Inconsistency Probability of Sparse Equations over F2

TL;DR

This paper analyzes the probability that sparse polynomial systems over GF(2) are inconsistent, revealing how hypergraph structure influences this probability and providing bounds and formulas for specific cases.

Contribution

It introduces a hypergraph-based approach to study inconsistency probability, deriving bounds and explicit formulas for special hypergraph structures, advancing understanding of sparse systems over GF(2).

Findings

Inconsistency probability depends on hypergraph structure.

Derived tight asymptotics for complete k-uniform hypergraphs.

Provided explicit formulas for paths and stars in 2-sparse systems.

Abstract

Let n denote the number of variables and m the number of equations in a sparse polynomial system over the binary field. We study the inconsistency probability of randomly generated sparse polynomial systems over the binary field, where each equation depends on at most k variables and the number of variables grows. Associating the system with a hypergraph, we show that the inconsistency probability depends strongly on structural properties of this hypergraph, not only on n,m, and k. Using inclusion--exclusion, we derive general bounds and obtain tight asymptotics for complete k-uniform hypergraphs. In the 2-sparse case, we provide explicit formulas for paths and stars, characterize extremal trees and forests, and conjecture a formula for cycles. These results have implications for SAT solving and cryptanalysis.