253,951 research outputs found
Optimal normal bases
Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe
Optimal normal bases in GF(pn)
AbstractIn this paper the use of normal bases for multiplication in the finite fields GF(pn) is examined. We introduce the concept of an optimal normal basis in order to reduce the hardware complexity of multiplying field elements. Constructions for these bases in GF(2n) and extensions of the results to GF(pn) are presented. This work has applications in crytography and coding theory since a reduction in the complexity of multiplying and exponentiating elements of GF(2n) is achieved for many values of n, some prime
The Gaussian normal basis and its trace basis over finite fields
AbstractIt is well known that normal bases are useful for implementations of finite fields in various applications including coding theory, cryptography, signal processing, and so on. In particular, optimal normal bases are desirable. When no optimal normal basis exists, it is useful to have normal bases with low complexity. In this paper, we study the type k(⩾1) Gaussian normal basis N of the finite field extension Fqn/Fq, which is a classical normal basis with low complexity. By studying the multiplication table of N, we obtain the dual basis of N and the trace basis of N via arbitrary medium subfields Fqm/Fq with m|n and 1⩽m⩽n. And then we determine all self-dual Gaussian normal bases. As an application, we obtain the precise multiplication table and the complexity of the type 2 Gaussian normal basis and then determine all optimal type 2 Gaussian normal bases
Efficient multiplication in binary fields
The thesis discusses the basics of efficient multiplication in finite fields,
especially in binary fields. There are two broad approaches: polynomial representation and normal bases, used in software and hardware implementations,
respectively. Due to the advantages of normal bases of low complexity, there is
also a brief introduction to constructing optimal normal bases. Furthermore, as
irreducible polynomials are of fundamental importance for finite fields, the thesis
concludes with some irreducibility test
Normal and optimal normal bases in finite fields
Arithmetic operations in finite fields have many applications in cryptography, coding theory, and computer algebra. The realization of these operations can often be made more efficient by the normal basis representation of the field elements. This thesis is aimed at giving a survey of recent results concerning normal bases and efficient ways of multiplication, inversion, and exponentiation when the normal basis representation is used
Convergence rates for Bayesian density estimation of infinite-dimensional exponential families
We study the rate of convergence of posterior distributions in density
estimation problems for log-densities in periodic Sobolev classes characterized
by a smoothness parameter p. The posterior expected density provides a
nonparametric estimation procedure attaining the optimal minimax rate of
convergence under Hellinger loss if the posterior distribution achieves the
optimal rate over certain uniformity classes. A prior on the density class of
interest is induced by a prior on the coefficients of the trigonometric series
expansion of the log-density. We show that when p is known, the posterior
distribution of a Gaussian prior achieves the optimal rate provided the prior
variances die off sufficiently rapidly. For a mixture of normal distributions,
the mixing weights on the dimension of the exponential family are assumed to be
bounded below by an exponentially decreasing sequence. To avoid the use of
infinite bases, we develop priors that cut off the series at a
sample-size-dependent truncation point. When the degree of smoothness is
unknown, a finite mixture of normal priors indexed by the smoothness parameter,
which is also assigned a prior, produces the best rate. A rate-adaptive
estimator is derived.Comment: Published at http://dx.doi.org/10.1214/009053606000000911 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
- …