Recently, Park and Lee suggested a new framework for realizing Identity-Based Encryption (IBE) trapdoor called ‘two-equation-revocation’, and proposed a new IBE system that makes use of a Map-To-Point hash function. In this paper, we present a variant of the PL system by giving a simple way to remove the Map-To-Point hash function from the PL system. Our variant is proven to be secure under non-standard security assumptions, which results in the degradation of security. Instead, our variant can have several efficiency advantages over the PL system: (1) it provides receiver's anonymity, (2) it has no correctness error, (3) it has shorter ciphertext, and (4) it has faster encryption. As a result, (when not considering security assumptions and security losses) our variant is as efficient as the Boneh-Boyen and Sakai-Kasahara IBE systems that are considered as being the most practical ones.