Digital signature scheme on the $2 \times 2$ matrix algebra

The article considers the structure of the $2\times2$ matrix algebra set over a ground finite field $GF(p)$. It is shown that this algebra contains three types of commutative subalgebras of order $p^2,$ which differ in the value of the order of their multiplicative group. Formulas describing the number of subalgebras of every type are derived. A new post-quantum digital signature scheme is introduced based on a novel form of the hidden discrete logarithm problem. The scheme is characterized in using scalar multiplication as an additional operation masking the hidden cyclic group in which the basic exponentiation operation is performed when generating the public key. The advantage of the developed signature scheme is the comparatively high performance of the signature generation and verification algorithms as well as the possibility to implement a blind signature protocol on its base.