Modern Computer Arithmetic focuses on arbitrary-precision algorithms for efficiently performing arithmetic operations such as addition, multiplication and division, and their connections to topics such as modular arithmetic, greatest common divisors, the Fast Fourier Transform (FFT), and the computation of elementary and special functions. Brent and Zimmermann present algorithms that are ready to implement in your favorite language, while keeping a high-level description and avoiding too low-level or machine-dependent details. The book is intended for anyone interested in the design and implementation of efficient high-precision algorithms for computer arithmetic, and more generally efficient multiple-precision numerical algorithms. It may also be used in a graduate course in mathematics or computer science, for which exercises are included. These vary considerably in difficulty, from easy to small research projects, and expand on topics discussed in the text. Solutions are available from the authors.

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Modern Computer Arithmetic

This survey explains how some useful arithmetic operations can be sped up from quadratic time to essentially linear time.

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Fast multiplication and its applications

https://specfun.inria.fr/bostan/publications/BoSc05.pdf

Polynomial evaluation and interpolation on special sets of points

We study the complexity of computing one or several terms (not necessarily consecutive) in a recurrence with polynomial coefficients. As applications, we improve the best currently known upper bounds for factoring integers deterministically and for computing the Cartier-Manin operator of hyperelliptic curves.

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Linear Recurrences with Polynomial Coefficients and Application to Integer Factorization and Cartier-Manin Operator

This paper presents extremely fast algorithms for code-based public-key cryptography, including full protection against timing attacks. For example, at a 2128 security level, this paper achieves a reciprocal decryption throughput of just 60493 cycles (plus cipher cost etc.) on a single Ivy Bridge core. These algorithms rely on an additive FFT for fast root computation, a transposed additive FFT for fast syndrome computation, and a sorting network to avoid cache-timing attacks.

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R. Moenck

Papers

Fast modular transforms