The ElGamal algorithm, which can be used for both signature and encryption, is of importance in public-key cryptosystems. However, there has arisen an issue that different criteria of selecting a random number are used for the same algorithm. In the aspects of the sufficiency, necessity, security and computational overhead of parameter selection, this paper analyzes these criteria in a comparative manner and points out the insecurities in some textbook cryptographic schemes. Meanwhile, in order to enhance security a novel generalization of the ElGamal signature scheme is made by expanding the range of selecting random numbers at an acceptable cost of additional computation, and its feasibility is demonstrated. The ElGamal algorithm, which can be used for both signature and encryption, is of importance in public-key cryptosystems. However, there has arisen an issue that different criteria of selecting a random number are used for the same algorithm. In the aspects of the sufficiency, necessity, security and computational overhead of parameter selection, this, in order to enhance security a novel generalization of the ElGamal signature scheme is made by expanding the range of selecting random numbers at an acceptable cost of additional computation, and its feasibility is demonstrated.