44 research outputs found
On Index Calculus Algorithms for Subfield Curves
In this paper we further the study of index calculus methods for solving the elliptic curve discrete logarithm problem (ECDLP). We focus on the index calculus for subfield curves, also called Koblitz curves, defined over Fq with ECDLP in Fqn. Instead of accelerating the solution of polynomial systems during index calculus as was predominantly done in previous work, we define factor bases that are invariant under the q-power Frobenius automorphism of the field Fqn, reducing the number of polynomial systems that need to be solved. A reduction by a factor of 1/n is the best one could hope for. We show how to choose factor bases to achieve this, while simultaneously accelerating the linear algebra step of the index calculus method for Koblitz curves by a factor n2. Furthermore, we show how to use the Frobenius endomorphism to improve symmetry breaking for Koblitz curves. We provide constructions of factor bases with the desired properties, and we study their impact on the polynomial system solving costs experimentally.SCOPUS: cp.kinfo:eu-repo/semantics/publishe
On the Rapoport-Zink space for over a ramified prime
In this work, we study the supersingular locus of the Shimura variety
associated to the unitary group over a ramified prime. We
show that the associated Rapoport-Zink space is flat, and we give an explicit
description of the irreducible components of the reduction modulo of the
basic locus. In particular, we show that these are universally homeomorphic to
either a generalized Deligne-Lusztig variety for a symplectic group or to the
closure of a vector bundle over a classical Deligne-Lusztig variety for an
orthogonal group. Our results are confirmed in the group-theoretical setting by
the reduction method \`a la Deligne and Lusztig and the study of the admissible
set
Feynman integral relations from parametric annihilators
We study shift relations between Feynman integrals via the Mellin transform
through parametric annihilation operators. These contain the momentum space IBP
relations, which are well-known in the physics literature. Applying a result of
Loeser and Sabbah, we conclude that the number of master integrals is computed
by the Euler characteristic of the Lee-Pomeransky polynomial. We illustrate
techniques to compute this Euler characteristic in various examples and compare
it with numbers of master integrals obtained in previous works.Comment: v2: new section 3.1 added, several misprints corrected and additional
remark
Algorithms in Intersection Theory in the Plane
This thesis presents an algorithm to find the local structure of intersections of plane curves. More precisely, we address the question of describing the scheme of the quotient ring of a bivariate zero-dimensional ideal , \textit{i.e.} finding the points (maximal ideals of ) and describing the regular functions on those points. A natural way to address this problem is via Gr\"obner bases as they reduce the problem of finding the points to a problem of factorisation, and the sheaf of rings of regular functions can be studied with those bases through the division algorithm and localisation.
Let be an ideal generated by , a subset of with and a field. We present an algorithm that features a quadratic convergence to find a Gr\"obner basis of or its primary component at the origin.
We introduce an -adic Newton iteration to lift the lexicographic Gr\"obner basis of any finite intersection of zero-dimensional primary components of if is a \textit{good} maximal ideal. It relies on a structural result about the syzygies in such a basis due to Conca \textit{\&} Valla (2008), from which arises an explicit map between ideals in a stratum (or Gr\"obner cell) and points in the associated moduli space. We also qualify what makes a maximal ideal suitable for our filtration.
When the field is \textit{large enough}, endowed with an Archimedean or ultrametric valuation, and admits a fraction reconstruction algorithm, we use this result to give a complete -adic algorithm to recover , the Gr\"obner basis of . We observe that previous results of Lazard that use Hermite normal forms to compute Gr\"obner bases for ideals with two generators can be generalised to a set of generators. We use this result to obtain a bound on the height of the coefficients of and to control the probability of choosing a \textit{good} maximal ideal to build the -adic expansion of .
Inspired by Pardue (1994), we also give a constructive proof to
characterise a Zariski open set of (with action on ) that changes coordinates in such a way as to ensure the initial term ideal of a zero-dimensional becomes Borel-fixed when is sufficiently large. This sharpens our analysis
to obtain, when or , a complexity less than cubic in terms of the dimension of and softly linear in the height of the coefficients of .
We adapt the resulting method and present the analysis to find the -primary component of . We also discuss the transition towards other primary components via linear mappings, called \emph{untangling} and \emph{tangling}, introduced by van der Hoeven and Lecerf (2017). The two maps form one isomorphism to find points with an isomorphic local structure and, at the origin, bind them. We give a slightly faster tangling algorithm and discuss new applications of these techniques. We show how to extend these ideas to bivariate settings and give a bound on the arithmetic complexity for certain algebras
Parametric Toricity of Steady State Varieties of Reaction Networks
We study real steady state varieties of the dynamics of chemical reaction
networks. The dynamics are derived using mass action kinetics with parametric
reaction rates. The models studied are not inherently parametric in nature.
Rather, our interest in parameters is motivated by parameter uncertainty, as
reaction rates are typically either measured with limited precision or
estimated. We aim at detecting toricity and shifted toricity, using a framework
that has been recently introduced and studied for the non-parametric case over
both the real and the complex numbers. While toricity requires that the variety
specifies a subgroup of the direct power of the multiplicative group of the
underlying field, shifted toricity requires only a coset. In the non-parametric
case these requirements establish real decision problems. In the presence of
parameters we must go further and derive necessary and sufficient conditions in
the parameters for toricity or shifted toricity to hold. Technically, we use
real quantifier elimination methods. Our computations on biological networks
here once more confirm shifted toricity as a relevant concept, while toricity
holds only for degenerate parameter choices.Comment: Computations available as ancillary file
A Combinatorial Commutative Algebra Approach to Complete Decoding
Esta tesis pretende explorar el nexo de unión que existe entre la estructura algebraica de un código lineal y el proceso de descodificación completa. Sabemos que el proceso de descodificación completa para códigos lineales arbitrarios es NP-completo, incluso si se admite preprocesamiento de los datos. Nuestro objetivo es realizar un análisis algebraico del proceso de la descodificación, para ello asociamos diferentes estructuras matemáticas a ciertas familias de códigos. Desde el punto de vista computacional, nuestra descripción no proporciona un algoritmo eficiente pues nos enfrentamos a un problema de naturaleza NP. Sin embargo, proponemos algoritmos alternativos y nuevas técnicas que permiten relajar las condiciones del problema reduciendo los recursos de espacio y tiempo necesarios para manejar dicha estructura algebraica.Departamento de Algebra, GeometrÃa y TopologÃ
Tautological classes of matroids
We introduce certain torus-equivariant classes on permutohedral varieties
which we call "tautological classes of matroids" as a new geometric framework
for studying matroids. Using this framework, we unify and extend many recent
developments in matroid theory arising from its interaction with algebraic
geometry. We achieve this by establishing a Chow-theoretic description and a
log-concavity property for a 4-variable transformation of the Tutte polynomial,
and by establishing an exceptional Hirzebruch-Riemann-Roch-type formula for
permutohedral varieties that translates between K-theory and Chow theory.Comment: 69 pages; comments welcome. v2: minor edits, addition of subsection
10.