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Vestnik YuUrGU. Ser. Mat. Model. Progr., 2020, Volume 13, Issue 2, Pages 54–68 (Mi vyuru543)  

Mathematical Modelling

Commutative encryption method based on hidden logarithm problem

D. N. Moldovyan, N. A. Moldovyan, A. A. Moldovyan

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

Abstract: A candidate for post-quantum commutative encryption algorithm is proposed, which is based on the hidden discrete logarithm problem defined in a new 6-dimensional finite non-commutative associative algebra. The properties of the algebra are investigated in detail and used in the design of the proposed commutative cipher. The formulas describing the set of $p^2$ different global right-sided units contained in the algebra and local left-sided units are derived. Homomorphisms of two different types are considered and used in the commutative cipher. The encrypted message is represented in the form of a locally invertible element $T $ of the algebra and encryption procedure includes performing the exponentiation operation and homomorphism map followed by the left-sided multiplication by a randomly selected local right-sided unit. The introduced commutative cipher is secure to the known-plaintext attacks and has been used to develop the post-quantum no-key encryption protocol providing possibility to send securely a secret message via a public channel without using any pre-agreed key. The proposed commutative encryption algorithm is characterized in using the single-use keys that are selected at random directly during the encryption process.

Keywords: commutative encryption, probabilistic cipher, post-quantum cryptoscheme, no-key protocol, finite non-commutative algebra, associative algebra, global unit, right-sided unit.

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/mmp200205

Full text: PDF file (237 kB)
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UDC: 681.3
MSC: 94A60, 16Z05, 14G50, 11T71, 16S50 \hfill 10.14529/mmp200205
Received: 17.06.2019
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Citation: D. N. Moldovyan, N. A. Moldovyan, A. A. Moldovyan, “Commutative encryption method based on hidden logarithm problem”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 13:2 (2020), 54–68

Citation in format AMSBIB
\Bibitem{MolMolMol20}
\by D.~N.~Moldovyan, N.~A.~Moldovyan, A.~A.~Moldovyan
\paper Commutative encryption method based on hidden logarithm problem
\jour Vestnik YuUrGU. Ser. Mat. Model. Progr.
\yr 2020
\vol 13
\issue 2
\pages 54--68
\mathnet{http://mi.mathnet.ru/vyuru543}
\crossref{https://doi.org/10.14529/mmp200205}


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