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Hugues Randriambololona

Publications and source records attributed to Hugues Randriambololona.

12 recordsLinked to original sources
arXiv2025

The syzygy distinguisher

We present a new distinguisher for alternant and Goppa codes, whose complexity is subexponential in the error-correcting capability, hence better than that of generic decoding algorithms. Moreover it does not suffer from the strong regime limitations of the previous distinguishers or structure recovery algorithms: in particular, it applies to the codes used in the Classic McEliece candidate for postquantum cryptography standardization. The invariants that allow us to distinguish are graded Betti numbers of the homogeneous coordinate ring of a shortening of the dual code. Since its introduction in 1978, this is the first time an analysis (in the CPA model) of the McEliece cryptosystem breaks the exponential barrier.

cs.CR↗
arXiv2020

The quadratic hull of a code and the geometric view on multiplication algorithms

We introduce the notion of quadratic hull of a linear code, and give some of its properties. We then show that any symmetric bilinear multiplication algorithm for a finite-dimensional algebra over a field can be obtained by evaluation-interpolation at simple points (i.e. of degree and multiplicity 1) on a naturally associated space, namely the quadratic hull of the corresponding code. This also provides a geometric answer to some questions such as: which linear maps actually are multiplication algorithms, or which codes come from supercodes (as asked by Shparlinski-Tsfasman-Vladut). We illustrate this with examples, in particular we describe the quadratic hull of all the optimal algorithms computed by Barbulescu-Detrey-Estibals-Zimmermann for small algebras. In our presentation we actually work with multiplication reductions. This is a generalization of multiplication algorithms, that allows for instance evaluation-interpolation at points of higher degree and/or with multiplicities, and also includes the recently introduced notion of "reverse multiplication-friendly embedding" from Cascudo-Cramer-Xing-Yang. All our results hold in this more general context.

cs.IT↗
arXiv2020

On the tensor rank of multiplication in finite extensions of finite fields and related issues in algebraic geometry

In this paper, we give a survey of the known results concerning the tensor rank of the multiplication in finite extensions of finite fields, enriched with some not published recent results as well as analyzes enhancing the qualitative understanding of the domain. In particular, we identify and clarify certain results not completely proved and we emphasis the link with open problems in number theory, algebraic geometry, and coding theory.

math.AG↗
arXiv2020

Algebraic geometry codes and some applications

This article surveys the development of the theory of algebraic geometry codes since their discovery in the late 70's. We summarize the major results on various problems such as: asymptotic parameters, improved estimates on the minimum distance, and decoding algorithms. In addition, we present various modern applications of these codes such as public-key cryptography, algebraic complexity theory, multiparty computation or distributed storage.

cs.IT↗
arXiv2018

Harder-Narasimhan theory for linear codes

In this text we develop some aspects of Harder-Narasimhan theory, slopes, semistability and canonical filtration, in the setting of combinatorial lattices. Of noticeable importance is the Harder-Narasimhan structure associated to a Galois connection between two lattices. It applies, in particular, to matroids. We then specialize this to linear codes. This could be done from at least three different approaches: using the sphere-packing analogy, or the geometric view, or the Galois connection construction just introduced. A remarkable fact is that these all lead to the same notion of semistability and canonical filtration. Relations to previous propositions towards a classification of codes, and to Wei's generalized Hamming weight hierarchy, are also discussed. Last, we study the important question of the preservation of semistability (or more generally the behaviour of slopes) under duality, and under tensor product. The former essentially follows from Wei's duality theorem for higher weights---and its matroid version---which we revisit in an appendix, developing analogues of the Riemann-Roch, Serre duality, Clifford, and gap and gonality sequence theorems for codes. Likewise the latter is closely related to the bound on higher weights of a tensor product, conjectured by Wei and Yang, and proved by Schaathun in the geometric language, which we reformulate directly in terms of codes. From this material we then derive semistability of tensor product.

math.CO↗
arXiv2016

On metric convexity, the discrete Hahn-Banach theorem, separating systems and sets of points forming only acute angles

