arXiv subjects

A. A. Chilikov

Publications and source records attributed to A. A. Chilikov.

3 recordsLinked to original sources

arXiv2020

Advanced attribute-based protocol based on the modified secret sharing scheme

We construct a new protocol for attribute-based encryption with the use of the modification of the standard secret sharing scheme. In the suggested modification of the secret sharing scheme, only one master key for each user is required that is achieved by linearly enlarging public parameters in the access formula. We then use this scheme for designing an attribute-based encryption protocol related to some access structure in terms of attributes. We demonstrate that the universe of possible attributes does not affect the resulting efficiency of the scheme. The security proofs for both constructions are provided.

cs.CR↗

arXiv2019

On algorithmic unsolvability of the problem embeddability of algebraic varieties over a field of characteristic zero

We show that for two afii varieties over an arbitrary field of characteristic zero, there is no general form of an algorithm for checking the presence of an embedding of one algebraic variety in another. Moreover, we establish this for affine varieties whose coordinate rings are given by generators and defining relations. Moreover, one of these varieties can be taken as an affine space, and in the case of a field of real numbers, an affine line.

math.AG↗

arXiv2019

Normal basises of algebras and Exponential Diophantine equations in rings of positive characteristic

In this paper we discourse basises of representable algebras. This question lead to arithmetic problems. We prove algorithmical solvability of exponential-Diophantine equations in rings represented by matrices over fields of positive characteristic. Consider the system of exponential-Diophantine equations $$ \sum\limits_{i=1}^s P_{ij}(n_1,\dots,n_t) b_{ij0} a_{ij1}^{n_1} b_{ij1} \dots a_{ijt}^{n_t}b_{ijt}=0 $$ where $b_{ijk},a_{ijk}$ are constants from matrix ring of characteristic $p$, $n_i$ are indeterminates. For any solution $(n_1,\dots,n_t)$ of the system we construct a word (over an alphabet containing $p^t$ symbols) ${\overline α_0},\dots,{\overline α_q}$ where ${\overline α_i}$ is a $t$-tuple $\langle n_1^{(i)},\dots,n_t^{(i)}\rangle$, $n^{(i)}$ is the $i$-th digit in the $p$-adic representation of $n$. The main result of this paper is as follows: the set of words corresponding in this sense to solutions of a system of exponential-Diophantine equations is a regular language (i.e. recognizable by a finite automaton). There exists an effective algorithm which calculates this language. This algorithm is constructed in the paper.

math.RA↗