Skip to main content
SpringerLink
Log in
Book cover

International Workshop on the Arithmetic of Finite Fields

WAIFI 2008: Arithmetic of Finite Fields pp 62–76Cite as

Efficient Finite Fields in the Maxima Computer Algebra System

  • Conference paper

  • 709 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5130))

Abstract

In this paper we present our implementation of finite fields in the free and open Maxima computer algebra system. In the first version of our package we focused our efforts on efficient computation of primitive elements and modular roots. Our optimizations involve some heuristic methods that use “modular composition” and the generalized Tonelli-Shanks algorithm. Other open and free systems such as GP/Pari do not include in their standard packages any support for finite fields. The computation of the primitive element in Maxima is now faster than in Axiom. Our package provides a more user-friendly interface for teaching than other comparable systems.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Brickell, E., Gordon, D., McCurley, K., Wilson, D.: Fast exponentiation with precomputation. In: Rueppel, R.A. (ed.) EUROCRYPT 1992. LNCS, vol. 658, pp. 200–207. Springer, Heidelberg (1993)

    Chapter  Google Scholar 

  2. Caruso, F., D’Aurizio, J., Mc Andrew, A.: On line manual on Finite Fields in Maxima (2007), http://www.dm.unipi.it/~caruso

  3. Chor, B., Rivest, R.L.: A knapsack-type public key cryptosystem based on arithmetic in finite fields. IEEE Trans. Inform. Theory 34(5, part 1) 901–909 (1988)

    Google Scholar 

  4. Cipolla, M.: Sulla risoluzione apiristica delle congruenze binomie secondo un modulo primo. Mathematische Annalen 63, 54–61 (1907)

    Article  MathSciNet  MATH  Google Scholar 

  5. Creutzig, C., Oevel, W.: MuPAD Tutorial, 2nd edn. Springer, Heidelberg (2004)

    Google Scholar 

  6. Daemen, J., Rijmen, V.: The design of Rijndael. In: Information Security and Cryptography. AES—the advanced encryption standard. Springer, Heidelberg (2002)

    Google Scholar 

  7. Pari Group. GP/Pari on line documentation (2003), http://pari.math.u-bordeaux.fr/

  8. Heck, A.: Introduction to Maple, 3rd edn. Springer, Heidelberg (2003)

    MATH  Google Scholar 

  9. Itoh, T., Tsujii, S.: A fast algorithm for computing multiplicative inverses in \(\mathbb{F}_{2^m}\) using normal bases. Inform. and Comput. 78(3), 171–177 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jenks, R.D., Sutor, R.S.: AXIOM. The scientific computation system, With a foreword by David V. Chudnovsky and Gregory V. Chudnovsky. Numerical Algorithms Group Ltd., Oxford (1992)

    MATH  Google Scholar 

  11. Lehmer, D.H.: Computer technology applied to the theory of numbers. In: Studies in Number Theory, pp. 117–151; Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.) (1969)

    Google Scholar 

  12. McAndrew, A.: Teaching cryptography with open-source software. In: SIGCSE 2008: Proceedings of the 39th SIGCSE technical symposium on Computer science education, pp. 325–329. ACM, New York (2008)

    Chapter  Google Scholar 

  13. McEliece, R.J.: Finite Fields for Computer Scientists and Engineers. Kluwer Academic Publishers, Boston (1987)

    MATH  Google Scholar 

  14. Menezes, A.J., van Oorschot, P.C., Vanstone, S.A.: Handbook of applied cryptography. CRC Press Series on Discrete Mathematics and its Applications. CRC Press, Boca Raton (1997) (With a foreword by Ronald L. Rivest)

    MATH  Google Scholar 

  15. Müller, S.: On the Computation of Square Roots in Finite Fields. Designs, Codes and Cryptography 31(3), 301–312 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Pohlig, S.C., Hellman, M.E.: An improved algorithm for computing logarithms over GF(p) and its cryptographic significance. IEEE Trans. Information Theory IT-24(1), 106–110 (1978)

    Article  MathSciNet  Google Scholar 

  17. Rowney, K.T., Silverman, R.D.: Finite field manipulations in Macsyma. SIGSAM Bull. 23(1), 39–48 (1989)

    Article  MATH  Google Scholar 

  18. Schelter, W.F., The Maxima Group: Maxima on line documentation (2001), http://maxima.sourceforge.net

  19. Shanks, D.: Five Number-Theoretic Algorithms. In: Proceedings of the Second Manitoba Conference on Numerical Mathematics, pp. 51–70 (1972)

    Google Scholar 

  20. Shoup, V.: New Algorithms for Finding Irredicible Polynomials Over Finite Fields. Mathematics of Computation 54(189), 435–447 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  21. Shoup, V.: Searching for Primitive Roots in Finite Fields. Math. Comp. 58(197), 369–380 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  22. Stickelberger, L.: Über eine neue Eigenschaft der Diskriminanten algebraischer Zahlkörper. In: Verhandlungen des ersten Internationalen Mathematiker-Kongresses, pp. 182–193 (1897)

    Google Scholar 

  23. Tonelli, A.: Bemerkung über die Auflösung quadratischer Congruenzen. Göttingen Nachrichten, 344–346 (1891)

    Google Scholar 

  24. Wang, Y.: On the least primitive root of a prime. Sci. Sinica 10, 1–14 (1961)

    MathSciNet  Google Scholar 

  25. Yan, S.Y.: Number Theory for Computing, 2nd edn. Springer, New York (2002)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica L. Tonelli, Università di Pisa, Pisa, Italy

    Fabrizio Caruso & Jacopo D’Aurizio

  2. School of Computer Science and Mathematics, Victoria University, Melbourne, Australia

    Alasdair McAndrew

Authors
  1. Fabrizio Caruso
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jacopo D’Aurizio
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alasdair McAndrew
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Joachim von zur Gathen José Luis Imaña Çetin Kaya Koç

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Caruso, F., D’Aurizio, J., McAndrew, A. (2008). Efficient Finite Fields in the Maxima Computer Algebra System. In: von zur Gathen, J., Imaña, J.L., Koç, Ç.K. (eds) Arithmetic of Finite Fields. WAIFI 2008. Lecture Notes in Computer Science, vol 5130. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69499-1_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-69499-1_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-69498-4

  • Online ISBN: 978-3-540-69499-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation