6,703 research outputs found
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
Discrete phase-space structure of -qubit mutually unbiased bases
We work out the phase-space structure for a system of qubits. We replace
the field of real numbers that label the axes of the continuous phase space by
the finite field \Gal{2^n} and investigate the geometrical structures
compatible with the notion of unbiasedness. These consist of bundles of
discrete curves intersecting only at the origin and satisfying certain
additional properties. We provide a simple classification of such curves and
study in detail the four- and eight-dimensional cases, analyzing also the
effect of local transformations. In this way, we provide a comprehensive
phase-space approach to the construction of mutually unbiased bases for
qubits.Comment: Title changed. Improved version. Accepted for publication in Annals
of Physic
An Approximation Problem in Multiplicatively Invariant Spaces
Let be Hilbert space and a -finite
measure space. Multiplicatively invariant (MI) spaces are closed subspaces of that are invariant under point-wise multiplication by
functions in a fix subset of Given a finite set of data
in this paper we prove the
existence and construct an MI space that best fits , in the
least squares sense. MI spaces are related to shift invariant (SI) spaces via a
fiberization map, which allows us to solve an approximation problem for SI
spaces in the context of locally compact abelian groups. On the other hand, we
introduce the notion of decomposable MI spaces (MI spaces that can be
decomposed into an orthogonal sum of MI subspaces) and solve the approximation
problem for the class of these spaces. Since SI spaces having extra invariance
are in one-to-one relation to decomposable MI spaces, we also solve our
approximation problem for this class of SI spaces. Finally we prove that
translation invariant spaces are in correspondence with totally decomposable MI
spaces.Comment: 18 pages, To appear in Contemporary Mathematic
Construction of self-dual normal bases and their complexity
Recent work of Pickett has given a construction of self-dual normal bases for
extensions of finite fields, whenever they exist. In this article we present
these results in an explicit and constructive manner and apply them, through
computer search, to identify the lowest complexity of self-dual normal bases
for extensions of low degree. Comparisons to similar searches amongst normal
bases show that the lowest complexity is often achieved from a self-dual normal
basis
Composite Cyclotomic Fourier Transforms with Reduced Complexities
Discrete Fourier transforms~(DFTs) over finite fields have widespread
applications in digital communication and storage systems. Hence, reducing the
computational complexities of DFTs is of great significance. Recently proposed
cyclotomic fast Fourier transforms (CFFTs) are promising due to their low
multiplicative complexities. Unfortunately, there are two issues with CFFTs:
(1) they rely on efficient short cyclic convolution algorithms, which has not
been investigated thoroughly yet, and (2) they have very high additive
complexities when directly implemented. In this paper, we address both issues.
One of the main contributions of this paper is efficient bilinear 11-point
cyclic convolution algorithms, which allow us to construct CFFTs over
GF. The other main contribution of this paper is that we propose
composite cyclotomic Fourier transforms (CCFTs). In comparison to previously
proposed fast Fourier transforms, our CCFTs achieve lower overall complexities
for moderate to long lengths, and the improvement significantly increases as
the length grows. Our 2047-point and 4095-point CCFTs are also first efficient
DFTs of such lengths to the best of our knowledge. Finally, our CCFTs are also
advantageous for hardware implementations due to their regular and modular
structure.Comment: submitted to IEEE trans on Signal Processin
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