In this paper, we consider what condition is sufficient for random inputs to secure probabilistic public-key encryption schemes. Although a framework given in  enables us to discuss uniformly and comprehensively security notions of public-key encryption schemes even for the case where cryptographically weak pseudorandom generator is used as random nonce generator to encrypt single plaintext messages, the results are rather theoretical. Here we naturally generalize the framework in order to handle security for the situation where we want to encrypt many messages with the same key. We extend some results w.r.t. single message security in  – separation results between security notions and a non-trivial sufficient condition for the equivalence between security notions – to multiple messages security. Besides the generalization, we show another separation between security notions for k-tuple messages and for (k+1)-tuple messages. The natural generalization, obtained here, rather improves to understand the security of public-key encryption schemes and eases the discussion of the security of practical public-key encryption schemes. In other words, the framework contributes to elucidating the role of randomness in public-key encryption scheme. As application of results in the generalized framework, we consider compatibility between the ElGamal encryption scheme and some sequence generators. Especially, we consider the applicability of the linear congruential generator (LCG) to the ElGamal encryption scheme.