A p-adic quasi-quadratic point counting algorithm

In this article we give an algorithm for the computation of the number of rational points on the Jacobian variety of a generic ordinary hyperelliptic curve defined over a finite field of cardinality $q$ with time complexity $O(n^{2+o(1)})$ and space complexity $O(n^2)$, where $n=\log(q)$. In the latter complexity estimate the genus and the characteristic are assumed as fixed. Our algorithm forms a generalization of both, the AGM algorithm of J.-F. Mestre and the canonical lifting method of T. Satoh. We canonically lift a certain arithmetic invariant of the Jacobian of the hyperelliptic curve in terms of theta constants. The theta null values are computed with respect to a semi-canonical theta structure of level $2^\nu p$ where $\nu >0$ is an integer and p=\mathrm{char}(\F_q)>2. The results of this paper suggest a global positive answer to the question whether there exists a quasi-quadratic time algorithm for the computation of the number of rational points on a generic ordinary abelian variety defined over a finite field.Comment: 32 page

Sur les pinceaux de courbes définis par une fonction régulière

RésuméDans cet article nous généralisons beaucoup de résultats sur la topologie d'une fonction polynomiale, f:C2→C dans le cas d'une fonction régulière f:U→C.AbstractIn this article, we generalize many of the results on the topology of a polynomial function, f:C2→C in the case of a regular function f:U→C

Computing isogenies between Abelian Varieties

47 pagesInternational audienceWe describe an efficient algorithm for the computation of isogenies between abelian varieties represented in the coordinate system provided by algebraic theta functions. We explain how to compute all the isogenies from an abelian variety whose kernel is isomorphic to a given abstract group. We also describe an analog of Vélu's formulas to compute an isogenis with prescribed kernels. All our algorithms rely in an essential manner on a generalization of the Riemann formulas. In order to improve the efficiency of our algorithms, we introduce a point compression algorithm that represents a point of level $4\ell$ of a $g$ dimensional abelian variety using only $g(g+1)/2\cdot 4^g$ coordinates. We also give formulas to compute the Weil and commutator pairing given input points in theta coordinates. All the algorithms presented in this paper work in general for any abelian variety defined over a field of odd characteristic

Linear Algebra over Z_p[[u]] and related rings

38 pagesInternational audienceLet R be a complete discrete valuation ring, S=R[[u]] and n a positive integer. The aim of this paper is to explain how to compute efficiently usual operations such as sum and intersection of sub-S-modules of S^d. As S is not principal, it is not possible to have a uniform bound on the number of generators of the modules resulting of these operations. We explain how to mitigate this problem, following an idea of Iwasawa, by computing an approximation of the result of these operations up to a quasi-isomorphism. In the course of the analysis of the p-adic and u-adic precisions of the computations, we have to introduce more general coefficient rings that may be interesting for their own sake. Being able to perform linear algebra operations modulo quasi-isomorphism with S-modules has applications in Iwasawa theory and p-adic Hodge theory

Meson Form-factors and Wave-functions with Wilson fermions

Results for semi-leptonic form-factors for processes like $D \to K l \nu$ and the Bethe-Salpeter amplitudes (BSA) for pion and rho mesons are presented. The form-factor data is consistent with previous calculations. We find that the long distance fall-off of BSA for both $\pi$ and $\rho$ is very well fit by an exponential, but surprisingly the effective mass governing this fall-off is lighter than the pion's. Lastly, by studying the dependence of $\rho$ polarization on separation direction we show that there is a measureable $l=2$ state in addition to $l=0$ in the BSA for the rho. (Talk presented by R. Gupta at LATTICE92. Latex needs macro package espcrc2.sty)Comment: 4 pages including 4 PS figure

A Tunable Broadcast Encryption Scheme

In this paper, we describe yet another broadcast encryption scheme for stateless receivers. The main difference between our scheme and the classical schemes derived from the complete subtree and its subsequent improvements is that in our scheme the group management is based upon a more adaptable data structure. In these classical schemes, users must be spread on a tree structure where each level of the tree is associated to some distinguishing property of the users. The fact that the underlying data structure is a fixed tree is a strong limitation for some applications where an operator wants to select users very dynamically following criterions with changing levels of priority. Our scheme may be thought as if in the complete subtree it would be possible to exchange the different level of the tree in order to make it very efficient to revoke or select a class of users. It is also very efficient in the cases where there exists very unbalanced groups of users. This scheme allows one to select or revoke users by sending ciphertexts of linear size with respect to the number of groups which is in general far less than the number of users. Moreover, by using a specific group repartition, it is possible to recover a tree structure in order to apply the classical methods which guarantee that our scheme is in general as efficient as a usual ones. We prove that our scheme is fully collusion secure in the generic group with pairing model

Semi-simplified modulo pp of semi-stable representations: an algorithmic approach

The aim of this paper is to present an algorithm the complexity of which is polynomial to compute the semi-simplified modulo $p$ of a semi-stable \Q_p-representation of the absolute Galois group of a $p$-adic field (\emph{i.e.} a finite extension of \Q_p). In order to do so, we use abundantly the $p$-adic Hodge theory and, in particular, the Breuil-Kisin modules theory.Comment: 35 pages, in Frenc

Recommendations for the Design and Validation of a Physical True Random Number Generator Integrated in an Electronic Device

These Recommendations describe essential elements of the design of a secure physical true random number generator (PTRNG) integrated in an electronic device. Based on these elements, we describe and justify requirements for the design, validation and testing of PTRNGs, which are intended to guarantee the security of generators aimed at cryptographic applications