The computation process of a Distributed Verifiable Random Function (DVRF) on some input specified by the user involves multiple, possibly malicious servers, and results in a publicly verifiable pseudorandom output to the user. Previous DVRF constructions assumed trusted generation of secret keys for the servers and imposed a threshold on the number of corrupted servers. In this paper we propose the first generic approach for building DVRFs, under much weaker setup assumptions, where we only require existence of a shared random string. More precisely, we first aim at constructions of Distributed Verifiable Unpredictable Functions (DVUF) that can then be converted to DVRF using inner products with a random string as specified by Micali, Rabin, and Vadhan (FOCS'99) for the non-distributed VUF/VRF case. Our main contribution are generic DVUF constructions from aggregate signatures that satisfy the property of uniqueness.We define uniqueness for two flavors of aggregate signatures (with public and sequential aggregation) and show that both flavors can be used to obtain DVUF. By proving uniqueness of existing pairing-based aggregate signature schemes we immediately obtain several concrete communication-efficient DVUF/DVRF instantiations.