Paper 2023/300 Original: eprint.iacr.org/2023/300 ↗

CNF Characterization of Sets over $\mathbb{Z}_2^n$ and Its Applications in Cryptography

Abstract

In recent years, the automatic search has been widely used to search differential characteristics and linear approximations with high probability/correlation. Among these methods, the automatic search with the Boolean Satisfiability Problem (SAT, in short) gradually becomes a powerful cryptanalysis tool in symmetric ciphers. A key problem in the automatic search method is how to fully characterize a set $S \subseteq \{0,1\}^n$ with as few Conjunctive Normal Form (CNF, in short) clauses as possible, which is called a full CNF characterization (FCC, in short) problem. In this work, we establish a complete theory to solve a best solution of a FCC problem. Specifically, we start from plain sets, which can be characterized by exactly one clause. By exploring mergeable sets, we find a sufficient and necessary condition for characterizing a plain set. Based on the properties of plain sets, we further provide an algorithm for solving a FCC problem with $S$, which can generate all the minimal plain closures of $S$ and produce a best FCC theoretically. We also apply our algorithm to S-boxes used in block ciphers to characterize their differential distribution tables and linear approximation tables. All of our FCCs are the best-known results at present.