A new form of the hidden discrete logarithm problem, called split logarithm problem, is introduced as primitive of practical post-quantum digital signature schemes, which is characterized in using two non-permutable elements A and B of a finite noncommutative associative algebra, which are used to compute generators Q = AB and G = BQ of two finite cyclic groups of prime order q. The public key is calculated as a triple of vectors (Y,Z, T ): Y = Qx, Z = Gw, and T = QaB−1Gb, where x, w, a, and b are random integers. Security of the signature scheme is defined by the computational difficulty of finding the pair of integers (x,w), although, using a quantum computer, one can easily find the ratio x/w mod q.