Generalized Galois and Fibonacci Matrices in Cryptographic Applications

Abstract

The terms of the Galois matrices , as well as those bijectively associated with them the Fibonacci matrices connect by the operator of the right-hand transposition (that is, transposition to the auxiliary diagonal), are borrowed from the theory of cryptography, in which generators of pseudorandom number (PRN) widely use according to Galois and Fibonacci schemes (in configuration). A distinctive feature of both the and matrices is that the identical binary sequences can programmatically calculate the sequences formed by the PRN generators. The latter's constructions are based on linear feedback shift registers, implemented by software or hardware methods in Galois and Fibonacci architecture. The proposed generalized Galois matrices, discussed in the Chapter, significantly expand the variety of PRN generators. That is achieved both by increasing the number of generating elements (in the classical version used a single element ) and since generalized generators can construct not only using PRN but also polynomials, not necessarily (as in classical generators), which are primitive. The listed features of generalized Galois matrices provide PRN generators with significantly higher cryptographic security than generators based on conventional matrices.