66,120 research outputs found
Optimality of the Width- Non-adjacent Form: General Characterisation and the Case of Imaginary Quadratic Bases
Efficient scalar multiplication in Abelian groups (which is an important
operation in public key cryptography) can be performed using digital
expansions. Apart from rational integer bases (double-and-add algorithm),
imaginary quadratic integer bases are of interest for elliptic curve
cryptography, because the Frobenius endomorphism fulfils a quadratic equation.
One strategy for improving the efficiency is to increase the digit set (at the
prize of additional precomputations). A common choice is the width\nbd-
non-adjacent form (\wNAF): each block of consecutive digits contains at
most one non-zero digit. Heuristically, this ensures a low weight, i.e.\ number
of non-zero digits, which translates in few costly curve operations. This paper
investigates the following question: Is the \wNAF{}-expansion optimal, where
optimality means minimising the weight over all possible expansions with the
same digit set?
The main characterisation of optimality of \wNAF{}s can be formulated in the
following more general setting: We consider an Abelian group together with an
endomorphism (e.g., multiplication by a base element in a ring) and a finite
digit set. We show that each group element has an optimal \wNAF{}-expansion if
and only if this is the case for each sum of two expansions of weight 1. This
leads both to an algorithmic criterion and to generic answers for various
cases.
Imaginary quadratic integers of trace at least 3 (in absolute value) have
optimal \wNAF{}s for . The same holds for the special case of base
and , which corresponds to Koblitz curves in
characteristic three. In the case of , optimality depends on
the parity of . Computational results for small trace are given
Group ring elements with large spectral density
Given an arbitrary d>0 we construct a group G and a group ring element S in
Z[G] such that the spectral measure mu of S has the property that mu((0,eps)) >
C/|log(eps)|^(1+d) for small eps. In particular the Novikov-Shubin invariant of
any such S is 0. The constructed examples show that the best known upper bounds
on mu((0,eps)) are not far from being optimal.Comment: 19 pages, v3: Changes suggested by a referee; Essentially this is the
version published in Math. An
Panphasia: a user guide
We make a very large realisation of a Gaussian white noise field, called
PANPHASIA, public by releasing software that computes this field. Panphasia is
designed specifically for setting up Gaussian initial conditions for
cosmological simulations and resimulations of structure formation. We make
available both software to compute the field itself and codes to illustrate
applications including a modified version of a public serial initial conditions
generator. We document the software and present the results of a few basic
tests of the field. The properties and method of construction of Panphasia are
described in full in a companion paper Jenkins 2013.Comment: 11 pages, 2 figures. Software to calculate Panphasia is available
from: http://icc.dur.ac.uk/Panphasia.ph
- âŠ