特征二中的确定性NC二次根计数

中文摘要 AI 辅助

计算布尔公式的满足赋值数量是理论计算机科学中的一个基本问题，其中$\\#3\text{-}\SAT$是标准的$\SharpP$-完全问题。更一般地，在$\F_2$上计算多项式方程组解的数量是$\SharpP$-完全的。这里我们关注更结构化的问题：计算单个多项式方程的解的数量。对于有限域上的多项式方程，Ehrenfeucht和Karpinski \cite{computationalcomplexityofxorandcountingproblems1990}展示了二次和三次之间的显著差异：二次根计数可在多项式时间内解决，而三次问题是$\SharpP$-完全的。他们的二次算法是顺序的。对于固定有限域，Ishai等人~\cite{ishai2012randomizing}后来在奇特征中给出了确定性并行算法，在特征二中给出了随机并行算法。我们给出了一个确定性$\NC$算法，用于精确计算特征二的每个固定有限域上二次多项式方程的解的数量。我们的算法分离根式，并使用绝对迹来实现区分两种非退化有限域类型的位，作为$\F_2$上二次形式的Arf不变量 \cite{arf1941untersuchungen}。然后使用Browder的行列式准则 \cite{browder2006complete}从整数矩阵中恢复该不变量。这取代了Ishai等人算法中的随机规范形式步骤。

英文摘要

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.

发表机构

• Saarland University（萨尔兰大学）
• University of Regensburg（雷根斯堡大学）

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