Deterministic NC Quadratic Root Counting in Characteristic Two

Abstract

Counting satisfying assignments of Boolean formulas is a basic problem in theoretical computer science, with $\#3\text{-}\SAT$ as the standard $\SharpP$-complete problem. More generally, counting the solutions of a system of polynomial equations over $\F_2$ is $\SharpP$-complete. Here we focus on the more structured problem of counting the solutions of a single polynomial equation. For polynomial equations over finite fields, Ehrenfeucht and Karpinski \cite{computationalcomplexityofxorandcountingproblems1990} showed a sharp difference between degrees two and three: quadratic root counting is solvable in polynomial time, while the degree-three problem is $\SharpP$-complete. Their quadratic algorithm is sequential. For fixed finite fields, Ishai et al.~\cite{ishai2012randomizing} later gave deterministic parallel algorithms in odd characteristic and randomized parallel algorithms in characteristic two. We give a deterministic $\NC$ algorithm for exactly counting the solutions of a quadratic polynomial equation over every fixed finite field of characteristic two. Our algorithm separates the radical and uses the absolute trace to realize the bit distinguishing the two nondegenerate finite-field types as the Arf invariant \cite{arf1941untersuchungen} of a quadratic form over $\F_2$. It then recovers that invariant from an integral matrix using Browder's determinant criterion \cite{browder2006complete}. This replaces the randomized canonical-form step in the algorithm of Ishai et al.