Adaptive Proofs of Knowledge in the Random Oracle Model

We formalise the notion of adaptive proofs of knowledge in the random oracle model, where the extractor has to recover witnesses for multiple, possibly adaptively chosen statements and proofs. We also discuss extensions to simulation soundness, as typically required for the “encrypt-then-prove” construction of strongly secure encryption from IND-CPA schemes. Utilizing our model we show three results: (1) Simulation-sound adaptive proofs exist. (2) The “encrypt-then-prove” construction with a simulation-sound adaptive proof yields CCA security. This appears to be a “folklore” result but which has never been proven in the random oracle model. As a corollary, we obtain a new class of CCA-secure encryption schemes. (3) We show that the Fiat-Shamir transformed Schnorr protocol is not adaptively secure and discuss the implications of this limitation. Our result not only separates adaptive proofs from proofs of knowledge, but also gives a strong hint why Signed ElGamal as the most prominent encrypt-then-prove example has not been proven CCA-secure without making further assumptions.