2,120 research outputs found
Efficient dot product over word-size finite fields
We want to achieve efficiency for the exact computation of the dot product of
two vectors over word-size finite fields. We therefore compare the practical
behaviors of a wide range of implementation techniques using different
representations. The techniques used include oating point representations,
discrete logarithms, tabulations, Montgomery reduction, delayed modulus
Bounds on the coefficients of the characteristic and minimal polynomials
This note presents absolute bounds on the size of the coefficients of the
characteristic and minimal polynomials depending on the size of the
coefficients of the associated matrix. Moreover, we present algorithms to
compute more precise input-dependant bounds on these coefficients. Such bounds
are e.g. useful to perform deterministic chinese remaindering of the
characteristic or minimal polynomial of an integer matrix
Computational linear algebra over finite fields
We present here algorithms for efficient computation of linear algebra
problems over finite fields
Matrix powers algorithms for trust evaluation in PKI architectures
This paper deals with the evaluation of trust in public-key infrastructures.
Different trust models have been proposed to interconnect the various PKI
components in order to propagate the trust between them. In this paper we
provide a new polynomial algorithm using linear algebra to assess trust
relationships in a network using different trust evaluation schemes. The
advantages are twofold: first the use of matrix computations instead of graph
algorithms provides an optimized computational solution; second, our algorithm
can be used for generic graphs, even in the presence of cycles. Our algorithm
is designed to evaluate the trust using all existing (finite) trust paths
between entities as a preliminary to any exchanges between PKIs. This can give
a precise evaluation of trust, and accelerate for instance cross-certificate
validation
An introspective algorithm for the integer determinant
We present an algorithm computing the determinant of an integer matrix A. The
algorithm is introspective in the sense that it uses several distinct
algorithms that run in a concurrent manner. During the course of the algorithm
partial results coming from distinct methods can be combined. Then, depending
on the current running time of each method, the algorithm can emphasize a
particular variant. With the use of very fast modular routines for linear
algebra, our implementation is an order of magnitude faster than other existing
implementations. Moreover, we prove that the expected complexity of our
algorithm is only O(n^3 log^{2.5}(n ||A||)) bit operations in the dense case
and O(Omega n^{1.5} log^2(n ||A||) + n^{2.5}log^3(n||A||)) in the sparse case,
where ||A|| is the largest entry in absolute value of the matrix and Omega is
the cost of matrix-vector multiplication in the case of a sparse matrix.Comment: Published in Transgressive Computing 2006, Grenade : Espagne (2006
Symmetric indefinite triangular factorization revealing the rank profile matrix
We present a novel recursive algorithm for reducing a symmetric matrix to a
triangular factorization which reveals the rank profile matrix. That is, the
algorithm computes a factorization where is a permutation matrix,
is lower triangular with a unit diagonal and is
symmetric block diagonal with and antidiagonal
blocks. The novel algorithm requires arithmetic
operations. Furthermore, experimental results demonstrate that our algorithm
can even be slightly more than twice as fast as the state of the art
unsymmetric Gaussian elimination in most cases, that is it achieves
approximately the same computational speed. By adapting the pivoting strategy
developed in the unsymmetric case, we show how to recover the rank profile
matrix from the permutation matrix and the support of the block-diagonal
matrix. There is an obstruction in characteristic for revealing the rank
profile matrix which requires to relax the shape of the block diagonal by
allowing the 2-dimensional blocks to have a non-zero bottom-right coefficient.
This relaxed decomposition can then be transformed into a standard
decomposition at a
negligible cost
Sequential products in effect categories
A new categorical framework is provided for dealing with multiple arguments
in a programming language with effects, for example in a language with
imperative features. Like related frameworks (Monads, Arrows, Freyd
categories), we distinguish two kinds of functions. In addition, we also
distinguish two kinds of equations. Then, we are able to define a kind of
product, that generalizes the usual categorical product. This yields a powerful
tool for deriving many results about languages with effects
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