Skip to main content
SpringerLink
Log in

Exponentiation in finite fields: Theory and practice

  • Conference paper
  • First Online:
Book cover Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 1997)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1255))

Abstract

Finally we want to outline the main properties for a fast software exponentiation algorithm in \(\mathbb{F}_{2^n }\)for large n∈ℕ:

  1. 1.

    The algorithm should use fast polynomial multiplication. Neither multiplication by multiplication tensors nor classical polynomial arithmetic is fast enough.

  2. 2.

    The algorithm should be based upon an addition chain for the exponent e with a small number of non-doubling steps.

  3. 3.

    The algorithm should offer a cheap way to compute \(\alpha ^{2^m }\)for m∈ℕ and \(\alpha \in \mathbb{F}_{2^n }\). Both Shoup's and Gao et al.'s algorithm achieve this.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • G. B. Agnew, R. C. Mullin, and S. A. Vanstone, Fast exponentiation in GF(2n). In Advances in Cryptology—EUROCRYPT '88, ed. C. G. Günther, vol. 330 of Lecture Notes in Computer Science, 251–255. Springer, Berlin, 1988.

    Google Scholar 

  • I. Bocharova and B. Kudryashov, Fast exponentiation in cryptography. In Proceedings Applied algebra, algebraic algorithms and error correcting codes: 11th International Symposium AAECC, ed. G. Cohen, Lecture notes in computer science 948, Berlin, 1995, Springer, 146–157.

    Google Scholar 

  • A. Brauer, On addition chains. Bull. Amer. Math. Soc.45 (1939), 736–739.

    Google Scholar 

  • R. P. Brent and H. T. Kung, Fast algorithms for manipulating formal power series. J. Assoc. Comput. Mach.25 (1978), 581–595.

    Google Scholar 

  • E. Brickell, D. Gordon, K. McCurley, and D. Wilson, Fast exponentiation with precompuation. In Advances in cryptology: Proceedings EUROCRYPT '92, ed. R. Rueppel, Lecture notes in computer science 658, Berlin, 1993, Springer, 200–207.

    Google Scholar 

  • D. G. Cantor, On arithmetical algorithms over finite fields. Journal of Combinatorial Theory, Series A50 (1989), 285–300.

    Google Scholar 

  • D. G. Cantor and E. Kaltofen, On fast multiplication of polynomials over arbitrary algebras. Acta. Inform.28 (1991), 693–701.

    Google Scholar 

  • D. Coppersmith and S. Winograd, Matrix multiplication via arithmetic progressions. J. Symb. Comp.9 (1990), 251–280.

    Google Scholar 

  • W. Diffie and M. E. Hellman, New directions in cryptography. IEEE Trans. Inform. Theory22 (1976), 644–654.

    Article  Google Scholar 

  • P. Downey, B. Leong, and R. Sethi, Computing sequences with addition chains. SIAM J. Comput.10(3) (1981), 638–646.

    Google Scholar 

  • T. ElGamal, A public key cryptosystem and a signature scheme based on discrete logarithms. IEEE Transactions on information theoryIT-31(4) (1985), 469–472.

    Article  Google Scholar 

  • S. Gao and H. W. Lenstra, Jr., Optimal normal bases. Designs, Codes, and Cryptography2 (1992), 315–323.

    Google Scholar 

  • S. Gao, J. von Zur Gathen, and D. Panario, Gauss periods and fast exponentiation in finite fields. In Proc. Latin '95, Valparaiso, Chile, Springer Lecture Notes in Computer Science 911, 1995, 311–322.

    Google Scholar 

  • J. von Zur Gathen, Efficient and optimal exponentiation in finite fields. Comput complexity1 (1991), 360–394.

    Google Scholar 

  • J. von Zur Gathen and J. Gerhard, Arithmetic and factorization of polynomials over \(\mathbb{F}_2\). In Proc. ISSAC '96, Zürich, Switzerland. ACM press, 1996, 1–9.

    Google Scholar 

  • Joachim von Zur Gathen and Sandra Schlink, Normal bases via general Gauss periods. Reihe Informatik tr-ri-96-177, Universität-Gesamthochschule Paderborn, 1996.

    Google Scholar 

  • J. von Zur Gathen and V. Shoup, Computing Frobenius maps and factoring polynomials. Computational complexity 2 (1992), 187–224.

    Google Scholar 

  • W. Geiselmann, Algebraische Algorithmenentwicklung am Beispiel der Arithmetik in endlichen Körpern. Dissertation, Universität Karlsruhe, Aachen, 1994.

    Google Scholar 

  • T. Itoh and S. Tsujii, A fast algorithm for computing multiplicative inverses in GF(2m) using normal bases. Information and Computation78 (1988), 171–177.

    Google Scholar 

  • D. Jungnickel, Finite Fields: Structure and Arithmetics. BI Wissenschaftsverlag, Mannheim, 1993.

