31,683 research outputs found
On Numerical Analysis in Residue Number Systems
Recent attempts to utilize residue number systems
in digital computers have raised numerous questions about
adapting the techniques of numerical analysis to residue
number systems. Among these questions are the fundamental
problems of how to compare the magnitudes of two numbers, how
to detect additive and multiplicative overflow, and how to
divide in residue number systems. These three problems are
treated in separate chapters of this thesis and methods are
developed therein whereby magnitude comparison, overflow
detection, and division can be performed in residue number
systems. In an additional chapter, the division method is
extended to provide an algorithm for the direct approximation
of square roots in residue number systems. Numerous
examples are provided illustrating the nature of the problems considered and showing the use of the solutions presented in
practical computations. In a final chapter are presented the
results of extensive trial calculations for which a conventional
digital computer was programmed to simulate the use
of the division and square root algorithms in approximating
quotients and square roots in residue number systems. These
results indicate that, in practice, these division and
square root algorithms usually converge to the quotient or
square root somewhat faster than is suggested by the theory
Generalised Mersenne Numbers Revisited
Generalised Mersenne Numbers (GMNs) were defined by Solinas in 1999 and
feature in the NIST (FIPS 186-2) and SECG standards for use in elliptic curve
cryptography. Their form is such that modular reduction is extremely efficient,
thus making them an attractive choice for modular multiplication
implementation. However, the issue of residue multiplication efficiency seems
to have been overlooked. Asymptotically, using a cyclic rather than a linear
convolution, residue multiplication modulo a Mersenne number is twice as fast
as integer multiplication; this property does not hold for prime GMNs, unless
they are of Mersenne's form. In this work we exploit an alternative
generalisation of Mersenne numbers for which an analogue of the above property
--- and hence the same efficiency ratio --- holds, even at bitlengths for which
schoolbook multiplication is optimal, while also maintaining very efficient
reduction. Moreover, our proposed primes are abundant at any bitlength, whereas
GMNs are extremely rare. Our multiplication and reduction algorithms can also
be easily parallelised, making our arithmetic particularly suitable for
hardware implementation. Furthermore, the field representation we propose also
naturally protects against side-channel attacks, including timing attacks,
simple power analysis and differential power analysis, which is essential in
many cryptographic scenarios, in constrast to GMNs.Comment: 32 pages. Accepted to Mathematics of Computatio
Systematic redundant residue number system codes: analytical upper bound and iterative decoding performance over AWGN and Rayleigh channels
The novel family of redundant residue number system (RRNS) codes is studied. RRNS codes constitute maximumâminimum distance block codes, exhibiting identical distance properties to ReedâSolomon codes. Binary to RRNS symbol-mapping methods are proposed, in order to implement both systematic and nonsystematic RRNS codes. Furthermore, the upper-bound performance of systematic RRNS codes is investigated, when maximum-likelihood (ML) soft decoding is invoked. The classic Chase algorithm achieving near-ML soft decoding is introduced for the first time for RRNS codes, in order to decrease the complexity of the ML soft decoding. Furthermore, the modified Chase algorithm is employed to accept soft inputs, as well as to provide soft outputs, assisting in the turbo decoding of RRNS codes by using the soft-input/soft-output Chase algorithm. Index TermsâRedundant residue number system (RRNS), residue number system (RNS), turbo detection
Solution of the Dirac equation in lattice QCD using a domain decomposition method
Efficient algorithms for the solution of partial differential equations on
parallel computers are often based on domain decomposition methods. Schwarz
preconditioners combined with standard Krylov space solvers are widely used in
this context, and such a combination is shown here to perform very well in the
case of the Wilson--Dirac equation in lattice QCD. In particular, with respect
to even-odd preconditioned solvers, the communication overhead is significantly
reduced, which allows the computational work to be distributed over a large
number of processors with only small parallelization losses.Comment: Plain TeX source, 21 pages, figures include
A Fast Algorithm for Determining the Existence and Value of Integer Roots of N
We show that all perfect odd integer squares not divisible by 3, can be
usefully written as sqrt(N) = a + 18p, where the constant a is determined by
the basic properties of N. The equation can be solved deterministically by an
efficient four step algorithm that is solely based on integer arithmetic. There
is no required multiplication or division by multiple digit integers, nor does
the algorithm need a seed value. It finds the integer p when N is a perfect
square, and certifies N as a non-square when the algorithm terminates without a
solution. The number of iterations scales approximately as log(sqrt(N)/2) for
square roots. The paper also outlines how one of the methods discussed for
squares can be extended to finding an arbitrary root of N. Finally, we present
a rule that distinguishes products of twin primes from squares.Comment: 12 pages, 8 figure
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