Reducing public-key size in an anonymous credential system for CNF formulas with constant-size proofs

To enhance the user privacy at authentications, an anonymous credential system has been designed to allow a user to anonymously convince a verifier of the possession of a certificate issued by the issuing authority. In this system, a user can prove relations on his/her attributes embedded into the certificate. Previously, we have proposed an anonymous credential system with constant-size proofs and faster proof generations for CNF formulas than the existing system. In the system, in the verification, some value should be included in some set Φ. Unfortunately, this system has a problem of requiring long public parameters that is proportional to the size of the set Φ. In this paper, we propose an extension to reduce the public key by separating this set Φ into two sets Φ₁ and Φ₂ with logarithmic sizes such that u₁ ϵ Φ₁ and u₂ ϵΦ₂ if and only if u₁ + u₂ ϵ Φ. Furthermore, we evaluate the efficiency based on the implementation.