CSIDH on the surface

For primes p≡3mod4, we show that setting up CSIDH on the surface, i.e., using supersingular elliptic curves with endomorphism ring Z[(1+−p−−−√)/2], amounts to just a few sign switches in the underlying arithmetic. If p≡7mod8 then horizontal 2-isogenies can be used to help compute the class group action. The formulas we derive for these 2-isogenies are very efficient (they basically amount to a single exponentiation in Fp) and allow for a noticeable speed-up, e.g., our resulting CSURF-512 protocol runs about 5.68% faster than CSIDH-512. This improvement is completely orthogonal to all previous speed-ups, constant-time measures and construction of cryptographic primitives that have appeared in the literature so far. At the same time, moving to the surface gets rid of the redundant factor Z3 of the acting ideal-class group, which is present in the case of CSIDH and offers no extra security

Safe-Error Attacks on SIKE and CSIDH

The isogeny-based post-quantum schemes SIKE (NIST PQC round 3 alternate candidate) and CSIDH (Asiacrypt 2018) have received only little attention with respect to their fault attack resilience so far. We aim to fill this gap and provide a better understanding of their vulnerability by analyzing their resistance towards safe-error attacks. We present four safe-error attacks, two against SIKE and two against a constant-time implementation of CSIDH that uses dummy isogenies. The attacks use targeted bitflips during the respective isogeny-graph traversals. All four attacks lead to full key recovery. By using voltage and clock glitching, we physically carried out two of the attacks - one against each scheme -, thus demonstrate that full key recovery is also possible in practice

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

Patient Zero and Patient Six: Zero-Value and Correlation Attacks on CSIDH and SIKE

Recent works have started side-channel analysis on SIKE and show the vulnerability of isogeny-based systems to zero-value attacks. In this work, we expand on such attacks by analyzing the behavior of the zero curve $E_0$ and six curve $E_6$ in CSIDH and SIKE. We demonstrate an attack on static-key CSIDH and SIKE implementations that recovers bits of the secret key by observing via zero-value-based resp. exploiting correlation-collision-based side-channel analysis whether secret isogeny walks pass over the zero or six curve. We apply this attack to fully recover secret keys of SIKE and two state-of-the-art CSIDH-based implementations: CTIDH and SQALE. We show the feasibility of exploiting side-channel information for the proposed attacks based on simulations with various realistic noise levels. Additionally, we discuss countermeasures to prevent zero-value and correlation-collision attacks against CSIDH and SIKE in our attacker model

Threshold Schemes from Isogeny Assumptions

We initiate the study of threshold schemes based on the Hard Homogeneous Spaces (HHS) framework of Couveignes. Quantum-resistant HHS based on supersingular isogeny graphs have recently become usable thanks to the record class group precomputation performed for the signature scheme CSI-FiSh. Using the HHS equivalent of the technique of Shamir\u27s secret sharing in the exponents, we adapt isogeny based schemes to the threshold setting. In particular we present threshold versions of the CSIDH public key encryption, and the CSI-FiSh signature schemes. The main highlight is a threshold version of CSI-FiSh which runs almost as fast as the original scheme, for message sizes as low as 1880 B, public key sizes as low as 128 B, and thresholds up to 56; other speed-size-threshold compromises are possible

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

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