A simple generalization of the El-Gamal cryptosystem to non-abelian groups II

The MOR cryptosystem is a generalization of the ElGamal cryptosystem, where the discrete logarithm problem works in the automorphism group of a group G, instead of the group G itself. The framework for the MOR cryptosystem was first proposed by Paeng et al. [13]. Mahalanobis [10] used the group of unitriangular matrices for the MOR cryptosystem. That effort was successful: the MOR cryptosystem over the group of unitriangular matrices over q is as secure as the ElGamal cryptosystem over the finite field q. In this article we study the MOR cryptosystem over SL d q . If we assume that the only way to break the proposed MOR cryptosystem is to solve the discrete logarithm problem in the automorphism group, then it follows that the proposed MOR cryptosystem is as secure as the ElGamal cryptosystem over qd . This is a major improvement. This MOR cryptosystem works with matrices of degree d over q. To encrypt (decrypt) a plaintext (ciphertext) one works over the field q. To break this cryptosystem, one has to solve a discrete logarithm problem in qd . Even for a small positive integer d, this provides us with a considerable security advantage. There are some challenges in the implementation of this cryptosystem. Implementing matrix multiplication is hard. Though we have not reached the optimum speed for that [4], it might always stay harder than multiplication in a finite field. So one needs to find an optimum strategy to present the automorphisms