Cryptanalysis of a Diffie–Hellman's scheme analogue using conjugation and exponentiation on matrix platform

It is proved that the mixed generalized version of the Diffie–Hellman's protocol using matrix platform with the conjugation and exponentiation in a generic case admits computing the shared key in a polynomial time under assumption that the corresponding multiple discrete logarithm problem can be solved in a polynomial time. The computing algorithm uses the original method of linear decomposition and the approach by Menezes and others reducing the computation of the matrix exponent to the multiple discrete logarithm problem. The combination of these two approaches cannot be directly applied because the exponentiation is not automorphism. The proof of the main result is based on the analysis of belonging a monomial matrices to cosets of a matrix group by elementwise permutable subgroups. Thus, a similar question for the symmetric groups has to be studied. Fortunately, a number of results in this sphere is known.