Fast Ideal Arithmetic in Quadratic Fields

Ideal multiplication and reduction are fundamental operations on ideals and are used extensively in class group and infrastructure computations; hence, the efficiency of these operations is extremely important. In this thesis we focus on reduction in real quadratic fields and examine all of the known reduction algorithms, converting them whenever required to work with ideals of positive discriminant. We begin with the classical algorithms of Gauss and Lagrange and move on to the algorithm Rickert developed for the closely related case of positive definite binary quadratic forms. Given any reduction technique, we present a general method computing the relative generator necessary for infrastructure computations. Rickert’s algorithm along with an algorithm of Schönhage are adapted to ideals of real quadratic fields. We present a new method which combines the ideas of Lehmer, Williams, and others into a particularly simple algorithm. All of these algorithms have been implemented and compared with the Jacobson-Scheidler-Williams adaptation of NUCOMP in a cryptographic public key-exchange protocol. We conclude showing that Schönhage’s algorithm is asymptotically the fastest but not useful in practice, JSW-NUCOMP is the fastest practical method when multiplying and reducing, and the new algorithm is the fastest method when only reduction is required.