Modified Matrix Modular Cryptosystems

Abstract

The Basic Matrix Modular Cryptosystem (BMMC) is a public-key cryptosystem, which uses some matrix modular exponentiations in the matrix ring over the residue ring modulo n. The aim of this article is to decrease the number of these exponentiations and consequently to accelerate the execution of encryption algorithm. There are two ways to reach this aim. First way is to determine the large abelian subgroup in general linear group over the large residue ring and to choose the session keys in this subgroup, what will be to give the encryption without exponentiations. Other way is to use random integral exponent of the given matrix in the public key as session key and this will be the only exponentiation in encryption algorithm. A discussion about the security of built modifications made in the article shows that the level of security is high enough for an appropriate choice of parameters of the cryptosystems, namely, the lower bound for the selection of secure modulus n is 40-bit integer. Both modified cryptosystems are faster than BMMC and balanced with respect to a pair of security – efficiency, and BMMC is much faster than RSA.