arXiv subjects

Luca De Feo

Publications and source records attributed to Luca De Feo.

10 recordsLinked to original sources

arXiv2026

The SQInstructor: a guide to SQIsign and the Deuring Correspondence with level structures

We explore the use of level structures to generalize the SQIsign signature scheme. We give a general framework where, given the public key and the commitment, the challenge is to exhibit an isogeny between them with an additional requirement, namely to map a chosen level structure to another. We then instantiate the framework using 1-dimensional and 2-dimensional isogenies. In doing that we provide a new explicit Deuring correspondence for supersingular elliptic curves with level structures and solve new constrained norm equations.

math.NT↗

arXiv2019

Stronger and Faster Side-Channel Protections for CSIDH

CSIDH is a recent quantum-resistant primitive based on the difficulty of finding isogeny paths between supersingular curves. Recently, two constant-time versions of CSIDH have been proposed: first by Meyer, Campos and Reith, and then by Onuki, Aikawa, Yamazaki and Takagi. While both offer protection against timing attacks and simple power consumption analysis, they are vulnerable to more powerful attacks such as fault injections. In this work, we identify and repair two oversights in these algorithms that compromised their constant-time character. By exploiting Edwards arithmetic and optimal addition chains, we produce the fastest constant-time version of CSIDH to date. We then consider the stronger attack scenario of fault injection, which is relevant for the security of CSIDH static keys in embedded hardware. We propose and evaluate a dummy-free CSIDH algorithm. While these CSIDH variants are slower, their performance is still within a small constant factor of less-protected variants. Finally, we discuss derandomized CSIDH algorithms.

cs.CR↗

arXiv2019

Standard Lattices of Compatibly Embedded Finite Fields

Lattices of compatibly embedded finite fields are useful in computer algebra systems for managing many extensions of a finite field $\mathbb{F}_p$ at once. They can also be used to represent the algebraic closure $\bar{\mathbb{F}}_p$, and to represent all finite fields in a standard manner. The most well known constructions are Conway polynomials, and the Bosma-Cannon-Steel framework used in Magma. In this work, leveraging the theory of the Lenstra-Allombert isomorphism algorithm, we generalize both at the same time. Compared to Conway polynomials, our construction defines a much larger set of field extensions from a small pre-computed table; however it is provably as inefficient as Conway polynomials if one wants to represent all field extensions, and thus yields no asymptotic improvement for representing $\bar{\mathbb{F}}_p$. Compared to Bosma-Cannon-Steel lattices, it is considerably more efficient both in computation time and storage: all algorithms have at worst quadratic complexity, and storage is linear in the number of represented field extensions and their degrees. Our implementation written in C/Flint/Julia/Nemo shows that our construction in indeed practical.

math.NT↗

arXiv2018

Towards practical key exchange from ordinary isogeny graphs

We revisit the ordinary isogeny-graph based cryptosystems of Couveignes and Rostovtsev-Stolbunov, long dismissed as impractical. We give algorithmic improvements that accelerate key exchange in this framework, and explore the problem of generating suitable system parameters for contemporary pre-and post-quantum security that take advantage of these new algorithms. We also prove the session-key security of this key exchange in the Canetti-Krawczyk model, and the IND-CPA security of the related public-key encryption scheme, under reasonable assumptions on the hardness of computing isogeny walks. Our systems admit efficient key-validation techniques that yield CCA-secure encryp-tion, thus providing an important step towards efficient post-quantum non-interactive key exchange (NIKE).

cs.CR↗

arXiv2017

Mathematics of Isogeny Based Cryptography

These lectures notes were written for a summer school on Mathematics for post-quantum cryptography in Thiès, Senegal. They try to provide a guide for Masters' students to get through the vast literature on elliptic curves, without getting lost on their way to learning isogeny based cryptography. They are by no means a reference text on the theory of elliptic curves, nor on cryptography; students are encouraged to complement these notes with some of the books recommended in the bibliography. The presentation is divided in three parts, roughly corresponding to the three lectures given. In an effort to keep the reader interested, each part alternates between the fundamental theory of elliptic curves, and applications in cryptography. We often prefer to have the main ideas flow smoothly, rather than having a rigorous presentation as one would have in a more classical book. The reader will excuse us for the inaccuracies and the omissions.

cs.CR↗

arXiv2017

Computing isomorphisms and embeddings of finite fields

Let $\mathbb{F}_q$ be a finite field. Given two irreducible polynomials $f,g$ over $\mathbb{F}_q$, with $\mathrm{deg} f$ dividing $\mathrm{deg} g$, the finite field embedding problem asks to compute an explicit description of a field embedding of $\mathbb{F}_q[X]/f(X)$ into $\mathbb{F}_q[Y]/g(Y)$. When $\mathrm{deg} f = \mathrm{deg} g$, this is also known as the isomorphism problem. This problem, a special instance of polynomial factorization, plays a central role in computer algebra software. We review previous algorithms, due to Lenstra, Allombert, Rains, and Narayanan, and propose improvements and generalizations. Our detailed complexity analysis shows that our newly proposed variants are at least as efficient as previously known algorithms, and in many cases significantly better. We also implement most of the presented algorithms, compare them with the state of the art computer algebra software, and make the code available as open source. Our experiments show that our new variants consistently outperform available software.

cs.SC↗

arXiv2016

Explicit isogenies in quadratic time in any characteristic

Consider two elliptic curves $E,E'$ defined over the finite field $\mathbb{F}_q$, and suppose that there exists an isogeny $ψ$ between $E$ and $E'$. We propose an algorithm that determines $ψ$ from the knowledge of $E$, $E'$ and of its degree $r$, by using the structure of the $\ell$-torsion of the curves (where $\ell$ is a prime different from the characteristic $p$ of the base field). Our approach is inspired by a previous algorithm due to Couveignes, that involved computations using the $p$-torsion on the curves. The most refined version of that algorithm, due to De Feo, has a complexity of $\tilde{O}(r^2) p^{O(1)}$ base field operations. On the other hand, the cost of our algorithm is $\tilde{O}(r^2 + \sqrt{r} \log(q))$; this makes it an interesting alternative for the medium- and large-characteristic cases.

math.AG↗

arXiv2013

Fast algorithms for ell-adic towers over finite fields

Inspired by previous work of Shoup, Lenstra-De Smit and Couveignes-Lercier, we give fast algorithms to compute in (the first levels of) the ell-adic closure of a finite field. In many cases, our algorithms have quasi-linear complexity.

cs.SC↗

arXiv2010

Fast algorithms for computing isogenies between ordinary elliptic curves in small characteristic

The problem of computing an explicit isogeny between two given elliptic curves over F_q, originally motivated by point counting, has recently awaken new interest in the cryptology community thanks to the works of Teske and Rostovstev & Stolbunov. While the large characteristic case is well understood, only suboptimal algorithms are known in small characteristic; they are due to Couveignes, Lercier, Lercier & Joux and Lercier & Sirvent. In this paper we discuss the differences between them and run some comparative experiments. We also present the first complete implementation of Couveignes' second algorithm and present improvements that make it the algorithm having the best asymptotic complexity in the degree of the isogeny.

math.NT↗

arXiv2010

Fast Arithmetics in Artin-Schreier Towers over Finite Fields

An Artin-Schreier tower over the finite field F_p is a tower of field extensions generated by polynomials of the form X^p - X - a. Following Cantor and Couveignes, we give algorithms with quasi-linear time complexity for arithmetic operations in such towers. As an application, we present an implementation of Couveignes' algorithm for computing isogenies between elliptic curves using the p-torsion.

cs.SC↗