Structure of 6-dimensional finite non-commutative algebras with many single-sided units

Abstract

Finite Associative Noncommutative Algebras (FANAs) have gained considerable attention as a key foundational element for post-quantum (PQ) public-key (PK) cryptosystems, particularly those with a hidden group. These systems exploit the complexity of the hidden discrete logarithm problem (HDLP) and the challenge of solving large system of power equations. The structure of 6-dimensional FANAs over the finite field GF(p), which can include global single-sided units in different configurations (p², p³, and p⁴), plays an essential role in assessing the security of these cryptosystems. A novel PQ signature algorithm has been proposed based on FANAs with p^2 global single-sided units, while the others have been deemed less suitable for supporting the proposed algorithm. The decomposition of these algebras into isomorphic subalgebras, each with a global two-sided unit, significantly contributes to understanding the design of PQ cryptosystems that use FANAs with a large number of global singlesided units as their algebraic framework.