Advertisement

Gauss periods and fast exponentiation in finite fields

Extended abstract
  • Shuhong Gao
  • Joachim von zur Gathen
  • Daniel Panario
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 911)

Abstract

Gauss periods can be used to implement finite field arithmetic efficiently. For a small prime p and infinitely many integers n, exponentiation of an arbitrary element in F p n can be done with O(n2 loglog n) operations in F p , and exponentiation of a Gauss period with O(n2) operations in F p . Comparing to the previous estimate O(n2 log nloglog n), using polynomial bases, this shows that normal bases generated by Gauss periods offer some asymptotic computational advantage. Experimental results indicate that Gauss periods are often primitive elements in finite fields.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L.M. Adleman and H.W. Lenstra, Jr., “Finding irreducible polynomials over finite fields”, Proc. 18th Annual ACM Symp. on Theory of Computing (1986), 350–355.Google Scholar
  2. G.B. Agnew, R.C. Mullin, I.M. Onyszchuk and S.A. Vanstone, “An implementation for a fast public key cryptosystem”, J. of Cryptology3 (1991), 63–79.Google Scholar
  3. G.B. Agnew, R.C. Mullin and S.A. Vanstone, “An implementation of elliptic curve cryptosystems over F2155”, IEEE J. on Selected Areas in Communications11 (1993), 804–813.CrossRefGoogle Scholar
  4. E. Artin, Collected Papers, Addison-Wesley, 1965.Google Scholar
  5. D.W. Ash, I.F. Blake and S.A. Vanstone, “Low complexity normal bases”, Discrete Applied Math.25 (1989), 191–210.Google Scholar
  6. E. Bach and J. Shallit, “Factoring with cyclotomic polynomials”, Math. Comp.52 (1989), 201–219.Google Scholar
  7. E.R. Berlekamp, “Bit-serial Reed-Solomon encoders”, IEEE Trans. Info. Th.28 (1982), 869–874.Google Scholar
  8. I.F. Blake, S. Gao and R.C. Mullin, “Specific irreducible polynomials with linearly independent roots over finite fields”, submitted to Linear Algebra and Its Applications, 1993.Google Scholar
  9. E.F. Brickell, D.M. Gordon, K.S. McCurley and D.B. Wilson, “Fast exponentiation with precomputation”, in Proc. Eurocrypt'92, Balatonfured, Hungary, 1992.Google Scholar
  10. J. Brillhart, D.H. Lehmer, J.L. Selfridge, B. Tuckerman and S.S. Wagstaff, “Factorizations of b n±1, b=2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers”, Vol. 22 of Contemporary Mathematics, AMS, 1988, 2nd edition.Google Scholar
  11. D.G. Cantor and E. Kaltofen, “On fast multiplication of polynomials over arbitrary algebras”, Acta Inform. 28 (1991), 693–701.Google Scholar
  12. L. Carlitz, “Primitive roots in finite fields”, J. London Math. Soc.43 (1952), 373–382.Google Scholar
  13. H. Davenport, “Bases for finite fields”, J. London Math. Soc.43 (1968), 21–39.Google Scholar
  14. S. Gao and H.W. Lenstra, Jr., “Optimal normal bases”, Designs, Codes and Cryptography2 (1992), 315–323.Google Scholar
  15. S. Gao and S.A. Vanstone, “On orders of optimal normal basis generators”, 1994, to appear in Mathematics of Computation.Google Scholar
  16. J. von zur Gathen, “Efficient and optimal exponentiation in finite fields”, computational complexity1 (1991), 360–394.Google Scholar
  17. J. von zur Gathen and M. Giesbrecht, “Constructing normal bases in finite fields”, J. Symb. Comp.10 (1990), 547–570.Google Scholar
  18. C.F. Gauss, Disquisitiones Arithmeticae, Braunschweig, 1801. English Edition, Springer-Verlag, 1986.Google Scholar
  19. W. Geiselmann and D. Gollmann, “Symmetry and duality in normal basis multiplication”, AAECC-6, Lecture Notes in Computer Science 357 (1989), Springer-Verlag, 230–238.Google Scholar
  20. C. Hooley, “On Artin's conjecture”, J. reine angew. Math.226 (1967), 209–220.Google Scholar
  21. D. Jungnickel, Finite Fields: Structure and Arithmetics, Bibliographisches Institut, Mannheim, 1993.Google Scholar
  22. H.W. Lenstra, Jr. and R.J. Schoof, “Primitive normal bases for finite fields”, Math. Comp.48 (1987), 217–231.Google Scholar
  23. A.J. Menezes, I.F. Blake, X. Gao, R.C. Mullin, S.A. Vanstone and T. Yaghoobian, Applications of Finite Fields, Kluwer Academic Publishers, Boston-Dordrecht-Lancaster, 1993.Google Scholar
  24. R.C. Mullin, I.M. Onyszchuk, S.A. Vanstone and R.M. Wilson, “Optimal normal bases in GF(pn)”, Discrete Applied Math.22 (1988/1989), 149–161.Google Scholar
  25. M. Pohst and H. Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge University Press, 1989.Google Scholar
  26. A. Schönhage, “Schnelle Multiplikation von Polynomen über Körpern der Charakteristik 2”, Acta Inf.7 (1977), 395–398.Google Scholar
  27. A. Schönhage and V. Strassen, “Schnelle Multiplikation großer Zahlen”, Computing7 (1971), 281–292.Google Scholar
  28. V. Shoup, “Exponentiation in GF(2n) using fewer polynomial multiplications”, preprint, 1994.Google Scholar
  29. S.A. Stepanov and I.E. Shparlinskiy, “On construction of primitive elements and primitive normal bases in a finite field”, in Computational Number Theory, ed. A. Pethő, M.E. Pohst, H.C. Williams and H.G. Zimmer, 1991. (Proc. Colloq. Comp. Number Theory, Hungary, 1990).Google Scholar
  30. D.H. Stinson, “Some observations on parallel algorithms for fast exponentiation in GF(2n)”, SIAM J. Computing19 (1990), 711–717.Google Scholar
  31. T. Storer, Cyclotomy and Difference Sets, Markham, Chicago, 1967.Google Scholar
  32. V. Strassen, “Gaussian elimination is not optimal”, Numer. Mathematik13 (1969), 354–356.Google Scholar
  33. C.C. Wang, “An algorithm to design finite field multipliers using a self-dual normal basis”, IEEE Trans. Comput.38 (1989), 1457–1460.Google Scholar
  34. L.C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York, 1982.Google Scholar
  35. A. Wassermann, “Konstruktion von Normalbasen”, Bayreuther Mathematische Schriften31 (1990), 155–164.Google Scholar
  36. A. Wassermann, “Zur Arithmetik in endlichen Körpern”, Bayreuther Mathematische Schriften44 (1993), 147–251.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Shuhong Gao
    • 1
  • Joachim von zur Gathen
    • 1
  • Daniel Panario
    • 1
  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada

Personalised recommendations