On the parallel generation of the residues for the continued fraction factoring algorithm

Abstract

Abstract. In order to implement the continued fraction algorithm on a highly parallel computer, like the Massively Parallel Processor, it is necessary to be able to compute certain numbers which occur at widely-spaced intervals within the continued fraction expansion of fÑ~, where N is the number to be factored. In this paper several properties of the continued fraction expansion of a quadratic irrational are developed. These results are then applied to the development of a very simple algorithm for finding the widely-spaced numbers referred to above.

Use of the rabinowitsch polynomial to determine the class groups of a real quadratic field

“…In [7] we proved, The reader is referred to [8] for further details and proofs of the above results. 1.…”

A Lower Bound for the Class Number of a Real Quadratic Field of ERD-Type

“…It is well-known (cf. [12]) that / is an ideal of OK if and only if / has a representation as / = [a, b + CUJ] where a > 0, b > 0, c | b, c \ a and ac \ N(b + ecu). In fact, for a given /, the integers c and a are uniquely determined, where a is the least positive integer in /.…”

Voronoi's algorithm in purely cubic congruence function fields of unit rank 1

“…The method is based on computing chains of successive minima in the maximal order O of the field K. An implementation in purely cubic fields was given by Williams et al [20], and improvements based on Shanks' idea of the infrastructure of the set of reduced principal integral ideals in K [13] were given in [21] and [19]. In the case of a real quadratic number field, Voronoi's method reduces to the well-known continued fraction algorithm for quadratic irrationalities given in [22] and [19]. Buchmann [1] generalized Voronoi's ideas to arbitrary number fields of unit rank 1 and 2.…”

“…Since the number of reduced fractional ideals in K is at most O( √ ∆), where ∆ is the discriminant of K (see [1]), and is thus finite, one must obtain a reduced fractional ideal a n+1 so that a n+1 = a 1 = O after at most O( √ ∆) steps, in which case θ n+1 = is the fundamental unit of K. Thus, at the heart of Voronoi's algorithm lies the problem of computing the minimum adjacent to 1 in a reduced fractional ideal. Specific implementations describing how to accomplish this, together with numerical examples, were given for the real quadratic case in [22], the purely cubic case in [20], and the totally complex quartic case in [2].…”