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Éric Schost

Publications and source records attributed to Éric Schost.

At least 37 records · Page 2Linked to original sources
arXiv2019

Change of basis for m-primary ideals in one and two variables

Following recent work by van der Hoeven and Lecerf (ISSAC 2017), we discuss the complexity of linear mappings, called untangling and tangling by those authors, that arise in the context of computations with univariate polynomials. We give a slightly faster tangling algorithm and discuss new applications of these techniques. We show how to extend these ideas to bivariate settings, and use them to give bounds on the arithmetic complexity of certain algebras.

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arXiv2019

Implementations of efficient univariate polynomial matrix algorithms and application to bivariate resultants

Complexity bounds for many problems on matrices with univariate polynomial entries have been improved in the last few years. Still, for most related algorithms, efficient implementations are not available, which leaves open the question of the practical impact of these algorithms, e.g. on applications such as decoding some error-correcting codes and solving polynomial systems or structured linear systems. In this paper, we discuss implementation aspects for most fundamental operations: multiplication, truncated inversion, approximants, interpolants, kernels, linear system solving, determinant, and basis reduction. We focus on prime fields with a word-size modulus, relying on Shoup's C++ library NTL. Combining these new tools to implement variants of Villard's algorithm for the resultant of generic bivariate polynomials (ISSAC 2018), we get better performance than the state of the art for large parameters.

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arXiv2019

Quadratic Probabilistic Algorithms for Normal Bases

It is well known that for any finite Galois extension field $K/F$, with Galois group $G = \mathrm{Gal}(K/F)$, there exists an element $α\in K$ whose orbit $G\cdotα$ forms an $F$-basis of $K$. Such an element $α$ is called \emph{normal} and $G\cdotα$ is called a normal basis. In this paper we introduce a probabilistic algorithm for finding a normal element when $G$ is either a finite abelian or a metacyclic group. The algorithm is based on the fact that deciding whether a random element $α\in K$ is normal can be reduced to deciding whether $\sum_{σ\in G} σ(α)σ\in K[G]$ is invertible. In an algebraic model, the cost of our algorithm is quadratic in the size of $G$ for metacyclic $G$ and slightly subquadratic for abelian $G$.

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arXiv2018

Drinfeld Modules with Complex Multiplication, Hasse Invariants and Factoring Polynomials over Finite Fields

We present a novel randomized algorithm to factor polynomials over a finite field $\F_q$ of odd characteristic using rank $2$ Drinfeld modules with complex multiplication. The main idea is to compute a lift of the Hasse invariant (modulo the polynomial $f \in \F_q[x]$ to be factored) with respect to a random Drinfeld module $ϕ$ with complex multiplication. Factors of $f$ supported on prime ideals with supersingular reduction at $ϕ$ have vanishing Hasse invariant and can be separated from the rest. Incorporating a Drinfeld module analogue of Deligne's congruence, we devise an algorithm to compute the Hasse invariant lift, which turns out to be the crux of our algorithm. The resulting expected runtime of $n^{3/2+\varepsilon} (\log q)^{1+o(1)}+n^{1+\varepsilon} (\log q)^{2+o(1)}$ to factor polynomials of degree $n$ over $\F_q$ matches the fastest previously known algorithm, the Kedlaya-Umans implementation of the Kaltofen-Shoup algorithm.

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arXiv2018

Solving determinantal systems using homotopy techniques

Let $\K$ be a field of characteristic zero and $\Kbar$ be an algebraic closure of $\K$. Consider a sequence of polynomials$G=(g\_1,\dots,g\_s)$ in $\K[X\_1,\dots,X\_n]$, a polynomial matrix $\F=[f\_{i,j}] \in \K[X\_1,\dots,X\_n]^{p \times q}$, with $p \leq q$,and the algebraic set $V\_p(F, G)$ of points in $\KKbar$ at which all polynomials in $\G$ and all $p$-minors of $\F$vanish. Such polynomial systems appear naturally in e.g. polynomial optimization, computational geometry.We provide bounds on the number of isolated points in $V\_p(F, G)$ depending on the maxima of the degrees in rows (resp. columns) of $\F$. Next, we design homotopy algorithms for computing those points. These algorithms take advantage of the determinantal structure of the system defining $V\_p(F, G)$. In particular, the algorithms run in time that is polynomial in the bound on the number of isolated points.

