Collision algorithms for breaking megrelishvili protocol: Theory and numerical experiments

Megrelishvili protocol is a linear algebraic-based key exchange procedure between two parties. Unlike the well-known Diffie-Hellman protocol or its variants, the security of this protocol does not rely on the generic discrete logarithm problem in a particular group. In this article we investigate Megrelishvili Vector-Matrix Problem (MVMP) as a computational problem that underlies the security of Megrelishvili protocol. We propose two types of collision algorithms which solve MVMP faster than the exhaustive search approach. In particular, we prove that the running time of our algorithms asymptotically outperforms that of the exhaustive search approach in solving the MVMP in FJ by a factor of (1/n logg) · qn/2. Moreover, we also conducted numerical experiments to test the actual running time of our algorithms. The results from theoretical and experimental analyses show that our algorithms are generally faster than the brute-force approach.