Fast computation of permanents over $\mathbb{F}_3$ via $\mathbb{F}_2$ arithmetic

TL;DR

This paper introduces a novel method for efficiently computing matrix permanents over _3 by leveraging _2 arithmetic, resulting in significant speedups and insights into the distribution of permanents.

Contribution

The authors develop a fast _3 permanent computation algorithm using _2 operations, with practical implementation and performance analysis.

Findings

Achieved approximately 80x speedup in permanent computation.

Demonstrated the distribution of permanents tends to uniform as matrix size increases.

Provided Julia implementation and Monte Carlo simulation results.

Abstract

We present a method of representing an element of $\mathbb{F}_3^n$ as an element of $\mathbb{F}_n^2 \times \mathbb{F}_n^2$ which in practice will be a pair of unsigned integers. We show how to do addition, subtraction and pointwise multiplication and division of such vectors quickly using primitive binary operations (and, or, xor). We use this machinery to develop a fast algorithm for computing the permanent of a matrix in $\mathbb{F}_3^{n\times n}$. We present Julia code for a natural implementation of the permanent and show that our improved implementation gives, roughly, a factor of 80 speedup for problems of practical size. Using this improved code, we perform Monte Carlo simulations that suggest that the distribution of $\mbox{perm}(A)$ tends to the uniform distribution as $n \to \infty$.