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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.

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arXiv2017

On matrices with displacement structure: generalized operators and faster algorithms

For matrices with displacement structure, basic operations like multiplication, inversion, and linear system solving can all be expressed in terms of the following task: evaluate the product $\mathsf{A}\mathsf{B}$, where $\mathsf{A}$ is a structured $n \times n$ matrix of displacement rank $α$, and $\mathsf{B}$ is an arbitrary $n\timesα$ matrix. Given $\mathsf{B}$ and a so-called "generator" of $\mathsf{A}$, this product is classically computed with a cost ranging from $O(α^2 \mathscr{M}(n))$ to $O(α^2 \mathscr{M}(n)\log(n))$ arithmetic operations, depending on the type of structure of $\mathsf{A}$; here, $\mathscr{M}$ is a cost function for polynomial multiplication. In this paper, we first generalize classical displacement operators, based on block diagonal matrices with companion diagonal blocks, and then design fast algorithms to perform the task above for this extended class of structured matrices. The cost of these algorithms ranges from $O(α^{ω-1} \mathscr{M}(n))$ to $O(α^{ω-1} \mathscr{M}(n)\log(n))$, with $ω$ such that two $n \times n$ matrices over a field can be multiplied using $O(n^ω)$ field operations. By combining this result with classical randomized regularization techniques, we obtain faster Las Vegas algorithms for structured inversion and linear system solving.

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arXiv2016

A Fast Algorithm for Computing the Truncated Resultant

Let P and Q be two polynomials in K[x, y] with degree at most d, where K is a field. Denoting by R $\in$ K[x] the resultant of P and Q with respect to y, we present an algorithm to compute R mod x^k in O~(kd) arithmetic operations in K, where the O~ notation indicates that we omit polylogarithmic factors. This is an improvement over state-of-the-art algorithms that require to compute R in O~(d^3) operations before computing its first k coefficients.

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arXiv2016

Computing minimal interpolation bases

We consider the problem of computing univariate polynomial matrices over a field that represent minimal solution bases for a general interpolation problem, some forms of which are the vector M-Padé approximation problem in [Van Barel and Bultheel, Numerical Algorithms 3, 1992] and the rational interpolation problem in [Beckermann and Labahn, SIAM J. Matrix Anal. Appl. 22, 2000]. Particular instances of this problem include the bivariate interpolation steps of Guruswami-Sudan hard-decision and Kötter-Vardy soft-decision decodings of Reed-Solomon codes, the multivariate interpolation step of list-decoding of folded Reed-Solomon codes, and Hermite-Padé approximation. In the mentioned references, the problem is solved using iterative algorithms based on recurrence relations. Here, we discuss a fast, divide-and-conquer version of this recurrence, taking advantage of fast matrix computations over the scalars and over the polynomials. This new algorithm is deterministic, and for computing shifted minimal bases of relations between $m$ vectors of size $σ$ it uses $O~( m^{ω-1} (σ+ |s|) )$ field operations, where $ω$ is the exponent of matrix multiplication, and $|s|$ is the sum of the entries of the input shift $s$, with $\min(s) = 0$. This complexity bound improves in particular on earlier algorithms in the case of bivariate interpolation for soft decoding, while matching fastest existing algorithms for simultaneous Hermite-Padé approximation.

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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.

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arXiv2015

A Fast Algorithm for Computing the p-Curvature

We design an algorithm for computing the $p$-curvature of a differential system in positive characteristic $p$. For a system of dimension $r$ with coefficients of degree at most $d$, its complexity is $\softO (p d r^ω)$ operations in the ground field (where $ω$ denotes the exponent of matrix multiplication), whereas the size of the output is about $p d r^2$. Our algorithm is then quasi-optimal assuming that matrix multiplication is (\emph{i.e.} $ω= 2$). The main theoretical input we are using is the existence of a well-suited ring of series with divided powers for which an analogue of the Cauchy--Lipschitz Theorem holds.

