worst case. We present two algorithms, a randomized algorithm with running time q n+o (n) • q −n/O(k) time if q ≤ 2 4ekd , and q n+o(n) • ( log q 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(kq 6/7d ) . For the important special case of quadratic equations in F 2 , our randomized algorithm has running time O(2 0.8765n ).For systems over F 2 we also consider the case where the input polynomials do not have bounded degree, but instead can be efficiently represented as a ΣΠΣ circuit, i.e., a sum of products of sums of variables. For this case we present a deterministic algorithm running in time 2 n−δ n 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 ΣΠΣ 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)).

A Crossbred Algorithm for Solving Boolean Polynomial Systems

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Additive-error fine-grained quantum supremacy

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How many qubits are needed for quantum computational supremacy?

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