Investigation of the Practical Possibility of Solving Problems on Generalized Cellular Automata Associated with Cryptanalysis by Mean Algebraic Methods

Abstract

A number of previous author’s papers proposed methods for constructing various cryptographic algorithms, including block ciphers and cryptographic hash functions, based on generalized cellular automata. This one is aimed at studying a possibility to use the algebraic cryptanalysis methods related to the construction of Grobner bases for the generalized cellular automata to be applied in cryptography, i.e. this paper studies the possibility for using algebraic cryptanalysis methods to solve the problems of inversion of a generalized cellular automaton and recovering the key of such an automaton. If the cryptographic algorithm is represented as a system of polynomial equations over a certain finite field, then its breach is reduced to solving this system with respect to the key. Although the problem of solving a system of polynomial equations in a finite field is NP-difficult in the general case, the solution of a particular system can have low computational cost. Cryptanalysis based on the construction of a system of polynomial equations that links plain text, cipher-text and key, and its solution by algebraic methods, is usually called algebraic cryptanalysis. Among the main methods to solve systems of polynomial equations are those to construct Grobner bases. Cryptanalysis of ciphers and hash functions based on generalized cellular automata can be reduced to various problems. We will consider two such problems: the problem of inversion of a generalized cellular automaton, which, in case we know the values of the cells after k iterations, enables us to find the initial values. And the task of recovering the key, which is to find the initial values of the remaining cells, using the cell values after k steps and the initial values of a part of the cells. A computational experiment was carried out to solve the two problems above stated in order to determine the maximum size of a generalized cellular automaton for which the solution of these problems was possible. Using a Python language program, random 6-regular Ramanujan graphs with the appropriate number of vertices were generated. For each graph, was generated a system of equations that describes the k steps of the corresponding generalized cellular automaton. For the systems obtained, the Grebner bases were constructed using the Fouger algorithm F4, the Magma system v2.21-5, and the Polybori 0.8.3 library. The experiments were carried out both for the inversion task and for the key recovery task. We used a 16-core 16 GB RAM Intel Xeon E5-2690 computer, OS Linux. The article presents the results of experiments that confirm that the algebraic cryptanalysis of block ciphers and hash functions based on generalized cellular automata with the number of cells used in practice (of the order of several hundred or more) available tool based on the use of Grobner bases, is impossible.