Abstract
In 1989, Shanks introduced the NUCOMP algorithm [10] for computing the reduced composite of two positive definite binary quadratic forms of discriminant Δ. Essentially by applying reduction before composing the two forms, the intermediate operands are reduced from size O(Δ) to O(Δ 1/2) in most cases and at worst to O(Δ 3/4). Shanks made use of this to extend the capabilities of his hand-held calculator to computations involving forms with discriminants with as many as 20 decimal digits, even though his calculator had only some 10 digits precision. Improvements by Atkin (described in [3], [4]) have also made NUCOMP very effective for computations with forms of larger discriminant.
The first author acknowledges the assistance of an Australian Research Council International Research Exchange Grant, held by the second author, in allowing him to visit Macquarie University, Sydney, thus initiating the present work.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. Buchmann and H.C. Williams, A key-exchange system based on imaginary quadratic fields, Journal of Cryptology 1 (1988), 107–118.
D.G. Cantor, Computing in the Jacobian of a hyperelliptic curve, Math. Comp. 48 (1987), no. 177, 95–101.
H. Cohen, A course in computational algebraic number theory, Springer-Verlag, Berlin, 1993.
S. Düllmann, Ein Algorithmus zur Bestimmung der Klassengruppe positiv definiter binärer quadratischer Formen, Ph.D. thesis, Universität des Saarlandes, Saarbrücken, Germany, 1991.
M.J. Jacobson, Jr., Subexponential class group computation in quadratic orders, Ph.D. thesis, Technische Universität Darmstadt, Darmstadt, Germany, 1999.
M.J. Jacobson, Jr., R. Scheidler, and H.C. Williams, The efficiency and security of a real quadratic field based key exchange protocol, Public-Key Cryptography and Computational Number Theory (Warsaw, Poland), de Gruyter, 2001.
N. Koblitz, Hyperelliptic crypto systems, Journal of Cryptology 1 (1989), 139–150.
R. Scheidler, J. Buchmann, and H.C. Williams, A key-exchange protocol using real quadratic fields, Journal of Cryptology 7 (1994), 171–199.
R. Scheidler, A. Stein, and H.C. Williams, Key-exchange in real quadratic congruence function fields, Designs, Codes and Cryptography 7 (1996), 153–174.
D. Shanks, On Gauss and composition I, II, Proc. NATO ASI on Number Theory and Applications (R.A. Mollin, ed.), Kluwer Academic Press, 1989, pp. 163–179.
V. Shoup, NTL: A library for doing number theory, Software, 2001; see http://-www.shoup.net/ntl.
A. Stein, Sharp upper bounds for arithmetics in hyperelliptic function fields, J. Ramanujan Math. Soc. 16 (2001), no. 2, 1–86.
A.J. van der Poorten, A note on NUCOMP, to appear in Math. Comp.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Jacobson, M.J., van der Poorten, A.J. (2002). Computational Aspects of NUCOMP. In: Fieker, C., Kohel, D.R. (eds) Algorithmic Number Theory. ANTS 2002. Lecture Notes in Computer Science, vol 2369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45455-1_10
Download citation
DOI: https://doi.org/10.1007/3-540-45455-1_10
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43863-2
Online ISBN: 978-3-540-45455-7
eBook Packages: Springer Book Archive