CCA-Secure Public Key Encryption without Group-Dependent Hash Functions

So far, in almost all of the practical public key encryption schemes, hash functions which are dependent on underlying cyclic groups are necessary, e.g., H : {0, 1}* -> Z(p) where p is the order of the underlying cyclic group, and it could be required to construct a dedicated hash function for each public key. The motivation of this note is derived from the following two facts: 1). there is an important technical gap between hashing to a specific prime-order group and hashing to a certain length bit sequence, and this could cause a security hole: 2). surprisingly, to our best knowledge, there is no explicit induction that one could use the simple construction, instead of tailor-made hash functions. In this note, we investigate this issue and provide the first rigorous discussion that in many existing schemes, it is possible to replace such hash functions with a target collision resistant hash function H : {0, 1}* -> {0, 1}(k), where k is the security parameter. We think that it is very useful and could drastically save the cost for the hash function implementation in many practical cryptographic schemes.