The ElGamal cryptosystem is a public-key cryptosystem that is based on the discrete logarithm problem in a cyclic group. In other words, the security of the ElGamal cryptosystem relies on the fact that it is difficult to compute the discrete logarithm of a random element in the cyclic group.
Since the discrete logarithm problem is known to be hard in many different cyclic groups, the ElGamal cryptosystem can be used with any cyclic group, as long as certain mathematical properties are satisfied. These properties include:
The group is finite: The cyclic group used in the ElGamal cryptosystem must be finite, which means that it has a finite number of elements.
The group is generated by a primitive element: A primitive element is an element of the cyclic group that generates all other elements of the group. In other words, every element of the group can be expressed as a power of the primitive element. The ElGamal cryptosystem relies on the fact that it is difficult to compute discrete logarithms with respect to the primitive element.
The group operation is efficiently computable: The ElGamal cryptosystem requires that the group operation (i.e., multiplication or addition) be efficiently computable. This means that it should be possible to perform the group operation in a reasonable amount of time using a computer.