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Vestnik YuUrGU. Ser. Mat. Model. Progr., 2019, Volume 12, Issue 1, Pages 66–81 (Mi vyuru472)  

Mathematical Modelling

Finite non-commutative associative algebras as carriers of hidden discrete logarithm problem

N. A. Moldovyan, A. A. Moldovyan

St. Petersburg Institute for Informatics and Automation of Russian Academy of Sciences, St. Petersburg, Russian Federation

Abstract: The article introduces new finite algebras attractive as carriers of the discrete logarithm problem in a hidden group. In particular new $4$-dimensional and $6$-dimensional finite non-commutative algebras with associative multiplication operation and their properties are described. It is also proposed a general method for defining finite non-commutative associative algebras of arbitrary even dimension $m\ge 2$. Some of the considered algebras contain a global unit, but the other ones include no global unit element. In the last case the elements of the algebra are invertible locally relatively local bi-side units that act in the frame of some subsets of elements of algebra. For algebras of the last type there have been derived formulas describing the sets of the (right-side, left-side, and bi-side) local units. Algebras containing a large set of the global single-side (left-side and right-side) units and no global bi-side unit are also introduced. Since the known form of defining the hidden discrete logarithm problem uses invertibility of the elements of algebra relatively global unit, there are introduced new forms of defining this computationally difficult problem. The results of the article can be applied for designing public-key cryptographic algorithms and protocols, including the post-quantum ones. For the first time it is proposed a digital signature scheme based on the hidden discrete logarithm problem.

Keywords: finite associative algebra, non-commutative algebra, global unit, left-side units, local unit, local invertibility, discrete logarithm problem, public-key cryptoscheme, digital signature, post-quantum cryptography.

Funding Agency Grant Number
Russian Foundation for Basic Research 18-07-00932_a
The reported study was partially funded by Russian Foundation for Basic Research (project no. 18-07-00932-a).


DOI: https://doi.org/10.14529/mmp190106

Full text: PDF file (240 kB)
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UDC: 512.624.5
MSC: 94A60, 16Z05, 14G50, 11T71, 16S50
Received: 11.09.2018
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Citation: N. A. Moldovyan, A. A. Moldovyan, “Finite non-commutative associative algebras as carriers of hidden discrete logarithm problem”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 12:1 (2019), 66–81

Citation in format AMSBIB
\Bibitem{MolMol19}
\by N.~A.~Moldovyan, A.~A.~Moldovyan
\paper Finite non-commutative associative algebras as carriers of hidden discrete logarithm problem
\jour Vestnik YuUrGU. Ser. Mat. Model. Progr.
\yr 2019
\vol 12
\issue 1
\pages 66--81
\mathnet{http://mi.mathnet.ru/vyuru472}
\crossref{https://doi.org/10.14529/mmp190106}
\elib{http://elibrary.ru/item.asp?id=37092204}


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