Structure of a $4$-dimensional algebra and generating parameters of the hidden discrete logarithm problem

Structure of a $4$-dimensional algebra and generating parameters of the hidden discrete logarithm problem the field $GF(p)$ is studied in connection with using it as algebraic support of the hidden discrete logarithm problem that is an attractive primitive of post-quantum signature schemes. It is shown that each invertible $4$-dimensional vector that is not a scalar vector is included in a unique commutative group representing a subset of algebraic elements. Three types of commutative groups are contained in the algebra and formulas for computing the order and the number of groups are derived for each type. The obtained results are used to develop algorithms for generating parameters of digital signature schemes based on computational difficulty of the hidden logarithm problem.