Complete signature randomization in an algebraic cryptoscheme with a hidden group (Q6558350)

Digital signature schemes with platforms on finite non-commutative associative algebras are introduced. Those algebras are realized through vector unital multiplication operations on finite dimensional vector spaces over finite fields \(\mathbb{F}_p\), with \(p\) prime. In particular, the author proposes digital signature schemes on a 4-dimensional algebra whose commutative subalgebras he has classified in a previous paper [\textit{D. Moldovyan} et al., Quasigroups Relat. Syst. 30, No. 1, 133--140 (2022; Zbl 1493.94039)]. For the commutative subalgebras possessing two-dimensional cyclicity, namely those having two commutative generators, a hidden discrete logarithm problem is used to propose a first digital signature scheme. However, the resulting signatures vary within \(O(p^2)\) possibilities, and the author points this as a weak condition. In order to obtain \(O(p^4)\) possible signatures, or to get a ``complete signature randomization'', the author proposes a second signature scheme. Both schemes are proven to be correct, and their robustness is based on the hardness of the corresponding hidden discrete logarithm problem. However, the author asserts: ``the assessment of the security level of the proposed algorithm is quite rough and applies only to direct attacks related to solving a system of quadratic vector equations connecting elements of public and private keys. Obviously, further analysis of resistance to attacks of various types is required. At the moment, we only claim that the randomization technique used ensures sufficient completeness of the signature randomization''. The paper is hard to read since the notation is rather awful. This is a continuation of a long line of research undertaken by the author.