The Discrete Logarithm problem in the ElGamal cryptosystem over the abelian group U(n) Where n= p^m,or 2p^m

TL;DR

This paper explores the discrete logarithm problem in the ElGamal cryptosystem over specific abelian groups U(n), proposing that these groups offer comparable or enhanced security compared to traditional finite field groups.

Contribution

It introduces a novel approach to ElGamal cryptosystem using U(n) groups where n=p^m or 2p^m, demonstrating potential for increased security.

Findings

Maintains equivalent or better security than original ElGamal system

Increases the complexity of discrete logarithm solutions

Suggests enhanced resistance to cryptanalysis

Abstract

This study is mainly about the discrete logarithm problem in the ElGamal cryptosystem over the abelian group U(n) where n is one of the following forms p^m, or 2p^m where p is an odd large prime and m is a positive integer. It is another good way to deal with the ElGamal Cryptosystem using that abelian group U(n)={x: x is a positive integer such that x<n and gcd(n,x)=1} in the setting of the discrete logarithm problem . Since I show in this paper that this new study maintains equivalent (or better) security with the original ElGamal cryptosystem(invented by Taher ElGamal in 1985)[1], that works over the finite cyclic group of the finite field. It gives a better security because theoretically ElGamal Cryptosystem with U(p^m) or with U(2p^m) is much more secure since the possible solutions for the discrete logarithm will be increased, and that would make this cryptosystem is hard to broken even with thousands of years.