The improved version of the author's previously declared asymmetric cipher protocol based on matrix power function (MPF) is presented. Proposed modification avoids discrete logarithm attack (DLA) which could be applied to the previously declared protocol. This attack allows us to transform the initial system of MPF equations to so-called matrix multivariate quadratic (MMQ) system of equations, which is a system representing a subclass of multivariate quadratic (MQ) systems of equations. We are making a conjecture that avoidance of DLA in protocol, presented here, should increase its security, since an attempt to solve the initial system of MPF equations would appear to be no less complex than solving the system of MMQ equations. No algorithms are known to solve such a system of equations. Security parameters and their secure values are defined. Security analysis against chosen plaintext attack (CPA) and chosen ciphertext attack (CCA) is presented. Measures taken to prevent DLA attack increase the security of this protocol with respect to the previously declated protocol.