In this paper, we consider the security of the Sakalauskas-Tvarijonas-Raulynaitis (STR) key exchange protocol. We perform an analysis by exploring various cases of the canonical form of the publicly known matrix using elements of linear algebra and number theory. Additionally, we consider the multiplicative order of matrices and show how these two factors affect the security of the considered protocol. We show that regardless of the choice of publicly known matrix, the considered protocol is secure under the discrete logarithm assumption. In other words, if at least one of the secret exponents is found, then the STR protocol can be broken in polynomial time.

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.