Fast integer multiplication using generalized Fermat primes

For almost 35 years, Sch{\"o}nhage-Strassen's algorithm has been the fastest algorithm known for multiplying integers, with a time complexity O(n $\times$ log n $\times$ log log n) for multiplying n-bit inputs. In 2007, F{\"u}rer proved that there exists K > 1 and an algorithm performing this operation in O(n $\times$ log n $\times$ K log n). Recent work by Harvey, van der Hoeven, and Lecerf showed that this complexity estimate can be improved in order to get K = 8, and conjecturally K = 4. Using an alternative algorithm, which relies on arithmetic modulo generalized Fermat primes, we obtain conjecturally the same result K = 4 via a careful complexity analysis in the deterministic multitape Turing model

Isogeny graphs with maximal real multiplication

An isogeny graph is a graph whose vertices are principally polarized abelian varieties and whose edges are isogenies between these varieties. In his thesis, Kohel described the structure of isogeny graphs for elliptic curves and showed that one may compute the endomorphism ring of an elliptic curve defined over a finite field by using a depth first search algorithm in the graph. In dimension 2, the structure of isogeny graphs is less understood and existing algorithms for computing endomorphism rings are very expensive. Our setting considers genus 2 jacobians with complex multiplication, with the assumptions that the real multiplication subring is maximal and has class number one. We fully describe the isogeny graphs in that case. Over finite fields, we derive a depth first search algorithm for computing endomorphism rings locally at prime numbers, if the real multiplication is maximal. To the best of our knowledge, this is the first DFS-based algorithm in genus 2

Interactive certificate for the verification of Wiedemann's Krylov sequence: application to the certification of the determinant, the minimal and the characteristic polynomials of sparse matrices

Certificates to a linear algebra computation are additional data structures for each output, which can be used by a-possibly randomized- verification algorithm that proves the correctness of each output. Wiede-mann's algorithm projects the Krylov sequence obtained by repeatedly multiplying a vector by a matrix to obtain a linearly recurrent sequence. The minimal polynomial of this sequence divides the minimal polynomial of the matrix. For instance, if the $n\times n$ input matrix is sparse with n 1+o(1) non-zero entries, the computation of the sequence is quadratic in the dimension of the matrix while the computation of the minimal polynomial is n 1+o(1), once that projected Krylov sequence is obtained. In this paper we give algorithms that compute certificates for the Krylov sequence of sparse or structured $n\times n$ matrices over an abstract field, whose Monte Carlo verification complexity can be made essentially linear. As an application this gives certificates for the determinant, the minimal and characteristic polynomials of sparse or structured matrices at the same cost

Root optimization of polynomials in the number field sieve

The general number field sieve (GNFS) is the most efficient algorithm known for factoring large integers. It consists of several stages, the first one being polynomial selection. The quality of the chosen polynomials in polynomial selection can be modelled in terms of size and root properties. In this paper, we describe some algorithms for selecting polynomials with very good root properties.Comment: 16 pages, 18 reference

Turismo agroalimentario y nuevos metabolismos sociales de productos locales

Derivado de la crisis del sector agrícola, el turismo agroalimentario constituye una forma de apropiación social de los alimentos emblemáticos y un proceso reestructuración productiva del campo. A partir de una revisión de literatura, contrastada con la evidencia empírica, se realiza una comparación de tres escenarios cuyo común denominador es la vinculación entre alimentos y turismo. Se presentan los casos de la Ruta de la Sal Prehispánica (Puebla, México), la Ruta del Nopal (Ciudad de México) y una propuesta de Agroturismo Ancestral (Isla de Pascua, Chile). Se concluye que el turismo agroalimentario es una forma innovadora de metabolismo social del alimento, que implica aspectos materiales y simbólicos, con los que agrega valor a las actividades productivas tradicionales, a través de la incorporación de actividades económicas no agropecuarias. Destaca el carácter ambivalente de la actividad turística, cuyos resultados se orientan por procesos sociales y económicos de los ámbitos en que este fenómeno se reproduce

Computation of Discrete Logarithms in GF(2^607)

International audienceWe describe in this article how we have been able to extend the record for computations of discrete logarithms in characteristic 2 from the previous record over GF(2^503) to a newer mark of GF(2^607), using Coppersmith's algorithm. This has been made possible by several practical improvements to the algorithm. Although the computations have been carried out on fairly standard hardware, our opinion is that we are nearing the current limits of the manageable sizes for this algorithm, and that going substantially further will require deeper improvements to the method

Subquadratic computation of vector generating polynomials and improvement of the block Wiedemann algorithm

This paper describes a new algorithm for computing linear generators (vector generating polynomials) for matrix sequences, running in sub-quadratic time. This algorithm applies in particular to the sequential stage of Coppersmith's block Wiedemann algorithm. Experiments showed that our method can be substituted in place of the quadratic one proposed by Coppersmith, yielding important speedups even for realistic matrix sizes. The base fields we were interested in were finite fields of large characteristic. As an example, we have been able to compute a linear generator for a sequence of 4*4 matrices of length 242 304 defined over GF(2^607) in less than two days on one 667MHz alpha ev67 cpu

Resummation and Shower Studies

The transverse momentum spectra of the Z and Higgs bosons are studied, as probes of the consequences of multiple parton emissions in hadronic events. Emphasis is put on constraints, present in showers, that go beyond conventional leading log. It is shown that, if such constraints are relaxed, better agreement can be obtained with experimental data and with resummation descriptions.Comment: 6 pages, LaTeX, 3 eps figures, submitted to the proceedings of the Workshop on Physics at TeV Colliders, Les Houches, France, 26 May -- 6 June 200