Analysis of the conjugacy search problem of the Diffie-Hellman protocol based on the SL(2,3) subgroup

The purpose of the Diffie-Hellman key exchange is to establish a shared secret key given that the protocol works in an abelian group setting. In this paper, conditions are presented to attain a shared secret key over a nonabelian group for the well-known Diffie-Hellman method. To attain the shared secret key, a subgroup of a nonabelian group will be presented in the abelian configuration by incorporating group theory concept such as normal subgroup and cyclic subgroup. The protocol introduced on the nonabelian group, namely the special linear group, addresses the conjugacy search problem, aiming to enhance the security of the existing protocol. Examples in special linear group SL(2,3) are presented to illustrate the implementation of Diffie-Hellman key exchange protocol.