In this work, we introduce two post-quantum Verifiable Random Function (VRF) constructions based on abelian group actions and isogeny group actions with a twist. The former relies on the standard group action Decisional Diffie-Hellman (GA-DDH) assumption. VRFs serve as cryptographic tools allowing users to generate pseudorandom outputs along with publicly verifiable proofs. Moreover, the residual pseudorandomness of VRFs ensures the pseudorandomness of unrevealed inputs, even when multiple outputs and proofs are disclosed. Our work aims at addressing the growing demand for post-quantum VRFs, as existing constructions based on elliptic curve cryptography (ECC) or classical DDH-type assumptions are vulnerable to quantum threats.

In our contributions, our two VRF constructions, rooted in number-theoretic pseudorandom functions, are both simple and secure over the random oracle model. We introduce a new proof system for the factorization of group actions and set elements, serving as the proofs for our VRFs. The first proposal is based on the standard GA-DDH problem, and for its security proof, we introduce the (group action) master Decisional Diffie-Hellman problem over group actions, proving its equivalence to the standard GA-DDH problem. In the second construction, we leverage quadratic twists to enhance efficiency, reducing the key size and the proof sizes, expanding input size. The scheme is based on the square GA-DDH problem.

Moreover, we employ advanced techniques from the isogeny literature to optimize the proof size to 39KB and 34KB using CSIDH-512 without compromising VRF notions. The schemes feature fast evaluations but exhibit slower proof generation. To the best of our knowledge, these constructions represent the first two provably secure VRFs based on isogenies.

In this work we study algebraic and generic models for group actions, and extend them to the universal composability (UC) framework of Canetti (FOCS 2001). We revisit the constructions of Duman et al. (PKC 2023) integrating the type-safe model by Zhandry (Crypto 2022), adapted to the group action setting, and formally define an algebraic action model (AAM). This model restricts the power of the adversary in a similar fashion to the algebraic group model (AGM). By imposing algebraic behaviour to the adversary and environment of the UC framework, we construct the UC-AAM. Finally, we instantiate UC-AAM with isogeny-based assumptions, in particular the CSIDH action with twists, obtaining the explicit isogeny model, UC-EI; we observe that, under certain assumptions, this model is "closer" to standard UC than the UC-AGM, even though there still exists an important separation. We demonstrate the utility of our definitions by proving UC-EI security for the passive-secure oblivious transfer protocol described by Lai et al. (Eurocrypt 2021), hence providing the first concretely efficient two-message isogeny-based OT protocol in the random oracle model against malicious adversaries.