There are considered attacks on cryptoschemes based on the recently proposed hard problem called hidden conjugacy search problem (HCSP), dened over nite non-commutative groups. It is shown that using homomorphisms of the non-commutative nite group into nite elds GF(ps), s > 1, in some cases the HCSP can be reduced to mtwo independent problems: discrete logarithm and conjugacy search problem. Two methods for preventing such attacks are proposed. In the rst method there are used elements of the order p. The second method uses non-invertible elements and relates to dening the HCSP over the nite non-commutative ring.