Transformation Method for Solving System of Boolean Algebraic Equations

Abstract

In recent years, various methods and directions for solving a system of Boolean algebraic equations have been invented, and now they are being very actively investigated. One of these directions is the method of transforming a system of Boolean algebraic equations, given over a ring of Boolean polynomials, into systems of equations over a field of real numbers, and various optimization methods can be applied to these systems. In this paper, we propose a new transformation method for Solving Systems of Boolean Algebraic Equations (SBAE). The essence of the proposed method is that firstly, SBAE written with logical operations are transformed (approximated) in a system of harmonic-polynomial equations in the unit n-dimensional cube Kn with the usual operations of addition and multiplication of numbers. Secondly, a transformed (approximated) system in Kn is solved by using the optimization method. We substantiated the correctness and the right to exist of the proposed method with reliable evidence. Based on this work, plans for further research to improve the proposed method are outlined.

Polylinear Transformation Method for Solving Systems of Logical Equations

“…Solving a system of logical equations has many applications, such as synthesis, output data coding, the state assignment of finite automata, the modeling and testing of digital networks, automatic test pattern generation and the determination of the initial state in circuits, timing analysis, and the generation of delayed failure tests for combinational circuits. The solution of a system of logical equations in the field of cryptography is used to analyze and break block ciphers, since they can be reduced to the problem of solving large-scale systems of logical equations [1][2][3][4][5][6]. This is because, for a specific cipher, algebraic cryptanalysis consists of two stages: the transformation of the cipher into a system of polynomial equations (usually over Boolean ring), and the solution of the resulting system of polynomial equations [7][8][9].…”

“…Many of the applied algorithms for solving a system of logical equations or the Boolean satisfiability problem (SAT) that have been developed so far solve the problem in the Boolean domain. However, other areas have recently been developed and improved [11,12,14]. One of these directions is transformation into the real continuous domain.…”

“…The real continuous domain is a richer domain to work with, as it involves many, well developed algorithms. Already in a real continuous domain, the transformed system can be reduced to a numerical optimization problem, making it possible to apply, analyze and combine such methods as the steepest descent algorithm, Newton's method and the coordinate descent algorithm [1][2][3][4][5][6]14].…”

“…More recently, Barotov Dostonjon et al [14] proposed a new idea with their interesting formulas for solving systems of logical equations. The essence of their proposed method is that systems of logical equations are transformed into systems of harmonicpolynomial equations in a unit n-dimensional cube K n with the usual operations of the addition and multiplication of numbers, and the transformed system in K n is solved by an optimization method.…”

“…Since the proposed idea is new and under development, some of its open problems have not been solved [14]. Firstly, the corresponding system of harmonic-polynomial equations was found and the equivalence of the systems was established only when each polynomial of the system of logical equations consisted of pairwise coprime monomials.…”