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


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).


Elliptic curves Hyperelliptic curves Frobenius expansions 

Mathematics Subject Classification (2000)

11T71 11C08 


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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

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