A Faster Constant-time Algorithm of CSIDH keeping Two Points

At ASIACRYPT 2018, Castryck, Lange, Martindale, Panny and Renes proposed CSIDH, which is a key-exchange protocol based on isogenies between elliptic curves, and a candidate for post-quantum cryptography. However, the implementation by Castryck et al. is not constant-time. Specifically, a part of the secret key could be recovered by the side-channel Attacks. Recently, Meyer, Campos and Reith proposed a constant-time implementation of CSIDH by introducing dummy isogenies and taking secret exponents only from intervals of non-negative integers. Their non-negative intervals make the calculation cost of their implementation of CSIDH twice that of the worst case of the standard (variable-time) implementation of CSIDH. In this paper, we propose a more efficient constant-time algorithm that takes secret exponents from intervals symmetric with respect to the zero. For using these intervals, we need to keep two torsion points in an elliptic curve and calculation for these points. We evaluate the costs of our implementation and that of Meyer et al. in terms of the number of operations on a finite prime field. Our evaluation shows that our constant-time implementation of CSIDH reduces the calculation cost by 28.23% compared with the implementation by Mayer et al. We also implemented our algorithm by extending the implementation in C of Meyer et al. (originally from Castryck et al.). Then our implementation achieved 152.8 million clock cycles, which is about 29.03% faster than that of Meyer et al. and confirms the above reduction ratio in our cost evaluation

Optimal strategies for CSIDH

Since its proposal in Asiacrypt 2018, the commutative isogeny-based key exchange protocol (CSIDH) has spurred considerable attention to improving its performance and re-evaluating its classical and quantum security guarantees. In this paper we discuss how the optimal strategies employed by the Supersingular Isogeny Diffie-Hellman (SIDH) key agreement protocol can be naturally extended to CSIDH. Furthermore, we report a software library that achieves moderate but noticeable performance speedups when compared against state-of-the-art implementations of CSIDH-512, which is the most popular CSIDH instantiation. We also report an estimated number of field operations for larger instantiations of this protocol, namely, CSIDH-1024 and CSIDH-1792

Effective Pairings in Isogeny-based Cryptography

Pairings are useful tools in isogeny-based cryptography and have been used in SIDH/SIKE and other protocols. As a general technique, pairings can be used to move problems about points on curves to elements in finite fields. However, until now, their applicability was limited to curves over fields with primes of a specific shape and pairings seemed too costly for the type of primes that are nowadays often used in isogeny-based cryptography. We remove this roadblock by optimizing pairings for highly-composite degrees such as those encountered in CSIDH and SQISign. This makes the general technique viable again: We apply our low-cost pairing to problems of general interest, such as supersingularity verification and finding full-torsion points, and show that we can outperform current methods, in some cases up to four times faster than the state-of-the-art. Furthermore, we analyze how pairings can be used to improve deterministic and dummy-free CSIDH. Finally, we provide a constant-time implementation (in Rust) that shows the practicality of these algorithms

Karatsuba-based square-root Vélu’s formulas applied to two isogeny-based protocols

At a combined computational expense of about $6{\ell}$ field operations, Vélu\u27s formulas are used to construct and evaluate degree-$\ell$ isogenies in the vast majority of isogeny-based cryptographic schemes. By adapting to Vélu\u27s formulas a baby-step giant-step approach, Bernstein, De Feo, Leroux, and Smith presented a procedure that can computes isogeny operations at a reduced cost of just $\tilde{O}(\sqrt{\ell})$ field operations. In this paper, we present a concrete computational analysis of these novel procedure along with several algorithmic tricks that helped us to further decrease its computational cost. We also report an optimized Python3-code implementation of several instantiations of two isogeny-based key-exchange protocols, namely, CSIDH and B-SIDH. Our software library uses a combination of the modified Vélu\u27s formulas and an adaptation of the optimal strategies commonly used in the SIDH/SIKE protocols to produce significant speedups. Compared to a traditional Vélu constant-time implementation of CSIDH, our experimental results report a saving of 5.357\%, 13.68\% and 25.938\% base field operations for CSIDH-512, CSIDH-1024, and CSIDH-1792, respectively. Additionally, we present the first optimized implementation of B-SIDH ever reported in the open literature

Fast and Frobenius: Rational Isogeny Evaluation over Finite Fields

Consider the problem of efficiently evaluating isogenies $\phi: E \to E/H$ of elliptic curves over a finite field $\mathbb{F}_q$, where the kernel $H = \langle G\rangle$ is a cyclic group of odd (prime) order: given $E$, $G$, and a point (or several points) $P$ on $E$, we want to compute $\phi(P)$. This problem is at the heart of efficient implementations of group-action- and isogeny-based post-quantum cryptosystems such as CSIDH. Algorithms based on V{\'e}lu's formulae give an efficient solution to this problem when the kernel generator $G$ is defined over $\mathbb{F}_q$. However, for general isogenies, $G$ is only defined over some extension $\mathbb{F}_{q^k}$, even though $\langle G\rangle$ as a whole (and thus $\phi$) is defined over the base field $\mathbb{F}_q$; and the performance of V{\'e}lu-style algorithms degrades rapidly as $k$ grows. In this article we revisit the isogeny-evaluation problem with a special focus on the case where $1 \le k \le 12$. We improve V{\'e}lu-style isogeny evaluation for many cases where $k = 1$ using special addition chains, and combine this with the action of Galois to give greater improvements when $k > 1$

Action de Groupe Supersingulières et Echange de Clés Post-quantique

Alice and Bob want to exchange information and make sure that an eavesdropper will not be able to listen to them, even with a quantum computer.To that aim they use cryptography and in particular a key-exchange protocol. These type of protocols rely on number theory and algebraic geometry. However current protocols are not quantum resistant, which is the reason why new cryptographic tools must be developed. One of these tools rely on isogenies, i.e. homomorphisms between elliptic curves. In this thesis the first contribution is an implementation of an isogeny-based key-exchange protocol resistant against side-channel attacks (timing and power consumption analysis, fault injection). We also generalize this protocol to a larger set of elliptic curves.Alice et Bob souhaitent échanger des informations sans qu’un attaquant, même muni d’un ordinateur quantique, puisse les entendre. Pour cela, ils ont recours à la cryptologie et en particulier à un protocole d’échange de clés. Ces protocoles reposent sur la théorie des nombres et la géométrie algébrique. Cependant les protocoles actuellement utilisés ne résistent pas aux attaques quantiques, c’est pourquoi il est nécessaire de développer de nouveaux outils cryptographiques. L’un de ces outils repose sur les isogénies, c’est-à-dire des homomorphismes entre des courbes elliptiques. Dans cette thèse nous proposons une implémentation d’un des protocoles d’échange de clés basé sur les isogénies qui résiste aux attaques par canaux auxiliaires (étude de la durée d’exécution, de la consommation de courant et injection de fautes). Nous généralisons également ce protocole à un plus grand ensemble de courbes elliptiques