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Inconsistency Probability of Sparse Equations over F2

P. Horak, I. Semaev

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.

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.

Paper Structure

This paper contains 16 sections, 18 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Problem
  3. Motivation
  4. Sparse equations in cryptanalysis
  5. Sparse equations and $k$-SAT problem
  6. Bounds on inconsistency probability
  7. Auxiliary results
  8. Inconsistency probability for systems with $2$ and $3$ equations
  9. Main Results
  10. Inconsistency Probability for a Complete $k$-hypergraph
  11. $2$-sparse system of equations
  12. Preliminaries
  13. Trees with extremal consistency probabilities
  14. Probability of a path
  15. Cycle-free graph with maximum probability
  16. ...and 1 more sections

Key Result

Proposition 1

Let (system) be a randomly generated system with $m$ equations, and $\left\vert X\right\vert =n$. Then

Theorems & Definitions (20)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Theorem 5
  • Theorem 6
  • Lemma 7
  • Theorem 8
  • Lemma 9
  • Claim 10
  • ...and 10 more