Possibility of decryption speed-up by parallel processing in CCA secure hashed ElGamal

In order to prove the ElGamal CCA(Chosen Ciphertext Attack) security in the random oracle model, it is necessary to use the group where ICDH(Interactive Computational Diffie Hellman) assumption holds. Until now, only bilinear group with complex algebraic structure has been known as the ICDH group. In this paper, we introduce the ICDH group with simple algebraic structure. In other words, we prove that ICDH assumption holds in the integer group with composite modulus. On the basis of this, we propose the CCA secure hashed ElGamal and its fast variant to speed up decryption by parallel processing. Our parallel scheme has the fastest decryption among all CCA secure PKE(Public Key Encryption) schemes implemented in integer group and gives the possibility that ElGamal protocol could be practical when the big modulus numbers are used to resist the quantum attack.