On the decisional problem based on matrix power function defined over non-commutative group

In this paper, we perform statistical analysis for the decisional problem which is fundamental for the security of the key exchange protocol based on matrix power function. We have proven previously that the considered decisional problem is NP-complete and hence our proposal could potentially be quantum-safe. However, we did not explore the dependence of the complexity of the considered problem on the security parameters. Here we show that for small matrices certain information could be gained from the distribution of the entries of the public key matrices. On the other hand, we show that as the size of the matrices grows, the public key matrices are indistinguishable from truly random matrices.