arXiv subjects

Antonin Leroux

Publications and source records attributed to Antonin Leroux.

5 recordsLinked to original sources

arXiv2025

Evaluation of Modular Polynomials from Supersingular Elliptic Curves

We present several new algorithms to evaluate modular polynomials of level $\ell$ modulo a prime $p$ on an input $j$. More precisely, we introduce two new generic algorithms, sharing the following similarities: they are based on a CRT approach; they make use of supersingular curves and the Deuring correspondence; and, their memory requirements are optimal. The first algorithm combines the ideas behind a hybrid algorithm of Sutherland in 2013 with a recent algorithm to compute modular polynomials using supersingular curves introduced in 2023 by Leroux. The complexity (holding around several plausible heuristic assumptions) of the resulting algorithm matches the $\Tilde{O}(\ell^3 \log^{3} \ell + \ell \log p)$ time complexity of the best known algorithm by Sutherland, but has an optimal memory requirement. Our second algorithm is based on a sub-algorithm that can evaluate modular polynomials efficiently on supersingular $j$-invariants defined over $\Fp$, and achieves heuristic complexity quadratic in both $\ell$ and $\log j$, and linear in $\log p$. In particular, it is the first generic algorithm with optimal memory requirement to obtain a quadratic complexity in~$\ell$. Additionally, we show how to adapt our method to the computation of other types of modular polynomials such as the one stemming from Weber's function. Finally, we provide an optimised implementation of the two algorithms detailed in this paper, though we emphasise that various modules in our codebase may find applications outside their use in this paper.

math.NT↗

arXiv2023

Computation of Hilbert class polynomials and modular polynomials from supersingular elliptic curves

We present several new heuristic algorithms to compute class polynomials and modular polynomials modulo a prime $p$ by revisiting the idea of working with supersingular elliptic curves. The best known algorithms to this date are based on ordinary curves, due to the supposed inefficiency of the supersingular case. While this was true a decade ago, the recent advances in the study of supersingular curves through the Deuring correspondence motivated by isogeny-based cryptography has provided all the tools to perform the necessary tasks efficiently.

math.NT↗

arXiv2023

Hidden Stabilizers, the Isogeny To Endomorphism Ring Problem and the Cryptanalysis of pSIDH

The Isogeny to Endomorphism Ring Problem (IsERP) asks to compute the endomorphism ring of the codomain of an isogeny between supersingular curves in characteristic $p$ given only a representation for this isogeny, i.e. some data and an algorithm to evaluate this isogeny on any torsion point. This problem plays a central role in isogeny-based cryptography; it underlies the security of pSIDH protocol (ASIACRYPT 2022) and it is at the heart of the recent attacks that broke the SIDH key exchange. Prior to this work, no efficient algorithm was known to solve IsERP for a generic isogeny degree, the hardest case seemingly when the degree is prime. In this paper, we introduce a new quantum polynomial-time algorithm to solve IsERP for isogenies whose degrees are odd and have $O(\log\log p)$ many prime factors. As main technical tools, our algorithm uses a quantum algorithm for computing hidden Borel subgroups, a group action on supersingular isogenies from EUROCRYPT 2021, various algorithms for the Deuring correspondence and a new algorithm to lift arbitrary quaternion order elements modulo an odd integer $N$ with $O(\log\log p)$ many prime factors to powersmooth elements. As a main consequence for cryptography, we obtain a quantum polynomial-time key recovery attack on pSIDH. The technical tools we use may also be of independent interest.

cs.CR↗

arXiv2020

Faster computation of isogenies of large prime degree

Let $\mathcal{E}/\mathbb{F}_q$ be an elliptic curve, and $P$ a point in $\mathcal{E}(\mathbb{F}_q)$ of prime order $\ell$. Vélu's formulae let us compute a quotient curve $\mathcal{E}' = \mathcal{E}/\langle{P}\rangle$ and rational maps defining a quotient isogeny $ϕ: \mathcal{E} \to \mathcal{E}'$ in $\tilde{O}(\ell)$ $\mathbb{F}_q$-operations, where the $\tilde{O}$ is uniform in $q$.This article shows how to compute $\mathcal{E}'$, and $ϕ(Q)$ for $Q$ in $\mathcal{E}(\mathbb{F}_q)$, using only $\tilde{O}(\sqrt{\ell})$ $\mathbb{F}_q$-operations, where the $\tilde{O}$ is again uniform in $q$.As an application, this article speeds up some computations used in the isogeny-based cryptosystems CSIDH and CSURF.

cs.CR↗

arXiv2018

Information gain ratio correction: Improving prediction with more balanced decision tree splits

Decision trees algorithms use a gain function to select the best split during the tree's induction. This function is crucial to obtain trees with high predictive accuracy. Some gain functions can suffer from a bias when it compares splits of different arities. Quinlan proposed a gain ratio in C4.5's information gain function to fix this bias. In this paper, we present an updated version of the gain ratio that performs better as it tries to fix the gain ratio's bias for unbalanced trees and some splits with low predictive interest.

stat.ML↗