Efficient Computation of Singular Moduli with Application in Cryptography

  • Harald Baier
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2138)


We present an implementation that turns out to be most efficient in practice to compute singular moduli within a fixed floating point precision. First, we show how to efficiently determine the Fourier coefficients of the modular function j and related functions γ2, f2, and η. Comparing several alternative methods for computing singular moduli, we show that in practice the computation via the η-function turns out to be the most efficient one. An important application with respect to cryptography is that we can speed up the generation of cryptographically strong elliptic curves using the Complex Multiplication Approach.


class group complex multiplication cryptography elliptic curve Fourier series modular function ring class polynomial singular modulus 


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  1. [AM93]
    A.O.L. Atkin and F. Morain. Elliptic curves and primality proving. Mathematics of Computation, 61:29–67, 1993.MATHCrossRefMathSciNetGoogle Scholar
  2. [Bai01]
    H. Baier. Efficient Computation of Fourier Series and Singular Moduli with Application in Cryptography. Technical Report, Darmstadt University of Technology, 2001.Google Scholar
  3. [BB00]
    H. Baier and J. Buchmann. Efficient Construction of Cryptographically Strong Elliptic Curves. In Progress in Cryptology-INDOCRYPT2000, LNCS 1977, pages 191–202, Berlin, 2000. Springer-Verlag.Google Scholar
  4. [BSI00]
    Geeignete Kryptoalgorithmen gemäß §17(2) SigV, April 2000. Bundesamt für Sicherheit in der Informationstechnik.Google Scholar
  5. [Cox89]
    D. Cox. Primes of the form x 2 + ny 2. John Wiley & Sons, 1989.Google Scholar
  6. [Kan]
    M. Kaneko. Traces of singular moduli and the Fourier coefficients of the elliptic modular function j(τ). private communicated.Google Scholar
  7. [Kob93]
    N. Koblitz. Introduction to Elliptic Curves and Modular Forms. Springer-Verlag, 1993.Google Scholar
  8. [LiDIA]
    LiDIA. A library for computational number theory. Darmstadt University of Technology. URL: http://www.informatik.tu-darmstadt.de/TI/LiDIA/Welcome.html.
  9. [LZ94]
    G.-J. Lay and H.G. Zimmer. Constructing elliptic curves with given group order over large finite fields. In Proceedings of ANTS I, LNCS 877, pages 250–263, 1994.Google Scholar
  10. [Mah76]
    K. Mahler. On a class of non-linear functional equations connected with modular functions. Journal of the Australian Mathematical Society, 22, Series A:65–120, 1976.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Harald Baier
    • 1
  1. 1.Department of Computer ScienceDarmstadt University of TechnologyDarmstadtGermany

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