220 research outputs found
Markov Chain Monte Carlo Algorithms for Lattice Gaussian Sampling
Sampling from a lattice Gaussian distribution is emerging as an important
problem in various areas such as coding and cryptography. The default sampling
algorithm --- Klein's algorithm yields a distribution close to the lattice
Gaussian only if the standard deviation is sufficiently large. In this paper,
we propose the Markov chain Monte Carlo (MCMC) method for lattice Gaussian
sampling when this condition is not satisfied. In particular, we present a
sampling algorithm based on Gibbs sampling, which converges to the target
lattice Gaussian distribution for any value of the standard deviation. To
improve the convergence rate, a more efficient algorithm referred to as
Gibbs-Klein sampling is proposed, which samples block by block using Klein's
algorithm. We show that Gibbs-Klein sampling yields a distribution close to the
target lattice Gaussian, under a less stringent condition than that of the
original Klein algorithm.Comment: 5 pages, 1 figure, IEEE International Symposium on Information
Theory(ISIT) 201
Time- and Space-Efficient Evaluation of Some Hypergeometric Constants
The currently best known algorithms for the numerical evaluation of
hypergeometric constants such as to decimal digits have time
complexity and space complexity of or .
Following work from Cheng, Gergel, Kim and Zima, we present a new algorithm
with the same asymptotic complexity, but more efficient in practice. Our
implementation of this algorithm improves slightly over existing programs for
the computation of , and we announce a new record of 2 billion digits for
Doctorates and the Berlin declaration
K. Doevendans, S. Hanrot 2003. Doctorates and the Berlin Declaration.USO-Built Report Series 2:11-17. The Berlin declaration of September 2003 hasplaced education for a doctorate even more strongly in the Bologna Process. This is also the case in domains, such as architecture, where the different countries of Europe are at variance as to the presence and content of such a degree. The meaning of this development for joint supervision and joint degrees within USO-Built summarizes the content of this chapter
Worst-Case Hermite-Korkine-Zolotarev Reduced Lattice Bases
The Hermite-Korkine-Zolotarev reduction plays a central role in strong
lattice reduction algorithms. By building upon a technique introduced by Ajtai,
we show the existence of Hermite-Korkine-Zolotarev reduced bases that are
arguably least reduced. We prove that for such bases, Kannan's algorithm
solving the shortest lattice vector problem requires
d^{\frac{d}{2\e}(1+o(1))} bit operations in dimension . This matches the
best complexity upper bound known for this algorithm. These bases also provide
lower bounds on Schnorr's constants and that are
essentially equal to the best upper bounds. Finally, we also show the existence
of particularly bad bases for Schnorr's hierarchy of reductions
Accelerating the CM method
Given a prime q and a negative discriminant D, the CM method constructs an
elliptic curve E/\Fq by obtaining a root of the Hilbert class polynomial H_D(X)
modulo q. We consider an approach based on a decomposition of the ring class
field defined by H_D, which we adapt to a CRT setting. This yields two
algorithms, each of which obtains a root of H_D mod q without necessarily
computing any of its coefficients. Heuristically, our approach uses
asymptotically less time and space than the standard CM method for almost all
D. Under the GRH, and reasonable assumptions about the size of log q relative
to |D|, we achieve a space complexity of O((m+n)log q) bits, where mn=h(D),
which may be as small as O(|D|^(1/4)log q). The practical efficiency of the
algorithms is demonstrated using |D| > 10^16 and q ~ 2^256, and also |D| >
10^15 and q ~ 2^33220. These examples are both an order of magnitude larger
than the best previous results obtained with the CM method.Comment: 36 pages, minor edits, to appear in the LMS Journal of Computation
and Mathematic
A long note on Mulders' short product
The short product of two power series is the meaningful part of the product of these objects, i.e., _i+j < n a_ib_j x^i+j. In , Mulders gives an algorithm to compute a short product faster than the full product in the case of Karatsuba's multiplication . This algorithm work by selecting a cutoff point k and performing a full kk product and two (n-k)(n-k) short products recursively. Mulders also gives an heuristically optimal cutoff point n. In this paper, we determine the optimal cutoff point in Mulders' algorithm. We also give a slightly more general description of Mulders' method
Floating-Point -Approximations
International audienceComputing good polynomial approximations to usual functions is an important topic for the computer evaluation of those functions. These approximations can be good under several criteria, the most desirable being probably that the relative error is as small as possible in the sense, i.e. everywhere on the interval under study. In the present paper, we investigate a simpler criterion, the case. Though finding a best polynomial -approximation with real coefficients is quite easy, we show that if the coefficients are restricted to be floating point numbers to some precision, the problem becomes a general instance of the CVP problem, and hence is NP-hard. We investigate the practical behaviour of exact and approximate algorithms for this problem. The conclusion is that it is possible in a short amount of time to obtain a relative or absolute best -approximation. The main applications are for large dimension, as a preliminary step of finding -approximations and for functions with large variations, for which relative best approximation is by far more interesting than absolute
Proceedings of the 7th Conference on Real Numbers and Computers (RNC'7)
These are the proceedings of RNC7
Fabrique dâune icĂŽne : La Madone de Bentalha. Entretien avec Juliette Hanrot
Dans la nuit du 22 au 23 septembre 1997, un massacre de civils est perpĂ©trĂ© par une centaine dâhommes Ă Bentalha, Ă une trentaine de kilomĂštres dâAlger. Le lendemain, un photographe algĂ©rien travaillant pour lâAFP, Hocine Zaourar, qui sâest rendu dans lâhĂŽpital qui avait accueilli les victimes, saisit lâexpression de douleur dâune femme. Cette image fait immĂ©diatement la une de plus de 750 journaux dans le monde. Un tel retentissement sâexplique par la force dramatique de la photo, mais surtout par son analogie frappante avec lâiconographie chrĂ©tienne de la douleur. Câest cette photo, baptisĂ©e par la presse Madone de Bentalha, quâanalyse lâouvrage de Juliette Hanrot, La Madone de Bentalha. Histoire dâune photographie, en en dĂ©ployant les multiples niveaux de signification tant sur le plan de lâhistoire immĂ©diate et des effets mĂ©diatiques que sur le plan de ses rĂ©sonnances culturelles et anthropologiques. Dans lâentretien quâelle a accordĂ© Ă Dominique ClĂ©venot, Juliette Hanrot revient sur la fabrique de cette icĂŽne pour interroger la relation que lâimage de presse entretient avec le contexte historique de lâĂ©vĂ©nement quâelle relate, mais aussi avec lâimaginaire du public qui la reçoit
- âŠ