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Jean Kieffer

Publications and source records attributed to Jean Kieffer.

15 recordsLinked to original sources
arXiv2025

Fast evaluation of Riemann theta functions in any dimension

We describe an algorithm to numerically evaluate Riemann theta functions in any dimension in quasi-linear time in terms of the required precision, uniformly on reduced input. This algorithm is implemented in the FLINT number theory library and vastly outperforms existing software. As an application, we evaluate the theta constants attached to certain special abelian varieties of dimension 6 to construct explicit polynomials of degree 65 over $\mathbb{Q}$ with conjectural Galois group $\mathrm{SL}_2(\mathbb{F}_{64})$.

math.NT↗
arXiv2024

The asymptotic distribution of Elkies primes for reductions of abelian varieties is Gaussian

We generalize the notion of Elkies primes for elliptic curves to the setting of abelian varieties with real multiplication (RM), and prove the following. Let $A$ be an abelian variety with RM over a number field whose attached Galois representation has large image. Then the number of Elkies primes (in a suitable range) for reductions of $A$ modulo primes converges weakly to a Gaussian distribution around its expected value. This refines and generalizes results obtained by Shparlinski and Sutherland in the case of non-CM elliptic curves, and has implications for the complexity of the SEA point counting algorithm for abelian surfaces over finite fields.

math.NT↗
arXiv2024

Spanning isogeny classes of principally polarized abelian surfaces with RM

We describe how isogenies between principally polarized abelian surfaces (PPAS) with real multiplication (RM) can be decomposed into elementary isogeny types. This leads to a strategy for enumerating the isogeny class of a given PPAS with RM. This note is part of an ongoing work with R. van Bommel, S. Chidambaram, and E. Costa and is not indended for separate publication.

math.NT↗
arXiv2023

Computing isogeny classes of typical principally polarized abelian surfaces over the rationals

We describe an efficient algorithm which, given a principally polarized (p.p.) abelian surface $A$ over $\mathbb{Q}$ with geometric endomorphism ring equal to $\mathbb{Z}$, computes all the other p.p. abelian surfaces over $\mathbb{Q}$ that are isogenous to $A$. This algorithm relies on explicit open image techniques for Galois representations, and we employ a combination of analytic and algebraic methods to efficiently prove or disprove the existence of isogenies. We illustrate the practicality of our algorithm by applying it to 1 440 894 isogeny classes of Jacobians of genus 2 curves.

math.NT↗
arXiv2022

Upper bounds on the heights of polynomials and rational fractions from their values

Let $F$ be a univariate polynomial or rational fraction of degree $d$ defined over a number field. We give bounds from above on the absolute logarithmic Weil height of $F$ in terms of the heights of its values at small integers: we review well-known bounds obtained from interpolation algorithms given values at $d+1$ (resp. $2d+1$) points, and obtain tighter results when considering a larger number of evaluation points.

math.NT↗
arXiv2022

Sign choices in the AGM for genus two theta constants

Existing algorithms to compute genus 2 theta constants in quasi-linear time use Borchardt sequences, an analogue of the arithmetic-geometric mean for four complex numbers. In this paper, we show that these Borchardt sequences are given by good choices of square roots only, as in the genus 1 case. This removes the sign indeterminacies in the algorithm without relying on numerical integration.

math.NT↗
arXiv2022

Certified Newton schemes for the evaluation of low-genus theta functions

Theta functions and theta constants in low genus, especially genus 1 and 2, can be evaluated at any given point in quasi-linear time in the required precision using Newton schemes based on Borchardt sequences. Our goal in this paper is to provide the necessary tools to implement these algorithms in a provably correct way. In particular, we obtain uniform and explicit convergence results in the case of theta constants in genus 1 and 2, and theta functions in genus 1: the associated Newton schemes will converge starting from approximations to N bits of precision for N=60, 300, and 1600 respectively, for all suitably reduced arguments. We also describe a uniform quasi-linear time algorithm to evaluate genus 2 theta constants on the Siegel fundamental domain. Our main tool is a detailed study of Borchardt means as multivariate analytic functions.

