Public Verifiable Key Encapsulation Mechanism under Factoring

The cipher text consistency check that can be implemented publicly has practical significance, and it is always an interested studying aspect to construct cryptographic algorithm without pairings in the field of cryptographic. On the other hand, it is one of the most important research topics to design CCA-secure PKE schemes with weaker assumptions. Especially, there have been no CCA-secure PKE scheme under factoring assumption, except for a recent work by Hofheinz, D., Kiltz, E. A fly in the ointment is that HK09´s scheme is not a public verifiable construction. Using the Gap quality that the Decisional Diffie-Hellman problem is easy but the Computational Diffie-Hellman problem is difficult as factoring, we instantiated HK09´s scheme over the group of signed quadratic residues, and realized public verifiable construct. It will be used to construct threshold schemes.