Tightly secure public-key cryptographic schemes enjoy the advantage that the selection of the security parameter can be optimal to achieve a certain security level. Security models in the multi-user setting with corruptions (MU-C) consider more realistic threats in practice. Many efforts have been devoted to constructing tightly MU-C secure schemes. To date, we have many concrete constructions. Nevertheless, the study on how to generally achieve tight security in public-key cryptography remains lacking. In this paper, we take an insight into the key generations in public-key cryptography. We first generalize the key generation algorithms of traditional schemes and discuss the requirements of achieving tight security. We notice that for some schemes (e.g. key-unique schemes), these requirements inherently cannot be satisfied and hence these schemes cannot achieve tight security. This is in accordance with the impossibility results of tight reductions by Bader et al. (EUROCRYPT 2016). To further study possible constructions, we extend the key generations of public-key cryptographic schemes to obtain a different framework. To demonstrate its applications, we illustrate how to construct tightly secure key-unique schemes under the extended framework. This circumvents the impossibility results of tight security for key-unique schemes.