    Google Scholar 

  • A. Karatsuba and Y. Ofman, УМНожеНИе МНогозНаЧНЫх чИсел На автОМатах. Dokl. Akad. Nauk USSR145 (1962), 293–294. Multiplication of multidigit numbers on automata, Soviet Physics-Doklady 7 (1963), 595–596.

    Google Scholar 

  • D. E. Knuth, The Art of Computer Programming, Vol.2, Seminumerical Algorithms. Addison-Wesley, Reading MA, 2 edition, 1981.

    Google Scholar 

  • D. H. Lehmer, Euclid's algorithm for large numbers. American Mathematical Monthly45 (1938), 227–233.

    Google Scholar 

  • R. Lidl and H. Niederreiter, Finite Fields. Encyclopedia of Mathematics and its Applications 20. Addison-Wesley, Reading MA, 1983.

    Google Scholar 

  • M. Lothaire, Combinatorics on Words. Addison-Wesley Reading, MA, 1983.

    Google Scholar 

  • Alfred J. Menezes, Ian F. Blake, XuHong Gao, Ronald C. Mullin, Scott A. Vanstone, and Tomik Yaghoobian, Applications af finite fields. Kluwer Academic Publishers, Norwell MA, 1993.

    Google Scholar 

  • R. C. Mullin, I. M. Onyszchuk, S. A. Vanstone, and R. M. Wilson, Optimal normal bases in GF(pn). Discrete Applied Math.22 (1989), 149–161.

    Google Scholar 

  • A. Odlyzko, Discrete logarithms and their cryptographic significance. In Advances in Cryptology, Proceedings of Eurocrypt 1984. Springer-Verlag, 1985, 224–314.

    Google Scholar 

  • R. L. Rivest, A. Shamir, and L. M. Adleman, A method for obtaining digital signatures and public-key cryptosystems. Comm. ACM21 (1978), 120–126.

    Article  Google Scholar 

  • P. de Rooij, Efficient exponentiation using precomputation and vector addition chains. In Advances in cryptology: Proceedings EUROCRYPT '94, ed. A. DeSantis, Lecture notes in computer science 950, Berlin, 1995, Springer, 389–399.

    Google Scholar 

  • A. Schönhage, Schnelle Berechnung von Kettenbruchentwicklungea. Acta Informatica1 (1971), 139–144.

    Google Scholar 

  • A. Schönhage, A lower bound for the length of addition chains. Theor. Computer Science1 (1975), 1–12.

    Google Scholar 

  • A. Schönhage, Schnelle Multiplikation von Polynomen über Körpern der Charakteristik 2. Acta Inf.7 (1977), 395–398.

    Google Scholar 

  • A. Schönhage and V. Strassen, Schnelle Multiplikation großer Zahlen. Computing7 (1971), 281–292.

    Google Scholar 

  • V. Shoup, Exponentiation in GF(2 n) using fewer polynomial multiplications. Preprint, 1994.

    Google Scholar 

  • D. R. Stinson, Some observations on parallel algorithms for fast exponentiation in GF(2n). SIAM J. Comput.19 (1990), 711–717.

    Google Scholar 

  • V. Strassen, Gaussian elimination is not optimal. Numer. Mathematik13 (1969), 354–356.

    Google Scholar 

  • V. Strassen, The computational complexity of continued fractions. SIAM J. Comput.12 (1983), 1–27.

    Google Scholar 

  • B. P. Tunstall, Synthesis of noiseless compression codes. Ph.D. dissertation, Georgia Inst. Technol., 1968.

    Google Scholar 

  • A. Wassermann, Zur Arithmetik in endlichen Körpern. Bayreuther Math. Schriften44 (1993), 147–251.

    Google Scholar 

  • Y. Yacobi, Exponentiating faster with addition chains. In Advances in cryptology: Proceedings EUROCRYPT '90, ed. I. Damgard, Lecture notes in computer science 473, Berlin, 1991, Springer, 222–229.

    Google Scholar 

  • J. Ziv and A. Lempel, Compression of individual sequences via variable-rate coding. IEEE Trans. Inform. TheoryIT-24(5) (1978), 530–536.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich 17 Mathematik-Informatik, Universität-GH Paderborn, D-33095, Paderborn, Germany

    Joachim von zur Gathen & Michael Nöcker

Authors
  1. Joachim von zur Gathen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Nöcker
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Teo Mora Harold Mattson

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

von zur Gathen, J., Nöcker, M. (1997). Exponentiation in finite fields: Theory and practice. In: Mora, T., Mattson, H. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1997. Lecture Notes in Computer Science, vol 1255. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63163-1_8

Download citation

  • DOI: https://doi.org/10.1007/3-540-63163-1_8

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63163-7

  • Online ISBN: 978-3-540-69193-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation