1,334 research outputs found
Numerical calculation of three-point branched covers of the projective line
We exhibit a numerical method to compute three-point branched covers of the
complex projective line. We develop algorithms for working explicitly with
Fuchsian triangle groups and their finite index subgroups, and we use these
algorithms to compute power series expansions of modular forms on these groups.Comment: 58 pages, 24 figures; referee's comments incorporate
Computing Hilbert class polynomials with the Chinese Remainder Theorem
We present a space-efficient algorithm to compute the Hilbert class
polynomial H_D(X) modulo a positive integer P, based on an explicit form of the
Chinese Remainder Theorem. Under the Generalized Riemann Hypothesis, the
algorithm uses O(|D|^(1/2+o(1))log P) space and has an expected running time of
O(|D|^(1+o(1)). We describe practical optimizations that allow us to handle
larger discriminants than other methods, with |D| as large as 10^13 and h(D) up
to 10^6. We apply these results to construct pairing-friendly elliptic curves
of prime order, using the CM method.Comment: 37 pages, corrected a typo that misstated the heuristic complexit
Vers une arithmétique efficace pour le chiffrement homomorphe basé sur le Ring-LWE
Fully homomorphic encryption is a kind of encryption offering the ability to manipulate encrypted data directly through their ciphertexts. In this way it is possible to process sensitive data without having to decrypt them beforehand, ensuring therefore the datas' confidentiality. At the numeric and cloud computing era this kind of encryption has the potential to considerably enhance privacy protection. However, because of its recent discovery by Gentry in 2009, we do not have enough hindsight about it yet. Therefore several uncertainties remain, in particular concerning its security and efficiency in practice, and should be clarified before an eventual widespread use. This thesis deals with this issue and focus on performance enhancement of this kind of encryption in practice. In this perspective we have been interested in the optimization of the arithmetic used by these schemes, either the arithmetic underlying the Ring Learning With Errors problem on which the security of these schemes is based on, or the arithmetic specific to the computations required by the procedures of some of these schemes. We have also considered the optimization of the computations required by some specific applications of homomorphic encryption, and in particular for the classification of private data, and we propose methods and innovative technics in order to perform these computations efficiently. We illustrate the efficiency of our different methods through different software implementations and comparisons to the related art.Le chiffrement totalement homomorphe est un type de chiffrement qui permet de manipuler directement des données chiffrées. De cette manière, il est possible de traiter des données sensibles sans avoir à les déchiffrer au préalable, permettant ainsi de préserver la confidentialité des données traitées. À l'époque du numérique à outrance et du "cloud computing" ce genre de chiffrement a le potentiel pour impacter considérablement la protection de la vie privée. Cependant, du fait de sa découverte récente par Gentry en 2009, nous manquons encore de recul à son propos. C'est pourquoi de nombreuses incertitudes demeurent, notamment concernant sa sécurité et son efficacité en pratique, et devront être éclaircies avant une éventuelle utilisation à large échelle.Cette thèse s'inscrit dans cette problématique et se concentre sur l'amélioration des performances de ce genre de chiffrement en pratique. Pour cela nous nous sommes intéressés à l'optimisation de l'arithmétique utilisée par ces schémas, qu'elle soit sous-jacente au problème du "Ring-Learning With Errors" sur lequel la sécurité des schémas considérés est basée, ou bien spécifique aux procédures de calculs requises par certains de ces schémas. Nous considérons également l'optimisation des calculs nécessaires à certaines applications possibles du chiffrement homomorphe, et en particulier la classification de données privées, de sorte à proposer des techniques de calculs innovantes ainsi que des méthodes pour effectuer ces calculs de manière efficace. L'efficacité de nos différentes méthodes est illustrée à travers des implémentations logicielles et des comparaisons aux techniques de l'état de l'art
Universal optimality of the and Leech lattices and interpolation formulas
We prove that the root lattice and the Leech lattice are universally
optimal among point configurations in Euclidean spaces of dimensions and
, respectively. In other words, they minimize energy for every potential
function that is a completely monotonic function of squared distance (for
example, inverse power laws or Gaussians), which is a strong form of robustness
not previously known for any configuration in more than one dimension. This
theorem implies their recently shown optimality as sphere packings, and broadly
generalizes it to allow for long-range interactions.
The proof uses sharp linear programming bounds for energy. To construct the
optimal auxiliary functions used to attain these bounds, we prove a new
interpolation theorem, which is of independent interest. It reconstructs a
radial Schwartz function from the values and radial derivatives of and
its Fourier transform at the radii for integers
in and in . To prove this
theorem, we construct an interpolation basis using integral transforms of
quasimodular forms, generalizing Viazovska's work on sphere packing and placing
it in the context of a more conceptual theory.Comment: 95 pages, 6 figure
Computing Igusa class polynomials
We bound the running time of an algorithm that computes the genus-two class
polynomials of a primitive quartic CM-field K. This is in fact the first
running time bound and even the first proof of correctness of any algorithm
that computes these polynomials.
Essential to bounding the running time is our bound on the height of the
polynomials, which is a combination of denominator bounds of Goren and Lauter
and our own absolute value bounds. The absolute value bounds are obtained by
combining Dupont's estimates of theta constants with an analysis of the shape
of CM period lattices.
The algorithm is basically the complex analytic method of Spallek and van
Wamelen, and we show that it finishes in time Otilde(Delta^(7/2)), where Delta
is the discriminant of K. We give a complete running time analysis of all parts
of the algorithm, and a proof of correctness including a rounding error
analysis. We also provide various improvements along the way.Comment: 31 pages (Various improvements to the exposition suggested by the
referee. For the most detailed exposition, see Chapter II of the author's
thesis http://hdl.handle.net/1887/15572
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