Algebraic Supports and New Forms of
the Hidden Discrete Logarithm Problem
for Post-quantum Public-key Cryptoschemes
Dmitriy Moldovyan1, Nashwan Al-Majmar2, and Alexander Moldovyan1
1St.
Petersburg Institute for Informatics and Automation of Russian Academy of
Sciences, Russia
2Computer Sciences
and Information Technology Department, Ibb University, Yemen
Abstract: This paper introduces two new forms of
the hidden discrete logarithm problem defined over a finite non-commutative
associative algebras containing a large set of global single-sided units. The proposed forms are promising
for development on their base practical post-quantum public key-agreement
schemes and are characterized in performing two different masking operations
over the output value of the base exponentiation operation that is executed in
framework of the public key computation. The masking operations represent
homomorphisms and each of them is mutually commutative with the exponentiation
operation. Parameters of the masking operations are used as private key
elements. A 6-dimensional algebra containing a set of p3 global left-sided units is used as
algebraic support of one of the hidden logarithm problem form and a
4-dimensional algebra with p2 global right-sided units is used to implement the
other form of the said problem. The result of this paper is the proposed two methods
for strengthened masking of the exponentiation operation and two new post-quantum public key-agreement
cryptoschemes.
Mathematics
subject classification: 94A60, 16Z05, 14G50, 11T71, 16S50.
Keywords: Finite associative algebra,
non-commutative algebra, right-sided unit, left-sided unit, global units,
discrete logarithm problem, post-quantum cryptography, public key-agreement.
Received December 23, 2019; accepted November 24, 2020