Verifiable random functions are pseudorandom functions producing publicly verifiable proofs for their outputs, allowing for efficient checks of the correctness of their computation. In this work, we introduce a new computational hypothesis, the \(n\text {-}\mathsf {Eigen}\text {-}\mathsf {Value} \) assumption, which can be seen as a particularization of the \(\mathcal {U}_{l,k}\)-\(\mathrm {MDDH}\) assumption for the case \(l=k+1\), and prove its equivalence with the \(n\text {-}\mathsf {Rank} \) problem. Based on the newly introduced computational hypothesis, we build the core of a verifiable random function having an exponentially large input space and reaching adaptive security under a static assumption. The final construction achieves shorter public and secret keys compared to the existing schemes reaching the same properties.