  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.
