Isogeny-based schemes often come with special requirements on the field of definition of the involved elliptic curves. For instance, the efficiency of SQIsign, a promising candidate in the NIST signature standardisation process, requires a large power of two and a large smooth integer $T$ to divide $p^2-1$ for its prime parameter $p$. We present two new methods that combine previous techniques for finding suitable primes: sieve-and-boost and XGCD-and-boost. We use these methods to find primes for the NIST submission of SQIsign. Furthermore, we show that our methods are flexible and can be adapted to find suitable parameters for other isogeny-based schemes such as AprèsSQI or POKE. For all three schemes, the parameters we present offer the best performance among all parameters proposed in the literature.

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.

This work introduces several algorithms related to the computation of orientations in endomorphism rings of supersingular elliptic curves. This problem is at the heart of several results regarding the security of oriented-curves in isogeny-based cryptography. Under the Deuring correspondence, it can be expressed purely in terms of quaternion and boils down to representing integers by ternary quadratic forms. Our main contribution is to show that there exist efficient algorithms to solve this problem for quadratic orders of discriminant $n$ up to $O(p^{4/3})$. Our approach improves upon previous results by increasing this bound from $O(p)$ to $O(p^{4/3})$ and removing some heuristics. We introduce several variants of our new algorithm and provide a careful analysis of their asymptotic running time (without heuristic when it is possible). The best proven asymptotic complexity of one of our variants is $O(n^{3/4}/p)$ in average. The best heuristic variant has a complexity of $O(p^{1/3})$ for big enough $n$. We then introduce several results regarding the computation of ideals between oriented orders. The first application of this is a simplification of the known reduction from vectorization to computing the endomorphism ring, removing the assumption on the factorization of the discriminant. As a second application, we relate the problem of computing fixed-degree isogenies between supersingular curves to the problem of computing orientations in endomorphism rings, and we show that for a large range of degree $d$, our new algorithms improve on the state-of-the-art, and in important special cases, the range of degree $d$ for which there exist a polynomial-time algorithm is increased. In the most special case we consider, when both curves are oriented by a small degree endomorphism, we show heuristically that our techniques allow the computation of isogenies of any degree, assuming they exist.

We prove that isogenies between Drinfeld F[x]-modules over a finite field can be computed in polynomial time. This breaks Drinfeld analogs of isogeny-based cryptosystems.

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.

We use theta groups to study $2$-isogenies between Kummer lines, with a particular focus on the Montgomery model. This allows us to recover known formulas, along with more efficient forms for translated isogenies, which require only $2S+2m_0$ for evaluation. We leverage these translated isogenies to build a hybrid ladder for scalar multiplication on Montgomery curves with rational $2$-torsion, which cost $3M+6S+2m_0$ per bit, compared to $5M+4S+1m_0$ for the standard Montgomery ladder.