# FINITE NON-COMMUTATIVE ASSOCIATIVE ALGEBRA WITH A COMPRESSING MULTIPLICATION OPERATION

A new concept and a novel method for formulating the congruence-logarithm problem are considered. This method is aimed at the development of post-quantum public-key cryptographic algorithms and protocols. As a carrier of the congruence-logarithm problem the finite non-communicative associative algebra with new properties is proposed. New particular properties of the proposed algebra are related to the compressive property of the multiplication operation, the mutual associativity of different modifications of the parametrizable multiplication operation, and to the absence of a global unity element, whereas different subsets of the algebra elements contain different local unity elements. The formulas describing the right, the left, and the bi-side local unity elements are presented. Methods for computing the local units and the global bi-side zero divisors and for defining the subsets of algebra elements, being the cyclic groups, are described.

**Authors:** I. K. Abrosimov, D. N. Moldovyan, N. A. Moldovyan

**Direction:** Informatics and Computer Technologies

**Keywords:** Finite non-commutative associative algebra, congruence-logarithm problem, post-quantum cryptographic protocols

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