Bidirectional proxy re-encryption schemes without pairing nor random oracle

Blaze et al. in 1998 introduced the concept of proxy re-encryption, which allows a proxy to transform a ciphertext computed under Alice´s public key into one that can be opened under Bob´s decryption key. In CCS´07, Canetti et al. proposed an IND-PRE-CCA2 bidirectional proxy re-encryption scheme, and later, Libert et al. proposed another IND-PRE-CCA2 secure unidirectional proxy re-encryption. However, both schemes rely on the costly bilinear pairings. In CANS´08, Deng et al. proposed the first IND-PRE-CCA2 secure proxy re-encryption without parings. Later, in PKC´09, Shao et al. proposed another IND-PRE-CCA2 secure proxy re-encryption without parings. But both of these schemes prove their security in the random oracle. In this paper, based on Hanaoka´s efficient chosen ciphertext secure public key encryption under the computational Diffie-Hellman assumption, we present two new interesting bidirectional proxy re-encryption schemes, which are restricted IND-PRE-CCA2 secure without pairing nor random oracle.