Verifiable Random Functions with Optimal Tightness

Verifiable random functions (VRFs), introduced by Micali, Rabin and Vadhan (FOCS’99), are the public-key equivalent of pseudo- random functions. A public verification key and proofs accompanying the output enable all parties to verify the correctness of the output. How- ever, all known standard model VRFs have a reduction loss that is much worse than what one would expect from known optimal constructions of closely related primitives like unique signatures. We show that: 1. Every security proof for a VRF that relies on a non-interactive assumption has to lose a factor of Q, where Q is the number of adver- sarial queries. To that end, we extend the meta-reduction technique of Bader et al. (EUROCRYPT’16) to also cover VRFs. 2. This raises the question: Is this bound optimal? We answer this ques- tion in the affirmative by presenting the first VRF with a reduction from the non-interactive qDBDHI assumption to the security of VRF that achieves this optimal loss. We thus paint a complete picture of the achievability of tight verifiable random functions: We show that a security loss of Q is unavoidable and present the first construction that achieves this bound.