Computing a square root for the number field sieve

Abstract

The number field sieve is a method proposed by Lenstra, Lenstra, Manasse and Pollard for integer factorization (this volume, pp. 11–42). 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. 50–94). 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.

Membre de l'Option Recherche du Corps des Ingénieurs de l'Armement.

The author wishes to thank H. W. Lenstra, Jr., for his suggestions and help with the writing of this paper. Part of this work formed a D.E.A. thesis at the Ecole Polytechnique under the direction of MM. Marc Chardin, Marc Giusti and Jacques Stern, in May 1991.