Periodica Mathematica Hungarica

, Volume 55, Issue 1, pp 81–96 | Cite as

The Hamming weight of the non-adjacent-form under various input statistics

  • Clemens HeubergerEmail author
  • Helmut Prodinger


The Hamming weight of the non-adjacent form is studied in relation to the Hamming weight of the standard binary expansion. In particular, we investigate the expected Hamming weight of the NAF of an n-digit binary expansion with k ones where k is either fixed or proportional to n. The expected Hamming weight of NAFs of binary expansions with large (≥ n/2) Hamming weight is studied. Finally, the covariance of the Hamming weights of the binary expansion and the NAF is computed. Asymptotically, these Hamming weights become independent and normally distributed.

Key words and phrases

Non-adjacent form binary expansion Hamming weight transducer generating function Omega operator singularity analysis quasi-power theorem multivariate asymptotics 

Mathematics subject classification number

11A63 68W40 68Q45 05A16 05A15 


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  1. [1]
    G. E. Andrews, P. Paule and A. Riese, MacMahon’s partition analysis: the Omega package, European J. Combin., 22 (2001), 887–904.zbMATHCrossRefGoogle Scholar
  2. [2]
    S. Arno and F. S. Wheeler, Signed digit representations of minimal hamming weight, IEEE Trans. Comp., 42 (1993), 1007–1010.CrossRefGoogle Scholar
  3. [3]
    E. A. Bender, Central and local limit theorems applied to asymptotic enumeration, J. Combinatorial Theory Ser. A, 15 (1973), 91–111.zbMATHCrossRefGoogle Scholar
  4. [4]
    E. A. Bender and L. B. Richmond, Central and local limit theorems applied to asymptotic enumeration. II. Multivariate generating functions, J. Combin. Theory Ser. A, 34 (1983), 255–265.zbMATHCrossRefGoogle Scholar
  5. [5]
    M. Drmota, Asymptotic distributions and a multivariate Darboux method in enumeration problems, J. Combin. Theory Ser. A, 67 (1994), 169–184.zbMATHCrossRefGoogle Scholar
  6. [6]
    Ph. Flajolet and A. Odlyzko, Singularity analysis of generating functions, SIAM J. Discrete Math., 3 (1990), 216–240.zbMATHCrossRefGoogle Scholar
  7. [7]
    Ph. Flajolet and R. Sedgewick, Analytic combinatorics, in preparation, preprint available at
  8. [8]
    P. J. Grabner, C. Heuberger and H. Prodinger, Subblock occurrences in signed digit representations, Glasgow Math. J., 45 (2003), 427–440.zbMATHCrossRefGoogle Scholar
  9. [9]
    P. J. Grabner, C. Heuberger, H. Prodinger and J. Thuswaldner, Analysis of linear combination algorithms in cryptography, ACM Trans. Algorithms, 1 (2005), 123–142.CrossRefGoogle Scholar
  10. [10]
    C. Heuberger, Hwang’s quasi-power-theorem in dimension two, preprint available at
  11. [11]
    C. Heuberger and H. Prodinger, On minimal expansions in redundant number systems: Algorithms and quantitative analysis, Computing, 66 (2001), 377–393.zbMATHCrossRefGoogle Scholar
  12. [12]
    C. Heuberger and H. Prodinger, Analysis of alternative digit sets for nonadjacent representations, Monatsh. Math., 147 (2006), 219–248.zbMATHCrossRefGoogle Scholar
  13. [13]
    H.-K. Hwang, On convergence rates in the central limit theorems for combinatorial structures, European J. Combin., 19 (1998), 329–343.zbMATHCrossRefGoogle Scholar
  14. [14]
    P. A. MacMahon, Combinatory analysis, Cambridge University Press, Cambridge, 1915–1916, (Reprinted: Chelsea, New York, 1960).zbMATHGoogle Scholar
  15. [15]
    F. Morain and J. Olivos, Speeding up the computations on an elliptic curve using addition-subtraction chains, RAIRO Inform. Théor. Appl., 24 (1990), 531–543.zbMATHGoogle Scholar
  16. [16]
    G. W. Reitwiesner, Binary arithmetic, Advances in computers, vol. 1, Academic Press, New York, 1960, 231–308.Google Scholar
  17. [17]
    J. M. Thuswaldner, Summatory functions of digital sums occurring in cryptography, Period. Math. Hungar., 38 (1999), 111–130.zbMATHCrossRefGoogle Scholar
  18. [18]
    J. H. Van Lint, Introduction to coding theory, 2nd ed., Graduate Texts in Mathematics, vol. 86, Springer, 1992.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Institut für Mathematik BTechnische Universität GrazGraz CityAustria
  2. 2.Department of MathematicsUniversity of StellenboschStellenboschSouth Africa

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