Verifiable random functions (VRFs) are pseudorandom functions where the function owner can prove that a generated output is correct relative to a committed key. In this paper we introduce the notion of an exponent-VRF (eVRF): a VRF that does not provide its output y explicitly, but instead provides \(Y = y \cdot G\) , where G is a generator of some finite cyclic group (or \(Y=g^y\) in multiplicative notation). We construct eVRFs from the Paillier encryption scheme and from DDH, both in the random-oracle model. We then show that an eVRF is a powerful tool that has many important applications in threshold cryptography. In particular, we construct (1) a one-round fully simulatable distributed key-generation protocol (after a single two-round initialization phase), (2) a two-round fully simulatable signing protocol for multiparty Schnorr with a deterministic variant, (3) a two-party ECDSA protocol that has a deterministic variant, (4) a threshold Schnorr signing protocol where the parties can later prove that they signed without being able to frame another group, and (5) an MPC-friendly and verifiable HD-derivation. All these applications are derived from this single new eVRF abstraction, and the resulting protocols are concretely efficient.