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arXiv2014

A Quadratically Convergent Algorithm for Structured Low-Rank Approximation

Structured Low-Rank Approximation is a problem arising in a wide range of applications in Numerical Analysis and Engineering Sciences. Given an input matrix $M$, the goal is to compute a matrix $M'$ of given rank $r$ in a linear or affine subspace $E$ of matrices (usually encoding a specific structure) such that the Frobenius distance $\lVert M-M'\rVert$ is small. We propose a Newton-like iteration for solving this problem, whose main feature is that it converges locally quadratically to such a matrix under mild transversality assumptions between the manifold of matrices of rank $r$ and the linear/affine subspace $E$. We also show that the distance between the limit of the iteration and the optimal solution of the problem is quadratic in the distance between the input matrix and the manifold of rank $r$ matrices in $E$. To illustrate the applicability of this algorithm, we propose a Maple implementation and give experimental results for several applicative problems that can be modeled by Structured Low-Rank Approximation: univariate approximate GCDs (Sylvester matrices), low-rank Matrix completion (coordinate spaces) and denoising procedures (Hankel matrices). Experimental results give evidence that this all-purpose algorithm is competitive with state-of-the-art numerical methods dedicated to these problems.

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arXiv2014

A baby step-giant step roadmap algorithm for general algebraic sets

Let $\mathrm{R}$ be a real closed field and $\mathrm{D} \subset \mathrm{R}$ an ordered domain. We give an algorithm that takes as input a polynomial $Q \in \mathrm{D}[X_1,\ldots,X_k]$, and computes a description of a roadmap of the set of zeros, $\mathrm{Zer}(Q,\mathrm{R}^k)$, of $Q$ in $\mathrm{R}^k$. The complexity of the algorithm, measured by the number of arithmetic operations in the ordered domain $\mathrm{D}$, is bounded by $d^{O(k \sqrt{k})}$, where $d = \mathrm{deg}(Q)\ge 2$. As a consequence, there exist algorithms for computing the number of semi-algebraically connected components of a real algebraic set, $\mathrm{Zer}(Q,\mathrm{R}^k)$, whose complexity is also bounded by $d^{O(k \sqrt{k})}$, where $d = \mathrm{deg}(Q)\ge 2$. The best previously known algorithm for constructing a roadmap of a real algebraic subset of $\mathrm{R}^k$ defined by a polynomial of degree $d$ has complexity $d^{O(k^2)}$.

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arXiv2014

A fast algorithm for computing the characteristic polynomial of the p-curvature

We discuss theoretical and algorithmic questions related to the $p$-curvature of differential operators in characteristic $p$. Given such an operator $L$, and denoting by $\Chi(L)$ the characteristic polynomial of its $p$-curvature, we first prove a new, alternative, description of $\Chi(L)$. This description turns out to be particularly well suited to the fast computation of $\Chi(L)$ when $p$ is large: based on it, we design a new algorithm for computing $\Chi(L)$, whose cost with respect to $p$ is $\softO(p^{0.5})$ operations in the ground field. This is remarkable since, prior to this work, the fastest algorithms for this task, and even for the subtask of deciding nilpotency of the $p$-curvature, had merely slightly subquadratic complexity $\softO(p^{1.79})$.

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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.

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arXiv2012

Fast Computation of Common Left Multiples of Linear Ordinary Differential Operators

We study tight bounds and fast algorithms for LCLMs of several linear differential operators with polynomial coefficients. We analyze the arithmetic complexity of existing algorithms for LCLMs, as well as the size of their outputs. We propose a new algorithm that recasts the LCLM computation in a linear algebra problem on a polynomial matrix. This algorithm yields sharp bounds on the coefficient degrees of the LCLM, improving by one order of magnitude the best bounds obtained using previous algorithms. The complexity of the new algorithm is almost optimal, in the sense that it nearly matches the arithmetic size of the output.

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arXiv2011

On the complexity of computing with zero-dimensional triangular sets

We study the complexity of some fundamental operations for triangular sets in dimension zero. Using Las-Vegas algorithms, we prove that one can perform such operations as change of order, equiprojectable decomposition, or quasi-inverse computation with a cost that is essentially that of modular composition. Over an abstract field, this leads to a subquadratic cost (with respect to the degree of the underlying algebraic set). Over a finite field, in a boolean RAM model, we obtain a quasi-linear running time using Kedlaya and Umans' algorithm for modular composition. Conversely, we also show how to reduce the problem of modular composition to change of order for triangular sets, so that all these problems are essentially equivalent. Our algorithms are implemented in Maple; we present some experimental results.

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