Том 223, № 5 (2017)
- Жылы: 2017
- Мақалалар: 13
- URL: https://journals.rcsi.science/1072-3374/issue/view/14831
Article
Alexandr Alexandrovich Nechaev (7.8.1945–14.11.2014)
On the Multiplicative Groups of Free and Free Commutative Quasigroups
Аннотация
We study the structure of the multiplicative groups for the relatively free algebras of some quasigroup varieties, in particular, for the varieties of all quasigroups, all commutative quasigroups, and all TS-quasigroups. In all these cases, the corresponding multiplicative group is free. The work is dedicated to the glorious memory of Prof. Alexandr Alexandrovich Nechaev, who actively explored the possibility of application of quasigroups in cryptography during recent years.
On the Word Problem in the Free Quasigroups in the Varieties of Quasigroups Isotopic to Groups
Аннотация
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.
Some Homomorphic Cryptosystems Based on Nonassociative Structures
Аннотация
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 Prime Radical of Alternative Rings and Loops
Аннотация
A characterization of the prime radical of loops as the set of strongly Engel elements was given in our earlier paper. In this paper, some properties of the prime radical of loops are considered. Also a connection between the prime radical of the loop of units of an alternative ring and the prime radical of this ring is given.
Cryptographic Algorithms on Groups and Algebras
Аннотация
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.
On Binary Digit-Position Sequences over Galois Rings, Admitting an Effect of Reduction of Period
Аннотация
A class of binary digit-position sequences, obtained from the linear recurring sequence of maximal period over Galois rings of odd characteristics and admitting an effect of twofold reduction of period, has been found. A condition was found where sequences of some fixed linear recurring sequence of maximal period over Galois fields with such property are exhausted only by that class.
Experimental Study of the Hypothesis on the Order of a Random Element of the Matrix Modular Group
Аннотация
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.