New Public Key Cryptosystem Using Finite Non Abelian Groups

Abstract

Abstract. Most public key cryptosystems have been constructed based on abelian groups up to now. We propose a new public key cryptosystem built on finite non abelian groups in this paper. It is convertible to a scheme in which the encryption and decryption are much faster than other well-known public key cryptosystems, even without no message expansion. Furthermore a signature scheme can be easily derived from it, while it is difficult to find a signature scheme using a non abelian group.

On the Security of MOR Public Key Cryptosystem

“…In this section, generalizing the various ideas of [7,9,14], we show that it is possible to w-reduce(see the definition below) DLP(Inn(G)) to DLP(Inn(H)), where H is a maximal normal subgroup of G.…”

“…In [7,9], it is shown that DLP over inner automorphism groups of semidirect products can be reduced to DLP over inner automorphism groups of individual groups. For group extensions, a similar result can be derived.…”

“…At Crypto 2001, Paeng et al [8] proposed the MOR public key cryptosystem using finite non-abelian groups. For a 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, and there must be an efficient way to represent group elements as products of the specified generators of G. Furthermore, we expect the security of the MOR system to be something 'mor(e)' than that of DLP over G. Also it should be noted that the difficulty of DLP depends not only on the algebraic structure of the group, but also on how elements of the group are represented.…”

“…Despite of many cryptographic advantages(see [8]) of the MOR cryptosystem, the groups proposed so far have turned out to be unsatisfactory(see [7,9,14]). …”

“…Next, in Section 3, using the well-known theory of group extensions, we show that it is possible to 'reduce' the problem of finding the secret keys of MOR system over G to that of the MOR system over (smaller) subgroups H of G. Our method is a generalization of various attacks given in [7,9,14].…”