38,699 research outputs found
Isogenies of Elliptic Curves: A Computational Approach
Isogenies, the mappings of elliptic curves, have become a useful tool in
cryptology. These mathematical objects have been proposed for use in computing
pairings, constructing hash functions and random number generators, and
analyzing the reducibility of the elliptic curve discrete logarithm problem.
With such diverse uses, understanding these objects is important for anyone
interested in the field of elliptic curve cryptography. This paper, targeted at
an audience with a knowledge of the basic theory of elliptic curves, provides
an introduction to the necessary theoretical background for understanding what
isogenies are and their basic properties. This theoretical background is used
to explain some of the basic computational tasks associated with isogenies.
Herein, algorithms for computing isogenies are collected and presented with
proofs of correctness and complexity analyses. As opposed to the complex
analytic approach provided in most texts on the subject, the proofs in this
paper are primarily algebraic in nature. This provides alternate explanations
that some with a more concrete or computational bias may find more clear.Comment: Submitted as a Masters Thesis in the Mathematics department of the
University of Washingto
The computational complexity of traditional Lattice-Boltzmann methods for incompressible fluids
It is well-known that in fluid dynamics an alternative to customary direct
solution methods (based on the discretization of the fluid fields) is provided
by so-called \emph{particle simulation methods}. Particle simulation methods
rely typically on appropriate \emph{kinetic models} for the fluid equations
which permit the evaluation of the fluid fields in terms of suitable
expectation values (or \emph{momenta}) of the kinetic distribution function
being respectively and\textbf{\}
the position an velocity of a test particle with probability density
. These kinetic models can be continuous or discrete in
phase space, yielding respectively \emph{continuous} or \emph{discrete kinetic
models} for the fluids. However, also particle simulation methods may be biased
by an undesirable computational complexity. In particular, a fundamental issue
is to estimate the algorithmic complexity of numerical simulations based on
traditional LBM's (Lattice-Boltzmann methods; for review see Succi, 2001
\cite{Succi}). These methods, based on a discrete kinetic approach, represent
currently an interesting alternative to direct solution methods. Here we intend
to prove that for incompressible fluids fluids LBM's may present a high
complexity. The goal of the investigation is to present a detailed account of
the origin of the various complexity sources appearing in customary LBM's. The
result is relevant to establish possible strategies for improving the numerical
efficiency of existing numerical methods.Comment: Contributed paper at RGD26 (Kyoto, Japan, July 2008
Computational linear algebra over finite fields
We present here algorithms for efficient computation of linear algebra
problems over finite fields
Can local single-pass methods solve any stationary Hamilton-Jacobi-Bellman equation?
The use of local single-pass methods (like, e.g., the Fast Marching method)
has become popular in the solution of some Hamilton-Jacobi equations. The
prototype of these equations is the eikonal equation, for which the methods can
be applied saving CPU time and possibly memory allocation. Then, some natural
questions arise: can local single-pass methods solve any Hamilton-Jacobi
equation? If not, where the limit should be set? This paper tries to answer
these questions. In order to give a complete picture, we present an overview of
some fast methods available in literature and we briefly analyze their main
features. We also introduce some numerical tools and provide several numerical
tests which are intended to exhibit the limitations of the methods. We show
that the construction of a local single-pass method for general Hamilton-Jacobi
equations is very hard, if not impossible. Nevertheless, some special classes
of problems can be actually solved, making local single-pass methods very
useful from the practical point of view.Comment: 19 page
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