On the Security of MOR Public Key Cryptosystem

Abstract

Abstract. For a finite group G to be used in the MOR public key cryptosystem, it is necessary that the discrete logarithm problem(DLP) over the inner automorphism group Inn(G) of G must be computationally hard to solve. In this paper, under the assumption that the special conjugacy problem of G is easy, we show that the complexity of the MOR system over G is about log |G| times larger than that of DLP over G in a generic sense. We also introduce a group-theoretic method, called the group extension, to analyze the MOR cryptosystem. When G is considered as a group extension of H by a simple abelian group, we show that DLP over Inn(G) can be 'reduced' to DLP over Inn(H). On the other hand, we show that the reduction from DLP over Inn(G) to DLP over G is also possible for some groups. For example, when G is a nilpotent group, we obtain such a reduction by the central commutator attack.

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

“…Since the black box group algorithms work in any group, they will work in the automorphism group too, see [9,Theorem 1]. We have no way to prevent that.…”

The MOR cryptosystem and extra-special $p$-groups

“…In this section we describe the MOR cryptosystem [1,6] as automorphisms of a finite group G, however it can be generalized to other finitely generated algebraic structures easily. A description and a critical analysis of the MOR cryptosystem is in [3] and the references there.…”

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

“…In [4] the authors developed a central commutator attack ; they showed that inner automorphisms are not well suited for MOR cryptosystem; especially when the group is nilpotent.…”

“…The MOR cryptosystem has attracted a lot of attention and some well written papers [4,11,14]. In this article we propose a new group and a subgroup of the group of automorphisms for the MOR cryptosystem.…”