On the Low Weight Polynomial Multiple Problem

TL;DR

This paper explores the computational difficulty of finding low-weight multiples of polynomials, linking it to MAX-SAT problems, and confirms its complexity status through this relationship.

Contribution

It establishes a relationship between the LWPM problem and MAX-SAT, highlighting the problem's computational difficulty and providing new insights into its complexity.

Findings

LWPM problem is related to MAX-SAT.

Any MAX-SAT solver can solve LWPM instances.

The difficulty of LWPM is confirmed by this relationship.

Abstract

Finding a low-weight multiple (LWPM) of a given polynomial is very useful in the cryptanalysis of stream ciphers and arithmetic in finite fields. There is no known deterministic polynomial time complexity algorithm for solving this problem, and the most efficient algorithms are based on a time/memory trade-off. The widespread perception is that this problem is difficult. In this paper, we establish a relationship between the LWPM problem and the MAX-SAT problem of determining an assignment that maximizes the number of valid clauses of a system of affine Boolean clauses. This relationship shows that any algorithm that can compute the optimum of a MAX-SAT instance can also compute the optimum of an equivalent LWPM instance. It also confirms the perception that the LWPM problem is difficult.