The issue of the signature randomization in algebraic cryptoschemes with a hidden group, which are based on the computational difficulty of solving large systems of power equations, is considered. To ensure complete randomization of the signature, the technique of doubling the verification equation was used to specify the hidden group. A specific signature algorithm is proposed that uses 4-dimensional non-commutative associative algebra as an algebraic support. Known results on the study of the structure of this algebra were used in constructing the proposed algorithm and estimating its security. The question of implementing similar algorithms on finite non-commutative associative algebras of dimensions m > 6 is related to the open problem of studying their structure from the point of view of decomposition into a set of commutative subalgebras.