On the Low Weight Polynomial Multiple Problem

TL;DR

A relationship is established 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 that shows that any algorithm that can compute the optimum of a MAX-SAT instance can also compute the optimum of an equivalent LWPM instance.

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.

Figures (4)

• Figure 1: The reduction $(f,g)$ in Theorem \ref{['T-Red_to_MAX']}
• Figure 2: Probabilistic reduction from MAX-SAT to MIN-PM for matrices of size $40\times 30$
• Figure 3: Probabilistic reduction from MAX-SAT to MIN-PM for matrices of size $400\times 200$
• Figure 4: Probabilistic reduction from MAX-SAT to MIN-PM for matrices of size $1000\times 500$

Theorems & Definitions (5)

• Proposition 1
• proof
• Theorem 1
• proof
• Remark 1