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SAT Solving Using XOR-OR-AND Normal Forms

Bernhard Andraschko, Julian Danner, Martin Kreuzer

TL;DR

An algorithm which converts Boolean polynomials efficiently from their Algebraic Normal Form (ANF) to formulas in 2-XNF, and a new DPLL-based SAT solver for formulas in 2-XNF, which outperforms state-of-the-art alternative solving approaches.

Abstract

This paper introduces the XOR-OR-AND normal form (XNF) for logical formulas. It is a generalization of the well-known Conjunctive Normal Form (CNF) where literals are replaced by XORs of literals. As a first theoretic result, we show that every CNF formula is equisatisfiable to a formula in 2-XNF, i.e., a formula in XNF where each clause involves at most two XORs of literals. Subsequently, we present an algorithm which converts Boolean polynomials efficiently from their Algebraic Normal Form (ANF) to formulas in 2-XNF. Experiments with the cipher ASCON-128 show that cryptographic problems, which by design are based strongly on XOR-operations, can be represented using far fewer variables and clauses in 2-XNF than in CNF. In order to take advantage of this compact representation, new SAT solvers based on input formulas in 2-XNF need to be designed. By taking inspiration from graph-based 2-CNF SAT solving, we devise a new DPLL-based SAT solver for formulas in 2-XNF. Among others, we present advanced pre- and in-processing techniques. Finally, we give timings for random 2-XNF instances and instances related to key recovery attacks on round reduced ASCON-128, where our solver outperforms state-of-the-art alternative solving approaches.

SAT Solving Using XOR-OR-AND Normal Forms

TL;DR

An algorithm which converts Boolean polynomials efficiently from their Algebraic Normal Form (ANF) to formulas in 2-XNF, and a new DPLL-based SAT solver for formulas in 2-XNF, which outperforms state-of-the-art alternative solving approaches.

Abstract

This paper introduces the XOR-OR-AND normal form (XNF) for logical formulas. It is a generalization of the well-known Conjunctive Normal Form (CNF) where literals are replaced by XORs of literals. As a first theoretic result, we show that every CNF formula is equisatisfiable to a formula in 2-XNF, i.e., a formula in XNF where each clause involves at most two XORs of literals. Subsequently, we present an algorithm which converts Boolean polynomials efficiently from their Algebraic Normal Form (ANF) to formulas in 2-XNF. Experiments with the cipher ASCON-128 show that cryptographic problems, which by design are based strongly on XOR-operations, can be represented using far fewer variables and clauses in 2-XNF than in CNF. In order to take advantage of this compact representation, new SAT solvers based on input formulas in 2-XNF need to be designed. By taking inspiration from graph-based 2-CNF SAT solving, we devise a new DPLL-based SAT solver for formulas in 2-XNF. Among others, we present advanced pre- and in-processing techniques. Finally, we give timings for random 2-XNF instances and instances related to key recovery attacks on round reduced ASCON-128, where our solver outperforms state-of-the-art alternative solving approaches.
Paper Structure (12 sections, 17 theorems, 22 equations, 5 figures, 8 algorithms)

This paper contains 12 sections, 17 theorems, 22 equations, 5 figures, 8 algorithms.

Table of Contents

  1. Introduction
  2. The Ring of Boolean Polynomials
  3. Logical Representations of Boolean Polynomials
  4. Reduction of Formulas in XNF to 2-XNF
  5. 2-XNF Representations of Boolean Polynomials
  6. Graph-based 2-XNF SAT Solving
  7. Implication Graph Structures
  8. Pre-Processing Techniques
  9. In-Processing Techniques
  10. Decision Heuristics
  11. Graph-based 2-XNF DPLL-Solving
  12. Experiments and Timings

Key Result

Lemma 3.6

Let $L_1,L_2$ be two linerals, and let $Y$ be an additional logical variable. Then we have

Figures (5)

  • Figure 1: Implication graph $(V_0,E_0)$ from Example \ref{['ex:igs']}.
  • Figure 2: Implication graph $(V_1,E_1)$ from Example \ref{['ex:igs2']}.
  • Figure 3: Implication graph $(V_2,E_2)$ from Example \ref{['ex:igs3']}.
  • Figure 4: Cactus plots for the random benchmark suites.
  • Figure 5: Cactus plot for the benchmark suite consisting of $400$ satisfiable instances related to key-recovery attacks on round-reduced $\mathtt{Ascon\text{-}128}$.

Theorems & Definitions (65)

  • Definition 3.1
  • Definition 3.2: XOR-OR-AND Normal Form
  • Remark 3.3
  • Definition 3.4
  • Remark 3.5
  • Lemma 3.6
  • Proposition 3.7
  • proof
  • Example 3.8
  • Remark 3.9
  • ...and 55 more