Generalized Galois-Fibonacci Matrix Generators Pseudo-Random Sequences

Abstract

The article discusses various options for constructing binary generators of pseudo-random numbers (PRN) based on the so-called generalized Galois and Fibonacci matrices. The terms "Galois matrix" and "Fibonacci matrix" are borrowed from the theory of cryptography, in which the linear feedback shift registers (LFSR) generators of the PRN according to the Galois and Fibonacci schemes are widely used. The matrix generators generate identical PRN sequences as the LFSR generators. The transition from classical to generalized matrix PRN generators (PRNG) is accompanied by expanding the variety of generators, leading to a significant increase in their cryptographic resistance. This effect is achieved both due to the rise in the number of elements forming matrices and because generalized matrices are synthesized based on primitive generating polynomials and polynomials that are not necessarily primitive. Classical LFSR generators of PRN (and their matrix equivalents) have a significant drawback: they are susceptible to Berlekamp-Messi (BM) attacks. Generalized matrix PRNG is free from BM attack. The last property is a consequence of such a feature of the BM algorithm. This algorithm for cracking classical LFSR generators of PRN solves the problem of calculating the only unknown – a primitive polynomial generating the generator. For variants of generalized matrix PRNG, it becomes necessary to determine two unknown parameters: both an irreducible polynomial and a forming element that produces a generalized matrix. This problem turns out to be unsolvable for the BM algorithm since it is designed to calculate only one unknown parameter. The research results are generalized for solving PRNG problems over a Galois field of odd characteristics.