This package contains two C programs. The first program computes a lower bound on the genus of a curve defined over a fixed finite field, given its number of points. The second one computes an upper bound on the number of rational points of a curve of given genus, over a fixed finite field.
Ce package contient deux programmes C. L’un calcule une borne inférieure sur le genre d’une courbe définie sur un corps fini fixé possédant un nombre de points rationnels donné. L’autre calcule une borne supérieure sur le nombre de points rationnels d’une courbe de genre donné sur un corps fini fixé.
We propose an algorithm to find the roots of a univariate polynomial with coefficients in the function field of an algebraic curve (or more generally, in a discretely valued field). This method can be applied to the list decoding of algebraic-geometric codes when the Newton-Hensel method can’t be used.
Let k be a perfect field of any characteristic, X a geometrically irreducible curve defined over k and K its function field. Given a polynomial G over K and a divisor D, we propose an algorithm to find all roots of G in the Riemann-Roch space L(D). An application of this method is the root finding step in list decoding of algebraic-geometric codes, when the very efficient Newton-Hensel method can’t be used. Our algorithm has two steps:
1.) using a Newton-Puiseux method, compute sufficiently many terms of the roots of the polynomial’s image into some completion;
2.) pull-back its roots as functions of L(D) using linear algebra over k.
We generalize this method to discretely valued fields.
Logics course, BSC in Computer Science and Mathematics, University Paris XII (2001-2003)
Functional Programming Course, BSc in Computer Science, University Paris XII (2001-2003)
Objective Caml (aka Ocaml) is a wonderful programming language: flexible, safe, efficient, equipped with many libraries (graphics, network,…). It is often seen as a curiosity used only by theoretical computer scientists. In this article, we briefly browse the genealogical tree of this language to understand its main principles and to familiarize the reader with functional programming; then we present the different working modes and some of the original aspects of this language. We finish with a small example that features the graphics and Unix libraries and with the evocation of other developping tools that are available for Ocaml.
History’s Worst Software Bugs, by Simson Garfinkel, 2005.
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This talk is made for students who specialize in mathematics during their last year in high school. We present fundamental principles of cryptology an describe precisely the RSA public key cryptosystem, for ciphering and signature since it uses almost only elementary arithmetics (fast modular exponentiation, Euclid’s extended algorithm, Fermat-Euler’s Theorem…). We give elements of cryptanalysis of RSA and explain how to build fake smartcards. We conclude by mentioning some other cryptosystems.
This article is a non-technical introduction about the problem of detection and correction of errors in digital transmissions.
Sorry, this entry is only available in Français.
Sorry, this entry is only available in Français.