Implementing Basic Arithmetic in $\mathbb{F}_p$ via $\mathbb{F}_2$, and Its Application for Computing the Hamming Distance of Linear Codes

TL;DR

This paper introduces an efficient method for arithmetic in finite fields $ ext{F}_p$ using binary operations, enabling faster computation of Hamming distances in linear codes.

Contribution

It presents a novel approach to perform finite field arithmetic via binary operations, improving the efficiency of Hamming distance calculations for certain fields.

Findings

The new arithmetic method outperforms existing software in computing Hamming distances.

Implementation in C shows significant speedups on various processor architectures.

The approach enables new techniques like isometric addition to accelerate code analysis.

Abstract

We present a new general method for performing basic arithmetic in the finite field~$\mathbb{F}_p$ for any prime $p>2$ by using traditional binary operations over~$\mathbb{F}_2$. Our new approach is efficient and competitive with current state-of-art methods. We apply our new arithmetic method to the computation of the minimum Hamming distance of random linear codes for the fields $\mathbb{F}_3$ and $\mathbb{F}_7$. Our new arithmetic method allows to apply new techniques such as the isometric addition that accelerate the computation of the Hamming distance. We have developed implementations in the C programming language for computing the Hamming distance that clearly outperform both state-of-art licensed software and open-source software such as \textsc{Magma} and \textsc{GAP}/\textsc{Guava} on single-core processors, multicore processors, and shared-memory multiprocessors.