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Introduction: Development of post-quantum digital signature standards represents a current challenge in the area of cryptography. Recently, the signature schemes based on the hidden discrete logarithm problem had been proposed. Further development of this approach represents significant practical interest, since it provides possibility of designing practical signature schemes possessing small size of public key and signature. Purpose: Development of the method for designing post-quantum signature schemes and new forms of the hidden discrete logarithm problem, corresponding to the method. Results: A method for designing post-quantum signature schemes is proposed. The method consists in setting the dependence of the publickey elements on masking multipliers that eliminates the periodicity connected with the value of discrete logarithm of periodic functions constructed on the base of the public parameters of the cryptoscheme. Two novel forms for defining the hidden discrete logarithm problem in finite associative algebras are proposed. The first (second) form has allowed to use the finite commutative (non-commutative) algebra as algebraic support of the developed signature schemes. Practical relevance: Due to significantly smaller size of public key and signature and approximately equal performance in comparison with the known analogues, the developed signature algorithms represent interest as candidates for practical post-quantum cryptoschemes.

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Four-dimensional finite non-commutative associative algebras represent practical interest as algebraic support of post-quantum digital signature algorithms, especially algebras with two sided global unit, set by sparse basis vectors multiplication tables. A new algebra of the latter type, set over the field GF(p), is proposed and its structure is investigated. The studied algebra is described as a set of p2 + p + 1 commutative subalgebras of three different types. All subalgebras intersect strictly in the subset of scalar vectors. Formulas are derived for the number of subalgebras of each type.

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.