A new approach to the elgamal encryption scheme

The ElGamal encryption scheme can be used for both digital signatures and encryption, and its security results from the difficulty of calculating discrete logarithms in a finite field. This algorithm usually works in a multiplicative group of GF(p) and in this case the progress in the discrete logarithm problem forces the users of such a basic ElGamal public key cryptosystem to permanently increase a prime modulus p in order to ensure the desired security. But the task of finding a multiplicative group of GF(p) is unfeasible for an ordinary user. It is possible to overcome this inconvenience by forming an ElGamal encryption scheme which works in a multiplicative group of GF(p^m). Therefore, it is shown in the paper how to implement this cryptosystem for work in the multiplicative group of GF(p^m), in its subgroup, and in an algebraic system named the spurious multiplicative group of GF(p^m).