Finite non-commutative associative algebras as carriers of hidden discrete logarithm problem

The article introduces new finite algebras attractive as carriers of the discrete logarithm problem in a hidden group. In particular new $4$-dimensional and $6$-dimensional finite non-commutative algebras with associative multiplication operation and their properties are described. It is also proposed a general method for defining finite non-commutative associative algebras of arbitrary even dimension $m\ge 2$. Some of the considered algebras contain a global unit, but the other ones include no global unit element. In the last case the elements of the algebra are invertible locally relatively local bi-side units that act in the frame of some subsets of elements of algebra. For algebras of the last type there have been derived formulas describing the sets of the (right-side, left-side, and bi-side) local units. Algebras containing a large set of the global single-side (left-side and right-side) units and no global bi-side unit are also introduced. Since the known form of defining the hidden discrete logarithm problem uses invertibility of the elements of algebra relatively global unit, there are introduced new forms of defining this computationally difficult problem. The results of the article can be applied for designing public-key cryptographic algorithms and protocols, including the post-quantum ones. For the first time it is proposed a digital signature scheme based on the hidden discrete logarithm problem.