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

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, or 2p 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) or with U(2p) 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. Keywords— ElGamal Cryptosystem, The abelian group U(n), The Discrete Logarithm Problem over U(n), The ElGamal cryptosystem over U(n) : n =p, or 2p for a positive integer m and p is an odd large prime.