Algorithms for Solving Systems of Boolean Equations Based on the Transformation of Logical Expressions
Kabulov A. Babadzhanov A. Baizhumanov A. Saymanov I. Babadjanov A.
February 2026Multidisciplinary Digital Publishing Institute (MDPI)
Mathematics
2026#14Issue 4
This manuscript proves specific theorems for transforming Boolean expressions of logical formulas when moving from one basis to another, simplifying the solution of complex equations, especially for cryptographic applications. The paper develops methods for solving specific nonlinear systems of Boolean equations used in cryptographic S-boxes using transformations to simpler forms, such as disjunctive normal forms (DNFs) and Zhegalkin polynomials. The main contributions include a mathematical basis for transforming formulas, a complexity-reducing grouping method, and the RLSY program for practical implementation. A rigorous theory, cryptographic relevance, and a detailed description of the algorithm are proposed. The grouping method reduces the system complexity by a factor of (Formula presented.), as shown in a test example, improving computational efficiency. A solution to a special class of systems of nonlinear Boolean equations of the second degree, which are a logical model of algebraic cryptanalysis, is also proposed. Test examples of logical formula transformations are given.
algebraic-logical method , analytical transformation , Boolean equation , elementary conjunction , logical formulas , Zhegalkin polynomials
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School of Mathematics and Natural Sciences, New Uzbekistan University, Movarounnahr 1, Tashkent, 100007, Uzbekistan
Applied Mathematics and Intelligent Technologies Faculty, National University of Uzbekistan, Tashkent, 100174, Uzbekistan
Department of Algorithmization, Engineering Federation of Uzbekistan, Tashkent, 100003, Uzbekistan
Department of Mathematics, O. Zhanibekov South Kazakhstan State Pedagogical University, A. Baitursynov Street No. 13, Shymkent, 160012, Kazakhstan
Department of Applied Informatics, Kimyo International University in Tashkent, Shota Rustaveli Str. 156, Tashkent, 100121, Uzbekistan
Department of Information Systems, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, 21250, MD, United States
School of Mathematics and Natural Sciences
Applied Mathematics and Intelligent Technologies Faculty
Department of Algorithmization
Department of Mathematics
Department of Applied Informatics
Department of Information Systems
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