A Simple Generalization of the ElGamal Cryptosystem to Non-Abelian Groups II

Abstract

ABSTRACT. This is a study of the MOR cryptosystem using the special linear group over finite fields. The automorphism group of the special linear group is analyzed for this purpose. At our current state of knowledge, I show that this MOR cryptosystem has better security than the ElGamal cryptosystem over finite fields.

Analisis Sistem Kriptografi ElGamal Untuk Membentuk Sistem Kunci Publik Berbasis Grup Non-Komutatif

“…Walaupun begitu, beberapa sistem kriptografi asimetris yang dikembangkan masih dapat diretas oleh penyadap [5] [14,13]. Salah satu pengembangan pada sistem kriptografi ElGamal adalah penggunaan Non-Abelian Groups II agar sistem bekerja atas grup non-komutatif [15].…”

Gaussian elimination in split unitary groups with an application to public-key cryptography

“…It provides an interesting change in perspective in public-key cryptography -from finite cyclic groups to finite non-abelian groups. The MOR cryptosystem was studied for the special linear group in details by Mahalanobis [15]. For many other classical groups, except the orthogonal groups, the analysis of a MOR cryptosystem remains almost the same.…”

“…For many other classical groups, except the orthogonal groups, the analysis of a MOR cryptosystem remains almost the same. So we will remain brief in this paper and refer an interested reader to [15] (see also [16]).…”

“…Once, we have decided that the automorphism that we are going to use in the MOR cryptosystem will act by conjugation. Further analysis is straightforward and follows [15,Section 7]. Recall that we insisted that automorphisms in the MOR cryptosystem are presented as action on generators.…”

“…Thus the word problem in SU(2l, q 2 ) is completely solved as with SL(d, q) using only elementary matrices; this is particularly useful for the MOR cryptosystem. An analysis of a MOR cryptosystem similar to the MOR cryptosystem over SL(d, q) [15] will be done in the next section.…”