Toward CCA-secure bidirectional proxy re-encryption without pairing nor random oracle

Proxy re-encryption, introduced by Blaze et al. in 1998, allows a proxy to transform a ciphertext computed under Alice´s public key into one that can be opened under Bob´s decryption key. An open problem of constructing (bidirectional or unidirectional) IND-PRE-CCA2 secure proxy re-encryption scheme in the standard model yet without pairings was left until now. In this paper, based on Hanaoka´s efficient chosen ciphertext secure public key encryption under the computational Diffie-Hellman assumption, we try to solve this open problem by presenting two new bidirectional proxy re-encryption schemes, which are restricted IND-PRE-CCA2 secure without pairing nor random oracle.