Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T23:49:22.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Analysis of the Binary Asymmetric Joint Sparse Form

Published online by Cambridge University Press:  14 July 2014

CLEMENS HEUBERGER  and
SARA KROPF
CLEMENS HEUBERGER
Affiliation:
Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65–67, 9020 Klagenfurt am Wörthersee, Austria and Institut für Optimierung und Diskrete Mathematik (Math B), TU Graz, Steyrergasse 30, 8010 Graz, Austria (e-mail: [email protected])
SARA KROPF
Affiliation:
Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt am Wörthersee, Austria (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider redundant binary joint digital expansions of integer vectors. The redundancy is used to minimize the Hamming weight, i.e., the number of non-zero digit vectors. This leads to efficient linear combination algorithms in abelian groups, which are used in elliptic curve cryptography, for instance.

If the digit set is a set of contiguous integers containing zero, a special syntactical condition is known to minimize the weight. We analyse the optimal weight of all non-negative integer vectors with maximum entry less than N. The expectation and the variance are given with a main term and a periodic fluctuation in the second-order term. Finally, we prove asymptotic normality.

Keywords

Primary 11A63Secondary 94A6068W4060F05
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 6: Honouring the Memory of Philippe Flajolet - Part 2 , November 2014 , pp. 1087 - 1113
Copyright
Copyright © Cambridge University Press 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Avanzi, R. M. (2005) A note on the signed sliding window integer recoding and a left-to-right analogue. In Selected Areas in Cryptography (Handschuh, H. and Hasan, A., eds), Vol. 3357 of Lecture Notes in Computer Science, Springer, pp. 130–143.Google Scholar
[2]Barat, G. and Grabner, P. J. (2001) Distribution of binomial coefficients and digital functions. J. London Math. Soc. (2) 64 523–547.CrossRefGoogle Scholar
[3]Flajolet, P., Grabner, P., Kirschenhofer, P., Prodinger, H. and Tichy, R. F. (1994) Mellin transforms and asymptotics: Digital sums. Theoret. Comput. Sci. 123 291–314.Google Scholar
[4]Grabner, P. and Thuswaldner, J. (2000) On the sum of digits function for number systems with negative bases. Ramanujan J. 4 201–220.CrossRefGoogle Scholar
[5]Grabner, P. J., Heuberger, C. and Prodinger, H. (2004) Distribution results for low-weight binary representations for pairs of integers. Theoret. Comput. Sci. 319 307–331.CrossRefGoogle Scholar
[6]Heuberger, C. and Muir, J. A. (2006) Minimal weight and colexicographically minimal integer representations: Online resources. http://www.math.tugraz.at/~cheub/publications/colexi/.CrossRefGoogle Scholar
[7]Heuberger, C. and Muir, J. A. (2007) Minimal weight and colexicographically minimal integer representations. J. Math. Cryptol. 1 297–328.CrossRefGoogle Scholar
[8]Muir, J. A. and Stinson, D. R. (2006) Minimality and other properties of the width-w nonadjacent form. Math. Comp. 75 369–384.Google Scholar
[9]Straus, E. (1964) Addition chains of vectors (problem 5125). Amer. Math. Monthly 71 806–808.Google Scholar
[10]Tenenbaum, G. (1997) Sur la non-dérivabilité de fonctions périodiques associées à certaines formules sommatoires. In The Mathematics of Paul Erdős I, Vol. 13 of of Algorithms Combin., Springer, pp. 117–128.Google Scholar
[11]Vaaler, J. D. (1985) Some extremal functions in Fourier analysis. Bull. Amer. Math. Soc. (NS) 12 183–216.CrossRefGoogle Scholar