**Authors:** D. N. Moldovyan, A. A. Moldovyan, N. A. Moldovyan

**Keywords:** finite associative algebra, non-commutative algebra, global unit, local unit, right-sided unit, left-sided unit, discrete logarithm problem, public-key cryptoscheme,
digital signature, post-quantum cryptosystem.

### Abstract

A new form of the hidden discrete logarithm problem is proposed as cryptographic primitive for the development of the post-quantum signature schemes, which is characterized in
performing two masking operations over each of two elements from a hidden finite cyclic group used to compute the public-key elements. The latter is contained in the set of non-invertible elements of the finite non-commutative associative algebra with a two-sided unit. One of the said masking operations represents the automorphism-map operation and the other one is the left-sided (right-sided) multiplication by a local right-sided (left-sided) unit acting on the said hidden group. Two 4-dimensional algebras are considered as possible algebraic supports of the developed signature schemes. The formulas describing the sets of local left-sided and right-sided units are derived. Periodic functions set on the base of the public parameters of the signature scheme contain periods depending on the discrete logarithm value, but every of them takes on the values relating to different finite groups contained in the algebraic support. Therefore one can expect that the computational difficulty of breaking the introduced signature schemes on a hypothetic quantum computer is superpolinomial.

St. Petersburg Federal Research Center of

the Russian Academy of Sciences (SPC RAS),

St. Petersburg Institute for Informatics and

Automation of the Russian Academy of Sciences

14 Liniya, 39, St.Petersburg, 199178

Russia

E-mail:

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