Optimal logarithmic functions for lifting of a solution of an exponential congruence

This study is devoted to the problem of lifting of a solution of an exponential congruence in rings of integer algebraic numbers. To lift a solution in rings of integer rational numbers, Riesel suggested to use the Fermat quotients apparatus. With their use, the problem reduces to solution of a linear congruence modulo a prime number, and this congruence appears to be irreducible. In this paper we construct analogues of Fermat quotients in rings of integer algebraic numbers which also yield irreducible linear congruences for the problem of lifting of a solution in this case.