MOR Cryptosystem and classical Chevalley groups in odd characteristic

In this paper we study the MOR cryptosystem with finite Chevalley groups. There are four infinite families of finite classical Chevalley groups. These are: special linear groups SL(d, q), orthogonal groups O(d, q) and symplectic groups Sp(d, q). The family O(d, q) splits to two different families of Chevalley groups depending on the parity of d. The MOR cryptosystem over SL(d, q) was studied by the first author, “A simple generalization of the ElGamal cryptosystem to non-abelian groups II, Communications in Algebra 40 (2012), no. 9, 3583–3596”. In that case, the hardness of the MOR cryptosystem was found to be equivalent to the discrete logarithm problem in Fqd . In this paper, we show that the MOR cryptosystem over Sp(d, q) has the security of the discrete logarithm problem in Fqd . However, it seems likely that the security of the MOR cryptosystem for the family of orthogonal groups is F q 2 . We also develop an analog of row-column operations in orthogonal and symplectic groups.