This text has three parts. The first one is largely autobiographical, hence my use of the first person. There I recall how Gerard Cohen influenced important parts of my research. The second is of a more classic mathematical nature. I present a discrete analogue of the Hahn-Banach theorem, which serves as a basis for generalizing the notion of separating systems in the context of metric convexity. The third one aims at building a bridge between two communities of researchers, those interested in separating systems, and those interested in a certain question in combinatorial geometry --- sets of points forming only acute angles --- who seem not to be aware of each other, while they are working on precisely the same problem! Of course, these three themes are closely intertwined.

math.MG↗
arXiv2015

Linear independence of rank 1 matrices and the dimension of *-products of codes

We show that with high probability, random rank 1 matrices over a finite field are in (linearly) general position, at least provided their shape k x l is not excessively unbalanced. This translates into saying that the dimension of the *-product of two [n, k] and [n, l] random codes is equal to min(n, kl), as one would have expected. Our work is inspired by a similar result of Cascudo-Cramer-Mirandola-Zemor dealing with *-squares of codes, which it complements, especially regarding applications to the analysis of McEliece-type cryptosystems. We also briefly mention the case of higher *-powers, which require to take the Frobenius into account. We then conclude with some open problems.

cs.IT↗
arXiv2014

On products and powers of linear codes under componentwise multiplication

In this text we develop the formalism of products and powers of linear codes under componentwise multiplication. As an expanded version of the author's talk at AGCT-14, focus is put mostly on basic properties and descriptive statements that could otherwise probably not fit in a regular research paper. On the other hand, more advanced results and applications are only quickly mentioned with references to the literature. We also point out a few open problems. Our presentation alternates between two points of view, which the theory intertwines in an essential way: that of combinatorial coding, and that of algebraic geometry. In appendices that can be read independently, we investigate topics in multilinear algebra over finite fields, notably we establish a criterion for a symmetric multilinear map to admit a symmetric algorithm, or equivalently, for a symmetric tensor to decompose as a sum of elementary symmetric tensors.

cs.IT↗
arXiv2012

Asymptotically good binary linear codes with asymptotically good self-intersection spans

If C is a binary linear code, let C^2 be the linear code spanned by intersections of pairs of codewords of C. We construct an asymptotically good family of binary linear codes such that, for C ranging in this family, the C^2 also form an asymptotically good family. For this we use algebraic-geometry codes, concatenation, and a fair amount of bilinear algebra. More precisely, the two main ingredients used in our construction are, first, a description of the symmetric square of an odd degree extension field in terms only of field operations of small degree, and second, a recent result of Garcia-Stichtenoth-Bassa-Beelen on the number of points of curves on such an odd degree extension field.

cs.IT↗
arXiv2012

Bilinear complexity of algebras and the Chudnovsky-Chudnovsky interpolation method

We give new improvements to the Chudnovsky-Chudnovsky method that provides upper bounds on the bilinear complexity of multiplication in extensions of finite fields through interpolation on algebraic curves. Our approach features three independent key ingredients: (1) We allow asymmetry in the interpolation procedure. This allows to prove, via the usual cardinality argument, the existence of auxiliary divisors needed for the bounds, up to optimal degree. (2) We give an alternative proof for the existence of these auxiliary divisors, which is constructive, and works also in the symmetric case, although it requires the curves to have sufficiently many points. (3) We allow the method to deal not only with extensions of finite fields, but more generally with monogenous algebras over finite fields. This leads to sharper bounds, and is designed also to combine well with base field descent arguments in case the curves do not have sufficiently many points. As a main application of these techniques, we fix errors in, improve, and generalize, previous works of Shparlinski-Tsfasman-Vladut, Ballet, and Cenk-Ozbudak. Besides, generalities on interpolation systems, as well as on symmetric and asymmetric bilinear complexity, are also discussed.

cs.CC↗
arXiv2012

(2,1)-separating systems beyond the probabilistic bound

Building on previous results of Xing, we give new lower bounds on the rate of intersecting codes over large alphabets. The proof is constructive, and uses algebraic geometry, although nothing beyond the basic theory of linear systems on curves. Then, using these new bounds within a concatenation argument, we construct binary (2,1)-separating systems of asymptotic rate exceeding the one given by the probabilistic method, which was the best lower bound available up to now. This answers (negatively) the question of whether this probabilistic bound was exact, which has remained open for more than 30 years. (By the way, we also give a formulation of the separation property in terms of metric convexity, which may be an inspirational source for new research problems.)

math.CO↗