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This study is mainly about the discrete logarithm problem in the ElGamalcryptosystem over the abelian group U(n) where n is one of the following formsp^m, or 2p^m where p is an odd large prime and m is a positive integer. It isanother good way to deal with the ElGamal Cryptosystem using that abelian groupU(n)={x: x is a positive integer such that x<n and gcd(n,x)=1} in the settingof the discrete logarithm problem . Since I show in this paper that this newstudy maintains equivalent (or better) security with the original ElGamalcryptosystem(invented by Taher ElGamal in 1985)[1], that works over the finitecyclic group of the finite field. It gives a better security becausetheoretically ElGamal Cryptosystem with U(p^m) or with U(2p^m) is much moresecure since the possible solutions for the discrete logarithm will beincreased, and that would make this cryptosystem is hard to broken even withthousands of years.

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