Modern Computer Arithmetic

This electronic version is distributed under the terms and conditions of the Creative Commons license " Attribution-Noncommercial-No Derivative Works 3.0 ". You are free to copy, distribute and transmit this book under the following conditions: • Attribution. You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). • Noncommercial. You may not use this work for commercial purposes. For any reuse or distribution, you must make clear to others the license terms of this work. The best way to do this is with a link to the web page below. Any of the above conditions can be waived if you get permission from the copyright holder. Nothing in this license impairs or restricts the author's moral rights. Preface This is a book about algorithms for performing arithmetic, and their implementation on modern computers. We are concerned with software more than hardware — we do not cover computer architecture or the design of computer hardware since good books are already available on these topics. Instead we focus on 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 special functions. The algorithms that we present are mainly intended for arbitrary-precision arithmetic. That is, they are not limited by the computer wordsize of 32 or 64 bits, only by the memory and time available for the computation. We consider both integer and real (floating-point) computations. The book is divided into four main chapters, plus an appendix. Chapter 1 covers integer arithmetic. This has, of course, been considered in many other books and papers. However, there has been much recent progress, inspired in part by the application to public key cryptography, so most of the published books are now partly out of date or incomplete. Our aim has been to present the latest developments in a concise manner. Chapter 2 is concerned with the FFT and modular arithmetic, and their applications to computer arithmetic. We consider different number representations , fast algorithms for multiplication, division and exponentiation, and the use of the Chinese Remainder Theorem (CRT). Chapter 3 covers floating-point arithmetic. Our concern is with high-precision floating-point arithmetic, implemented in software if the precision provided by the hardware (typically …