International Association
for Cryptologic Research

We provide explicit radical N -isogeny formulae for all odd integers N . The formulae are compact closed-form expressions which require one Nth root computation and O(N) basic field operations. The formulae are highly efficient to compute a long chain of N -isogenies, and are extremely beneficial for speeding up certain cryptographic protocols such as CSIDH. Unfortunately, the formulae are conjectured, but we provide ample supporting evidence which strongly suggests their correctness. For CSIDH-512, we notice an additional 35% speed-up when using radical isogenies up to N = 199, compared to the work by Castryck, Decru, Houben and Vercauteren, which uses radical isogenies up to N = 19 only. The addition of our radical isogenies also speeds up the computation of larger class group actions in a comparable fashion.

<p> Recently, Geraud-Stewart and Naccache proposed two trapdoors based on matrix products. In this paper, we answer the call for cryptanalysis. We explore how using the trace and determinant of a matrix can be used to attack their constructions. We fully break their first construction in a polynomial-time attack. We show an information leak in the second construction using characteristic polynomials, and provide two attacks that decrease the bit security by about half. </p>

We present an efficient key recovery attack on the Supersingular Isogeny Diffie-Hellman protocol (SIDH). The attack is based on Kani's "reducibility criterion" for isogenies from products of elliptic curves and strongly relies on the torsion point images that Alice and Bob exchange during the protocol. If we assume knowledge of the endomorphism ring of the starting curve then the classical running time is polynomial in the input size (heuristically), apart from the factorization of a small number of integers that only depend on the system parameters. The attack is particularly fast and easy to implement if one of the parties uses 2-isogenies and the starting curve comes equipped with a non-scalar endomorphism of very small degree; this is the case for SIKE, the instantiation of SIDH that recently advanced to the fourth round of NIST's standardization effort for post-quantum cryptography. Our Magma implementation breaks SIKEp434, which aims at security level 1, in about ten minutes on a single core.

We address three main open problems concerning the use of radical isogenies, as presented by Castryck, Decru and Vercauteren at Asiacrypt 2020, in the computation of long chains of isogenies of fixed, small degree between elliptic curves over finite fields. Firstly, we present an interpolation method for finding radical isogeny formulae in a given degree N, which by-passes the need for factoring division polynomials over large function fields. Using this method, we are able to push the range for which we have formulae at our disposal from N ≤ 13 to N ≤ 37. Secondly, using a combination of known techniques and ad-hoc manipulations, we derived optimized versions of these formulae for N ≤ 19, with some instances performing more than twice as fast as their counterparts from 2020. Thirdly, we solve the problem of understanding the correct choice of radical when walking along the surface between supersingular elliptic curves over Fp with p ≡ 7 mod 8; this is non-trivial for even N and was only settled for N = 4 by Onuki and Moriya at PKC 2022. We give a conjectural statement for all even N and prove it for N ≤ 14. The speed-ups obtained from these techniques are substantial: using 16-isogenies, the computation of long chains of 2-isogenies over 512-bit prime fields can be improved by a factor 3, and the previous implementation of CSIDH using radical isogenies can be sped up by about 12%.

This paper introduces a new approach to computing isogenies called ``radical isogenies'' and a corresponding method to compute chains of $N$-isogenies that is very efficient for small $N$. The method is fully deterministic and completely avoids generating $N$-torsion points. It is based on explicit formulae for the coordinates of an $N$-torsion point $P'$ on the codomain of a cyclic $N$-isogeny $\varphi : E \to E'$, such that composing $\varphi$ with $E' \to E' / \langle P' \rangle$ yields a cyclic $N^2$-isogeny. These formulae are simple algebraic expressions in the coefficients of $E$, the coordinates of a generator $P$ of $\ker \varphi$, and an $N$th root $\sqrtN{\rho}$, where the radicand $\rho$ itself is given by an easily computable algebraic expression in the coefficients of $E$ and the coordinates of $P$. The formulae can be iterated and are particularly useful when computing chains of $N$-isogenies over a finite field $\F_q$ with $\gcd(q-1, N) = 1$, where taking an $N$th root is a simple exponentiation. Compared to the state-of-the-art, our method results in an order of magnitude speed-up for $N \leq 13$; for larger $N$, the advantage disappears due to the increasing complexity of the formulae. When applied to CSIDH, we obtain a speed-up of about $19 \%$ over the implementation by Bernstein, De Feo, Leroux and Smith for the CSURF-512 parameters.