arXiv subjects

Hugues Randriam

Publications and source records attributed to Hugues Randriam.

11 recordsLinked to original sources

arXiv2020

Interactive Oracle Proofs of Proximity to Algebraic Geometry Codes

In this work, we initiate the study of proximity testing to Algebraic Geometry (AG) codes. An AG code $C = C(\mathcal{X}, \mathcal{P}, D)$ over an algebraic curve $\mathcal{X}$ is a vector space associated to evaluations on $\mathcal{P}$ of functions in the Riemann-Roch space $L_\mathcal{X}(D)$. The problem of testing proximity to an error-correcting code $C$ consists in distinguishing between the case where an input word, given as an oracle, belongs to $C$ and the one where it is far from every codeword of $C$. AG codes are good candidates to construct short proof systems, but there exists no efficient proximity tests for them. We aim to fill this gap. We construct an Interactive Oracle Proof of Proximity (IOPP) for some families of AG codes by generalizing an IOPP for Reed-Solomon codes introduced by Ben-Sasson, Bentov, Horesh and Riabzev, known as the FRI protocol. We identify suitable requirements for designing efficient IOPP systems for AG codes. Our approach relies on a neat decomposition of the Riemann-Roch space of any invariant divisor under a group action on a curve into several explicit Riemann-Roch spaces on the quotient curve. We provide sufficient conditions on an AG code $C$ that allow to reduce a proximity testing problem for $C$ to a membership problem for a significantly smaller code $C'$. As concrete instantiations, we study AG codes on Kummer curves and curves in the Hermitian tower. The latter can be defined over polylogarithmic-size alphabet. We specialize the generic AG-IOPP construction to reach linear prover running time and logarithmic verification on Kummer curves, and quasilinear prover time with polylogarithmic verification on the Hermitian tower.

cs.IT↗

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↗

arXiv2019

Anticanonical codes from del Pezzo surfaces with Picard rank one

We construct algebraic geometric codes from del Pezzo surfaces and focus on the ones having Picard rank one and the codes associated to the anticanonical class. We give explicit constructions of del Pezzo surfaces of degree 4, 5 and 6, compute the parameters of the associated anticanonical codes and study their isomorphisms arising from the automorphisms of the surface. We obtain codes with excellent parameters and some of them turn out to beat the best known codes listed on the database codetable.

math.AG↗

arXiv2018

Gaps between prime numbers and tensor rank of multiplication in finite fields

We present effective upper bounds on the symmetric bilinear complexity of multiplication in extensions of a base finite field Fp2 of prime square order, obtained by combining estimates on gaps between prime numbers together with an optimal construction of auxiliary divisors for multiplication algorithms by evaluation-interpolation on curves. Most of this material dates back to a 2011 unpublished work of the author, but it still provides the best results on this topic at the present time. Then a few updates are given in order to take recent developments into account, including comparison with a similar work of Ballet and Zykin, generalization to classical bilinear complexity over Fp, and to short multiplication of polynomials, as well as a discussion of open questions on gaps between prime numbers or more generally values of certain arithmetic functions.

math.NT↗

arXiv2013

Simplicial Homology of Random Configurations

Given a Poisson process on a $d$-dimensional torus, its random geometric simplicial complex is the complex whose vertices are the points of the Poisson process and simplices are given by the uCech complex associated to the coverage of each point. By means of Malliavin calculus, we compute explicitly the n$th$ order moment of the number of $k$-simplices. The two first order moments of this quantity allow us to find the mean and the variance of the Euler caracteristic. Also, we show that the number of any connected geometric simplicial complex converges to the Gaussian law when the intensity of the Poisson point process tends to infinity. We use a concentration inequality to find bounds for the for the distribution of the Betti number of first order and the Euler characteristic in such simplicial complex.

math.PR↗

arXiv2013

New uniform and asymptotic upper bounds on the tensor rank of multiplication in extensions of finite fields

We obtain new uniform upper bounds for the (non necessarily symmetric) tensor rank of the multiplication in the extensions of the finite fields $\F_q$ for any prime or prime power $q\geq2$; moreover these uniform bounds lead to new asymptotic bounds as well. In addition, we also give purely asymptotic bounds which are substantially better by using a family of Shimura curves defined over $\F_q$, with an optimal ratio of $\F_{q^t}$-rational places to their genus where $q^t$ is a square.

math.AG↗

arXiv2013

Lower bounds on the minimum distance of long codes in the Lee metric

The Gilbert type bound for codes in the title is reviewed, both for small and large alphabets. Constructive lower bounds better than these existential bounds are derived from geometric codes, either over Fp or Fp2 ; or over even degree extensions of Fp: In the latter case the approach is concatena- tion with a good code for the Hamming metric as outer code and a short code for the Lee metric as an inner code. In the former case lower bounds on the minimum Lee distance are derived by algebraic geometric arguments inspired by results of Wu, Kuijper, Udaya (2007).

cs.IT↗

arXiv2011

Diviseurs de la forme 2D-G sans sections et rang de la multiplication dans les corps finis (Divisors of the form 2D-G without sections and bilinear complexity of multiplication in finite fields)

Let X be an algebraic curve, defined over a perfect field, and G a divisor on X. If X has sufficiently many points, we show how to construct a divisor D on X such that l(2D-G)=0, of essentially any degree such that this is compatible the Riemann-Roch theorem. We also generalize this construction to the case of a finite number of constraints, l(k_i.D-G_i)=0, where |k_i|\leq 2. Such a result was previously claimed by Shparlinski-Tsfasman-Vladut, in relation with the Chudnovsky-Chudnovsky method for estimating the bilinear complexity of the multiplication in finite fields based on interpolation on curves; unfortunately, as noted by Cascudo et al., their proof was flawed. So our work fixes the proof of Shparlinski-Tsfasman-Vladut and shows that their estimate m_q\leq 2(1+1/(A(q)-1)) holds, at least when A(q)\geq 5. We also fix a statement of Ballet that suffers from the same problem, and then we point out a few other possible applications.

math.AG↗

arXiv2009

Witness sets

Given a set C of binary n-tuples and c in C, how many bits of c suffice to distinguish it from the other elements in C? We shed new light on this old combinatorial problem and improve on previously known bounds.

math.CO↗

arXiv2004

A presentation of the quantum Fourier transform from a recursive viewpoint

In this note we construct a quantum Fourier transform circuit in a recursive way, by directly copying the 'divide and conquer' construction of the fast Fourier transform algorithm, rather than using the explicit formula that is given in most introductory texts to quantum computation. We do not pretend this presentation to be original, nor claim for any anteriority. The aim of this paper is purely pedagogical.

quant-ph↗

arXiv2004

Métriques de sous-quotient et théorème de Hilbert-Samuel arithmétique pour les faisceaux cohérents

The aim of this paper is twofold. First we prove a theorem of extension of sections of a coherent subquotient of a hermitian vector bundle on a complex analytic space with control of the norms, without any of the smoothness assumptions that were needed in previously known analogous results. Then we show how to associate an arithmetic Hilbert-Samuel function to a coherent sheaf on an arithmetic variety -- provided this coherent sheaf is a subquotient of a hermitian vector bundle -- and using the classical arithmetic Hilbert-Samuel theorem and our extension theorem, we give the leading term of the so constructed arithmetic Hilbert-Samuel function.

math.NT↗