On Several Verifiable Random Functions and the Q-Decisional Bilinear Diffie-Hellman Inversion Assumption

In 1999, Micali, Rabin and Vadhan introduced the notion of Verifiable Random Functions (VRF)$\backslash$citeFOCS:MicRabVad99. VRFs compute for a given input x and a secret key \$sk\$ a unique function value \$y=V\_sk (x)\$, and additionally a publicly verifiable proof $p$. Each owner of the corresponding public key \$pk\$ can use the proof to non-interactivly verify that the function value was computed correctly. Furthermore, the function value provides the property of pseudorandomness. Most constructions in the past are based on q-type assumptions. Since these assumptions get stronger for a larger factor q, it is desirable to show the existence of VRFs under static or general assumptions. In this work we will show for the constructions presented in $\backslash$citePKC:DodYam05 $\backslash$citeCCS:BonMonRag10 the equivalence of breaking the VRF and solving the underlying q-type assumption.