Let R be the regulator and let D be the absolute value of the discriminant of an order 0 of an algebraic number field of unit rank 1. It is shown how the infrastructure idea of Shanks can be used to decrease the number of binary operations needed to compute R from the best known 0(RD£) for most continued fraction methods to 0(Rll2De).These ideas can also be applied to significantly decrease the number of operations needed to determine whether or not any fractional ideal of 0 is principal. 1. Introduction. In [16] Shanks introduced an idea which has since been modified and extended by Lenstra [13], Schoof [15] and Williams [17]. This idea can be used to decrease the number of binary operations needed to compute the regulator of a real quadratic order of discriminant D from 0(Dll2+e) to 0(D1/4+£) for every s > 0. In [21] and [17] Williams et al. showed that Shanks' idea could be extended to complex cubic fields.In this note we show that it can be further extended to any order 0 of an algebraic number field 7 of unit rank one, i.e., to orders of real quadratic, complex cubic, and totally complex quartic fields.We present an algorithm which computes the regulator R of 0 in 0(Rll2De) binary operations. Here, D is the absolute value of the discriminant of 0. We also describe a method for testing an arbitrary (fractional) ideal o of 0 for principality. This technique requires a number of binary operations that is 0(Rl/2De +p(m)), where p(m) is a polynomial in the input length m of a.
The BabyStep Algorithm.Let 7 have degree n over the rationals Q, and suppose 7 has s real ¿¿-isomorphisms o\, Logf = log|£|i.
