Abstract

. The number field sieve is a method proposed by Lenstra, Lenstra, Manasse and Pollard for integer factorization (this volume,pp. 11--40). A heuristic analysis indicates that this method is asymptotically faster than any other existing one. It has had spectacular successes in factoring numbers of a special form. New technical difficulties arise when the method is adapted for general numbers (this volume, pp. 48--89). Among these is the need for computing the square root of a huge algebraic integer given as a product of hundreds of thousands of small ones. We present a method for computing such a square root that avoids excessively large numbers. It works only if the degree of the number field that is used is odd. The method is based on a careful use of the Chinese remainder theorem. 1. Introduction We begin by recalling the basic scheme of the number field sieve, cf. [7]. Let n be a positive integer that is not a power of a prime number. In order to factor n, we first find m...

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