The security analysis of the key exchange protocol based on the matrix power function defined over a family of non-commuting groups

In this paper, we revisited the previously proposed key exchange protocol based on the matrix power function. We prove that the entries of the public key matrices of both parties of the protocol are uniform. Using this result we defined a security game for our protocol and show that the malicious attacker cannot gain any significant advantage in winning this game by applying faithful representation or the linearization approaches. Moreover, we showed that the shared key is computationally indistinguishable from the imitation key if the security parameters are properly chosen.