math.NA↗
arXiv2022

Counting points on abelian surfaces over finite fields with Elkies's method

We generalize Elkies's method, an essential ingredient in the SEA algorithm to count points on elliptic curves over finite fields of large characteristic, to the setting of p.p. abelian surfaces. Under reasonable assumptions related to the distribution of Elkies primes, we obtain improvements over Schoof's method in two cases. If the abelian surface A over Fq has RM by a fixed quadratic field F, we reach the same asymptotic complexity Otilde(log4 q) as the SEA algorithm up to constant factors depending on F. If A is defined over a number field, we count points on A modulo sufficiently many primes in Otilde(log6 q) binary operations on average. Numerical experiments demonstrate the practical usability of our methods.

math.NT↗
arXiv2021

Degree and height estimates for modular equations on PEL Shimura varieties

We define modular equations in the setting of PEL Shimura varieties as equations describing Hecke correspondences, and prove upper bounds on their degrees and heights. This extends known results about elliptic modular polynomials, and implies complexity bounds for number-theoretic algorithms using these modular equations. In particular, we obtain tight degree bounds for modular equations of Siegel and Hilbert type for abelian surfaces.

math.AG↗
arXiv2020

Evaluating modular equations for abelian surfaces

We design efficient algorithms to evaluate modular equations of Siegel and Hilbert type for abelian surfaces over number fields or finite fields using complex approximations. Their output is provably correct when the associated graded ring of modular forms over Z is explicitly known; this includes the Siegel case, and the Hilbert case for the quadratic fields of discriminant 5 and 8. As part of the proofs, we establish new correctness and complexity results for certain key numerical algorithms on period matrices in genus 2, namely the reduction algorithm to the fundamental domain, the AGM method, and computing big period matrices and RM structures.

math.NT↗
arXiv2020

Computing isogenies from modular equations in genus two

We present an algorithm solving the following problem: given two genus 2 curves over a field k with isogenous Jacobians, compute such an isogeny explicitly. This isogeny can be either an l-isogeny or, in the real multiplication case, an isogeny with cyclic kernel; we require that k have large enough characteristic and that the curves be sufficiently generic. Our algorithm uses modular equations for these isogeny types, and makes essential use of an explicit Kodaira--Spencer isomorphism in genus 2.

math.AG↗
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↗
arXiv2018

Accelerating the Couveignes Rostovtsev Stolbunov key exchange protocol

We study a key exchange protocol based on isogenies between ordinary elliptic curves over a finite field, first mentioned by Couveignes and investigated by Rostovtsev and Stolbunov. After presenting the fundamental notions about elliptic curves, we present the theory of complex multiplication which is the theoretical basis of this cryptosystem. Modular curves, which are an essential tool in the computations, are also introduced. We then present the protocol itself and original ideas to boost its practical performances. Finally, we discuss our implementation and practical results.

math.NT↗
arXiv2004

Unstable Flow and Non-Monotonic Constitutive Equation of Transient Networks

We have measured the nonlinear rheological response of a model transient network over a large range of steady shear rates. The system is built up from an oil in water droplet microemulsion into which a telechelic polymer is incorporated. The phase behaviour is characterized which comprises a liquid-gas phase separation and a percolation threshold. The rheological measurements are performed in the one phase region above the percolation line. Shear thinning is observed for all samples, leading in most cases to an unstable stress response at intermediate shear rates. We built up a very simple mean field model which involves the reduction of the residence time of the stickers in the droplets due to the chain tensions at high shear. The computed constitutive equation is non-monotonic with a range where the stress is a decreasing function of the rate, a feature that indeed makes homogeneous flows unstable. The computed the flow curves compare well to the experiments.

cond-mat.soft↗