arXiv subjects

Christophe Levrat

Publications and source records attributed to Christophe Levrat.

7 recordsLinked to original sources

arXiv2026

Explicit Euclidean division algorithms for some degree 8 number rings

This article focuses on some rings of integers of number fields which are known to be norm-Euclidean domains, but for which no explicit algorithm computing the Euclidean division has yet been studied or implemented. The rings of integers we are interested in were proven to be Euclidean by H.W. Lenstra, Jr in 1978; they include the $n$-th cyclotomic rings for $n=15,20,24$. We present an algorithm performing Euclidean division in these rings based on Lenstra's proof and a closest vector computation by Conway and Sloane, and study its complexity. We give a complete implementation of the algorithm in SageMath. We also estimate the size of the remainders obtained when computing Euclidean divisions with this algorithm.

math.NT↗

arXiv2025

Effective Artin-Schreier-Witt theory for curves

We present an algorithm which, given a connected smooth projective curve $X$ over an algebraically closed field of characteristic $p>0$ and its Hasse--Witt matrix, as well as a positive integer $n$, computes all \'etale Galois covers of $X$ with group $\mathbb{Z}/p^n\mathbb{Z}$. We compute the complexity of this algorithm when $X$ is defined over a finite field, and provide a complete implementation in SageMath, as well as some explicit examples. We then apply this algorithm to the computation of the cohomology complex of a locally constant sheaf of $\mathbb{Z}/p^n\mathbb{Z}$-modules on such a curve.

math.NT↗

arXiv2025

A divide-and-conquer sumcheck protocol

We present a new sumcheck protocol called Fold-DCS (Fold-Divide-and-Conquer-Sumcheck) for multivariate polynomials based on a divide-and-conquer strategy. Its round complexity and soundness error are logarithmic in the number of variables, whereas they are linear in the classical sumcheck protocol. This drastic improvement in number of rounds and soundness comes at the expense of exchanging multivariate polynomials, which can be alleviated using polynomial commitment schemes. We first present Fold-DCS in the PIOP model, where the prover provides oracle access to a multivariate polynomial at each round. We then replace this oracle access in practice with a multivariate polynomial commitment scheme; we illustrate this with an adapted version of the recent commitment scheme Zeromorph [KT24], which allows us to replace most of the queries made by the verifier with a single batched evaluation check.

cs.IT↗

arXiv2025

Highway to Hull: An Algorithm for Solving the General Matrix Code Equivalence Problem

The matrix code equivalence problem consists, given two matrix spaces $\mathcal{C},\mathcal{D} \subset \mathbb{F}_q^{m\times n}$ of dimension $k$, in finding invertible matrices $P\in\mathrm{GL}_m(\mathbb{F}_q)$ and $Q\in\mathrm{GL}_n(\mathbb{F}_q)$ such that $\mathcal{D}=P\mathcal{C} Q^{-1}$. Recent signature schemes such as MEDS and ALTEQ relate their security to the hardness of this problem. Recent works by Narayanan, Qiao and Tang on the one hand and by Ran and Samardjiska on the other hand tackle this problem. The former is restricted to the ``cubic'' case $k = m =n$ and succeeds in $\widetilde{\mathcal{O}}(q^{\frac k 2})$ operations. The latter is an algebraic attack on the general problem whose complexity is not fully understood and which succeeds only on $\mathcal{O}(1/q)$ instances. We present a novel algorithm which solves the problem in the general case. Our approach consists in reducing the problem to the matrix code conjugacy problem, \emph{i.e.} the case $P=Q$. For the latter problem, similarly to the permutation code equivalence problem in Hamming metric, a natural invariant based on the \emph{Hull} of the code can be used. Next, the equivalence of codes can be deduced using a usual list collision argument. For $k=m=n$, our algorithm achieves the same time complexity as Narayanan \emph{et al.} but with a lower space complexity. Moreover, ours extends to a much broader range of parameters.

cs.CR↗

arXiv2023

Computing the cohomology of constructible \'etale sheaves on curves

We present an explicit expression of the cohomology complex of a constructible sheaf of abelian groups on the small \'etale site of an irreducible curve over an algebraically closed field, when the torsion of the sheaf is invertible in the field. This expression only involves finite groups, and is functorial in both the curve and the sheaf. In particular, we explain how to compute the Galois action on this complex. We also present an algorithm which computes it and study its complexity.

math.AG↗

arXiv2023

Curves are algebraic $K(\pi,1)$: theoretical and practical aspects

We prove that any geometrically connected curve $X$ over a field $k$ is an algebraic $K(\pi,1)$, as soon as its geometric irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible \'etale sheaf of $\mathbb{Z}/n\mathbb{Z}$-modules, with $n$ invertible in $k$, is canonically isomorphic to the cohomology of its corresponding $\pi_1(X)$-module. To this end, we explicitly construct some Galois coverings of $X$ corresponding to Galois coverings of the normalisation of its irreducible components. When $k$ is finite or separably closed, we explicitly describe finite quotients of $\pi_1(X)$ that allow to compute the cohomology groups of the sheaf, and give explicit descriptions of the cup products $H^1\times H^1\to H^2$ and $H^1\times H^2\to H^3$ in terms of finite group cohomology.

math.AG↗

arXiv2022

Calcul effectif de la cohomologie des faisceaux constructibles sur le site \'etale d'une courbe

This thesis deals with the algorithmic representation of constructible sheaves of abelian groups on the \'etale site of a variety over an algebraically closed field, as well as the explicit computation of their cohomology. We describe three representations of such sheaves on curves with at worst nodal singularities, as well as algorithms performing various operations (kernels and cokernels of morphisms, pullback and pushforward, internal Hom and tensor product) on these sheaves. We present an algorithm computing the cohomology complex of a locally constant constructible sheaf on a smooth or nodal curve, which in turn allows us to give an explicit description of the functor $\mathrm{R}\Gamma(X,-)\colon \mathrm{D}^b_c(X,\mathbb{Z}/n\mathbb{Z})\to \mathrm{D}^b_c(\mathbb{Z}/n\mathbb{Z})$. This description is functorial in the scheme $X$ and the given complex of constructible sheaves. In particular, if $X$ and the sheaf $\mathcal{F}$ are obtained by base change from a subfield, we describe the Galois action on the complex $\mathrm{R}\Gamma(X,\mathcal{F})$. We give precise bounds on the number of operations performed by the algorithm computing $\mathrm{R}\Gamma(X,\mathcal{F})$. We also give an explicit description of cup-products in the cohomology of locally constant constructible sheaves over smooth projective curves. Finally, we show how to use these algorithms in order to compute the cohomology groups of a constant sheaf on a smooth surface fibered over the projective line.

math.AG↗