Vol 223, No 5 (2017)

We prove that there exist non-Abelian group codes over an arbitrary finite field.

We consider the quasigroup varieties that are isotopic closures of the appropriate group varieties. We give the conditions for the word problem to be positively solvable simultaneously in the free algebras of the varieties of quasigroups and the corresponding groups.

A homomorphic encryption allows specific types of computations on ciphertext and generates an encrypted result that matches the result of operations performed on the plaintext. Some classic cryptosystems, e.g., RSA and ElGamal, allow homomorphic computation of only one operation. In 2009, C. Gentry suggested a model of a fully homomorphic algebraic system, i.e., a cryptosystem that supports both addition and multiplication operations. This cryptosystem is based on lattices. Later M. Dijk, C. Gentry, S. Halevi, and V. Vaikuntanathan suggested a fully homomorphic system based on integers. In a 2010 paper of A. V. Gribov, P. A. Zolotykh, and A. V. Mikhalev, a cryptosystem based on a quasigroup ring was constructed, developing an approach of S. K. Rososhek, and a homomorphic property of this system was investigated. An example of a quasigroup for which this system is homomorphic is given. Also a homomorphic property of the ElGamal cryptosystem based on a medial quasigroup is shown.

The paper concerns properties of fully indecomposable doubly stochastic matrices.

We analyze algorithms for open construction of a key on some noncommutative group. Algorithms of factorization and decomposition for associative algebras (of small dimension) are considered. A survey of applications (in particular, in cryptography) of so-called “hidden matrices” is given.

The hypothesis on the order of a random element of the matrix modular group is formulated as follows: a random element of a matrix group over the ring of residues modulo n with high probability has order greater than or equal to the value of the Euler function of n. If this hypothesis is correct, then it will be possible to significantly speed up the generation of the keys in the matrix modular cryptosystems, which will improve both efficiency and security of these cryptosystems. Experiments were carried out in five matrix modular groups by the scheme of the same type: first, a large sample of random elements of the group was formed, and then the orders of the elements of the sample were computed. Experimental results show that for all considered groups the orders of random elements satisfy the same probability distribution. Moreover, the probability that a random element of the group has “large order” (i.e., the order is greater than or equal to the value of the Euler function of n) was approximately the same in all considered groups, namely, about 0.85.