31,683 research outputs found

    On Numerical Analysis in Residue Number Systems

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    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

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    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

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    The novel family of redundant residue number system (RRNS) codes is studied. RRNS codes constitute maximum–minimum distance block codes, exhibiting identical distance properties to Reed–Solomon 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 Terms—Redundant residue number system (RRNS), residue number system (RNS), turbo detection

    Solution of the Dirac equation in lattice QCD using a domain decomposition method

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    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

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    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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