CNF Characterization of Sets over ℤ2n and Its Applications in Cryptography

• Published in: Journal of systems science and complexity Vol. 38; no. 4; pp. 1784 - 1802
• Main Authors: Hu, Xiaobo, Xu, Shengyuan, Tu, Yinzi, Feng, Xiutao, Zhang, Weihan
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2025; Springer Nature B.V
• Subjects: Algorithms; Approximation; Canonical forms; Complex Systems; Control; Cryptography; Mathematics; Mathematics and Statistics; Mathematics of Computing; Operations Research/Decision Theory; Search methods; Statistics; Systems Theory; Boolean satisfiability problem; Automatic cryptanalysis; full CNF characterization
• ISSN: 1009-6124; 1559-7067
• DOI: 10.1007/s11424-024-3168-2

In recent years, automatic search has been widely used to search differential characteristics and linear approximations with high probability/correlation in symmetric-key cryptanalyses. Among these methods, the automatic search with the Boolean satisfiability problem (SAT, in short) gradually becomes a powerful cryptanalysis tool. A key problem in the SAT-based automatic search method is how to fully characterize a set S ⊆ {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, the authors establish a complete theory to solve a best solution of an FCC problem. Specifically, the authors start from plain sets, which can be characterized by exactly one clause. By exploring mergeable sets, the authors give a sufficient and necessary condition for characterizing a plain set. Based on the properties of plain sets, the authors further provide an algorithm of solving an FCC problem for a given set S , which can generate all the minimal plain closures of S and produce a best FCC theoretically. As application, the authors apply the proposed algorithm to many common S -boxes used in block ciphers to characterize their differential distribution tables and linear approximation tables and get their FCCs, all of which are the best-known results at present.