International Association
for Cryptologic Research

Constructing a verifiable random function (VRF) with large input space and full adaptive security from a static complexity assumption, like decisional Diffie-Hellman for instance, has proven to be a challenging task. To date it is not even clear that such a VRF exists. Most known constructions either allow only a small input space of polynomially-bounded size, or do not achieve full adaptive security under a static complexity assumption.

The only known constructions without these restrictions are based on non-static, so-called \"q-type\" assumptions, which are parametrized by an integer q. Since q-type assumptions get stronger with larger q, it is desirable to have q as small as possible. In current constructions q is a polynomial (Hohenberger and Waters, Eurocrypt 2010) or at least linear (Boneh et al., CCS 2010) in the security parameter.

We construct a relatively simple and efficient verifiable random function, based on a q-type assumption where q is only logarithmic in the security parameter. We also describe a verifiable unpredictable function from a similar, but weaker assumption. Both constructions have full adaptive security and large input spaces.