Advertisement

Designs, Codes and Cryptography

, Volume 61, Issue 1, pp 71–89 | Cite as

On the distribution of the coefficients of normal forms for Frobenius expansions

  • Roberto Avanzi
  • Waldyr Dias BenitsJr
  • Steven D. GalbraithEmail author
  • James McKee
Article
  • 98 Downloads

Abstract

Frobenius expansions are representations of integers to an algebraic base which are sometimes useful for efficient (hyper)elliptic curve cryptography. The normal form of a Frobenius expansion is the polynomial with integer coefficients obtained by reducing a Frobenius expansion modulo the characteristic polynomial of Frobenius. We consider the distribution of the coefficients of reductions of Frobenius expansions and non-adjacent forms of Frobenius expansions (NAFs) to normal form. We give asymptotic bounds on the coefficients which improve on naive bounds, for both genus one and genus two. We also discuss the non-uniformity of the distribution of the coefficients (assuming a uniform distribution for Frobenius expansions).

Keywords

Elliptic curves Hyperelliptic curves Frobenius expansions 

Mathematics Subject Classification (2000)

11T71 11C08 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Avanzi R., Cohen H., Doche C., Frey G., Lange T., Nguyen K., Vercauteren F.: Handbook of elliptic and hyperelliptic cryptography. Chapman and Hall/CRC, Boca Raton (2006)zbMATHGoogle Scholar
  2. Avanzi R., Heuberger C., Prodinger H.: Minimality of the Hamming weight of the τ-NAF for Koblitz curves and improved combination with point halving. In: Preneel B., Tavares S.E. (eds.) SAC 2005, LNCS, vol. 3897, pp. 332–344. Springer, Heidelberg (2006)Google Scholar
  3. Avanzi R., Heuberger C., Prodinger H.: On Redundant τ-adic Expansions and Non-Adjacent Digit Sets. In: Biham E., Youssef A.M. (eds.) SAC 2006, LNCS, vol. 4356, pp. 285–301. Springer, Heidelberg (2007)Google Scholar
  4. Benits Jr. W.D.: Applications of Frobenius expansions in elliptic curve cryptography, PhD thesis, Royal Holloway University of London, London (2008).Google Scholar
  5. Brumley B.B., Järvinen K.: Koblitz curves and integer equivalents of Frobenius expansions. In: Adams C., Miri A., Wiener M. (eds.), SAC 2007, LNCS, vol. 4876, pp. 126–137, Springer, Heidelberg (2007).Google Scholar
  6. Ebeid N., Hasan M.: On τ-adic representations of integers. Des. Codes Cryptogr. 45(3), 271–296 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. Galbraith S.D., Ruprai R.S.: An improvement to the Gaudry–Schost algorithm for multidimensional discrete logarithm problems. In: Parker M. (ed.) Cryptography and Coding, LNCS, vol. 5921, pp. 368–382, Springer, Heidelberg (2009).Google Scholar
  8. Gaudry P., Schost E.: A low-memory parallel version of Matsuo, Chao and Tsujii’s algorithm. In: Buell D.A. (ed.) ANTS VI, LNCS, vol. 3076, pp. 208–222, Springer, Heidelberg (2004).Google Scholar
  9. Günther C., Lange T., Stein A.: Speeding up the arithmetic on Koblitz curves of genus two. In: Stinson D.R., Tavares S.E. (eds.), SAC 2000, LNCS, vol. 2012, pp. 106–117, Springer, Heidelberg (2000).Google Scholar
  10. Heuberger C.: Redundant τ-adic expansions II: non-optimality and chaotic behaviour. Math. Comput. Sci. 3(2), 141–157 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. Lange T.: Koblitz curve cryptosystems. Finite Field. Appl. 11(2), 200–229 (2005)zbMATHCrossRefGoogle Scholar
  12. Lange T., Shparlinski I.: Collisions in fast generation of ideal classes and points on hyperelliptic and elliptic curves. Appl. Algebra Eng. Commun. Comput. 15(5), 329–337 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  13. Lange T., Shparlinski I.: Distribution of some sequences of points on elliptic curves. J. Math. Cryptol. 1(1), 1–11 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  14. Koblitz N.: Elliptic curve cryptosystems. Math. Comp. 48(177), 203–209 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  15. Koblitz N.: Hyperelliptic cryptosystems. J. Cryptol. 1, 139–150 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  16. Koblitz N.: CM curves with good cryptographic properties. In: Feigenbaum J. (ed.) CRYPTO ’91, LNCS, vol. 576, pp. 279–287, Springer, Heidelberg (1992).Google Scholar
  17. Meier W., Staffelbach O.: Efficient multiplication on certain nonsupersingular elliptic curves. In: Brickell E.F. (ed.) CRYPTO ’92, LNCS, vol. 740, pp. 333–344, Springer, Heidelberg (1993).Google Scholar
  18. Müller V.: Fast Multiplication on Elliptic Curves over Small Fields of Characteristic Two. J. Cryptol. 11(4), 219–234 (1998)zbMATHCrossRefGoogle Scholar
  19. Silverman J.H.: The arithmetic of elliptic curves. Graduate texts in mathematics, vol. 106. Springer-Verlag, New York (1986)Google Scholar
  20. Smart N.P.: Elliptic curve cryptosystems over small fields of odd characteristic. J. Cryptol. 12(2), 141–151 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  21. Solinas J.: An improved algorithm for arithmetic on a family of elliptic curves. In: Kaliski Jr. B.S. (ed.), CRYPTO ’97, LNCS, vol. 1294, pp. 357–371, Springer, Heidelberg (1997).Google Scholar
  22. Solinas J.: Efficient arithmetic on Koblitz curves. Des. Codes Cryptogr. 19(2–3), 195–249 (2000)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Roberto Avanzi
    • 1
  • Waldyr Dias BenitsJr
    • 2
  • Steven D. Galbraith
    • 3
    Email author
  • James McKee
    • 4
  1. 1.Faculty of MathematicsRuhr-University BochumBochumGermany
  2. 2.Centro de Analises de Sistemas NavaisBrazilian NavyRio de JaneiroBrazil
  3. 3.Mathematics DepartmentAuckland UniversityAucklandNew Zealand
  4. 4.Mathematics Department, Royal HollowayUniversity of LondonEgham, SurreyUnited Kingdom

Personalised recommendations