The paper introduces a new form of the hidden discrete logarithm problem defined over finite non-commutative associative algebras containing two-sided global unit and sets of local left-sided and right-sided units. The proposed form is characterized in using a new mechanism for masking the finite cyclic group in which the base exponentiation operation is performed. Local units act in frame of subsets of non-invertible vectors and are used as elements of the private key in the proposed post-quantum digital signature scheme. A new 4-dimensional algebra is introduced as algebraic support of the proposed cryptoscheme. Formulas describing units of different types are derived.