Inconsistency Probability of Sparse Equations over F2
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.
Cite
@article{arxiv.2603.24890,
title = {Inconsistency Probability of Sparse Equations over F2},
author = {P. Horak and I. Semaev},
journal= {arXiv preprint arXiv:2603.24890},
year = {2026}
}