A conditional proxy re-encryption (CPRE) scheme enables the proxy to convert a ciphertext from Alice to Bob, if the ciphertext satisfies one condition set by Alice. To improve the issue of more fine-grained on the condition set, Fang, Wang, Ge and Ren proposed a new primitive named Interactive conditional PRE with fine grain policy (ICPRE-FG) in 2011, and left an open problem on how to construct CCA-secure ICPRE-FG without random oracles. In this paper, we answer this open problem affirmatively by presenting a new construction of CCA-secure key-policy attribute-based PRE (KP-ABPRE) without random oracles. In this paper, we enhance the security model of Fang's ICPRE-FG scheme by allowing the adversary to make some extra queries, which do not help them win the game trivially. Finally, we present a CCA-secure KP-ABPRE without random oracles under the 3-weak decisional bilinear Diffie-Hellman inversion(3-wDBDHI) assumption.