Distribution results for low-weight binary representations for pairs of integers

Authors:
Peter J. Grabner;Clemens Heuberger;Helmut Prodinger
Affiliations:
Institute für Mathematik A, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria;Institut für Mathematik B, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria;The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, P.O. Wits, Johannesburg 2050, South Africa
Venue:
Theoretical Computer Science - Combinatorics of the discrete plane and tilings
Year:
2004

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Abstract

We discuss an optimal method for the computation of linear combinations of elements of Abelian groups, which uses signed digit expansions. This has applications in elliptic curve cryptography. We compute the expected number of operations asymptotically (including a periodically oscillating second order term) and prove a central limit theorem. Apart from the usual right-to-left (i.e., least significant digit first) approach we also discuss a left-to-right computation of the expansions. This exhibits fractal structures that are studied in some detail.