Post-quantum signature scheme on matrix algebra

The paper considers the use of a finite multiplicative group of invertible matrices of dimension $2\times2$ set over the field $\mathrm{GF}(p)$ as algebraic carrier of the digital signature schemes based on the computational difficulty of the hidden discrete logarithm problem and satisfying the general criterion of post-quantum resistance. The existence of a sufficiently large number of commutative subgroups with two-dimensional cyclicity is shown. This fact is used in the construction of a specific signature scheme which is of interest as a post-quantum cryptosystem. In the introduced digital signature scheme, a new form of the hidden discrete logarithm problem is applied. The said form is characterized by the use of a commutative group with two-dimensional cyclicity as a hidden group and masking operations of two different types: ($i$) having the property of mutual commutativity with the exponentiation operation and ($ii$) free from this property. To ensure the correct operation of the cryptographic scheme, a special type of verification equation is used in the signature authentication procedure, and when generating a signature, one of the elements of the latter is calculated as a root of quadratic equation.