Skip to main content
SpringerLink
Log in

Applications of Arithmetical Geometry to Cryptographic Constructions

  • Conference paper
Finite Fields and Applications

Abstract

Public key cryptosystems are very important tools for data transmission. Their performance and security depend on the underlying crypto primitives. In this paper we describe one such primitive: The Discrete Logarithm (DL) in cyclic groups of prime order (Section 1).

To construct DL-systems we use methods from algebraic and arithmetic geometry and especially the theory of abelian varieties over finite fields. It is explained why Jacobian varieties of hyperelliptic curves of genus ≤ 4 are candidates for cryptographically “good” abelian varieties (Section 2).

In the third section we describe the (constructive and destructive) role played by Galois theory: Local and global Galois representation theory is used to count points on abelian varieties over finite fields and we give some applications of Weil descent and Tate duality.

I would like to thank the organizers for the opportunity to participate in the F q 5− conference, which, because of their great expertise and warm hospitality, became a very interesting and inspiring event.

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

Access this chapter

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 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L. Adleman, J. DeMarrais and M. Huang, A Subexponential Algorithm for Discrete Logarithm over the Rational Subgroup of the Jacobians of Large Genus Hyperelliptic Curves over Finite Fields, Algorithmic Number Theory — Proceedings of the First International Symposium, ANTS-I, Springer Verlag (1994), 28–40.

    Google Scholar 

  2. J. Basmaji, Ein Algorithmus zur Berechnung von Hecke-Operatoren und Anwendung auf modulare Kurven, Dissertation Essen (1996).

    Google Scholar 

  3. W. Difiie and M. Hellman, New directions in cryptography, IEEE Trans. Inform. Theory 22 (1976), 644–654.

    Article  Google Scholar 

  4. A. Enge, Computing discrete logarithms in high-genus hyperelliptic Jacobians in provably subexponential time, CORR 99-04, Univ. of Waterloo, (Febr. 1999), to appear in Math. Comp.

    Google Scholar 

  5. A. Enge and A. Stein, Smooth Ideals in Hyperelliptic Function Fields, CORR 2000-08, Univ. of Waterloo, (Febr. 2000).

    Google Scholar 

  6. A. Enge and P. Gaudry, A General Framework for Subexponential Discrete Logarithm Algorithms, Manuscript 19 pp (Febr. 2000).

    Google Scholar 

  7. G. Frey, Weil Restriction, ECC Waterloo (1998).

    Google Scholar 

  8. G. Frey, M. Müller and H.-G. Rück, The Tate Pairing and the Discrete Logarithm Applied to Elliptic Curve Cryptosystems, Trans, on Inf. Th., IEEE, 45 (1999), 1717–1719.

    Article  MATH  Google Scholar 

  9. G. Prey and H.-G. Rück, A Remark Concerning m-Divisibility and the Discrete Logarithm in the Divisor Class Group of Curves, Math. of Comp., 62 (1994), 865–874.

    Google Scholar 

  10. P. Gaudry, A variant of the Adleman-DeMarrais-Huang algorithm and its application to small genera, Laboratoire d’ Informatique Preprint LIX/RR/99/04 (1999).

    Google Scholar 

  11. P. Gaudry, F. Hess and N. Smart, Constructive and Destructive Facets of Weil Descent on Elliptic Curves, Preprint (2000).

    Google Scholar 

  12. S. Galbraith and N. Smart, A cryptographie appUcation of Weil descent, in: Codes and Cryptography, M. Walker ed., Springer LNCS 1746 (1999), 191–200.

    Google Scholar 

  13. J.-I. Igusa, The arithmetic variety of genus two, Ann. Math. 72 (1960), 612–649.

    Article  MATH  MathSciNet  Google Scholar 

  14. W. Kampkötter, Explizite Gleichungen für Jacobische Varietäten hyperelhptischer Kurven, Dissertation Essen (1991).

    Google Scholar 

  15. N. Koblitz, Hyperelliptic Cryptosystems, J. Cryptology 1 (1989),139–150.

    Article  MATH  MathSciNet  Google Scholar 

  16. U. Krieger, signature.c-Anwendungen hyperelliptischer Kurven in der Kryptographie, Diplomarbeit Essen (1997).

    Google Scholar 

  17. H. Lange and W. Ruppert, Complete systems of addition laws on abelian varieties. Invent. Math., 79 (1985), 603–610.

    Article  MATH  MathSciNet  Google Scholar 

  18. A.K. Lenstra and E.R. Verheul, Selecting Cryptographic Key Sizes, manuscript, (Oct.1999).

    Google Scholar 

  19. S. Lichtenbaum, Duality Theorems for Curves over p-adic fields, Inv. math., 7 (1969), 120–136.

    Article  MATH  MathSciNet  Google Scholar 

  20. J.-F. Mestre, Construction de courbes de genre 2 à partir de leurs modules, in: Progress in Math., 94, Birkhäuser Verlag (1991), 313–334.

    MathSciNet  Google Scholar 

  21. F. Morain, Calcul du nombre de points sur une courbe elliptique dans un corps fini: aspects algorithmiques, J. Th. des Nombres de Bordeaux, 7 (1995), 111–137.

    Article  MathSciNet  Google Scholar 

  22. A. Menezes, P. van Oorschot and S. Vanstone, Handbook of Applied Cryptography, CRC Press (1995).

    Google Scholar 

  23. V. Müller, A. Stein and A. Thiel, Computing discrete logarithms in real quadratic congruence function fields. Math. Comp., 68 (226) (1999), 807–822.

    Article  MATH  MathSciNet  Google Scholar 

  24. D. Mumford, On the equations defining abelian varieties I, Invent. Math., 1 (1966), 287–354.

    Article  MathSciNet  Google Scholar 

  25. D. Mumford, Abelian Varieties, Oxford University Press (1974).

    Google Scholar 

  26. D. Mumford, Tata lectures on Theta I, II + III, Birkhäuser Verlag (1982).

    Google Scholar 

  27. R. Murty, Exponents of Class Groups of Quadratic Fields, in: Proc. Conf. Number Th. and Arith. Geom. 19977, ed. G. Frey, Preprint 14 (1998), IEM, Essen.

    Google Scholar 

  28. U. Maurer and S. Wolf, Lower Bounds on Generic Algorithms in Groups, in: Advances in Cryptology-EURO CRYPT 98, K. Nyberg ed., Lecture Notes in Computer Science 1403, Springer Verlag (1998), 72–84.

    Google Scholar 

  29. N. Naumann, Weil-Restriktion Abelscher Varietäten, Diplomarbeit Essen (1999).

    Google Scholar 

  30. J. Neukirch, Algebraische Zahlentheorie, Springer (1992).

    Google Scholar 

  31. K. Nguyen, Brauer Groups of Local and Global Fields and Discrete Logarithms, in preparation.

    Google Scholar 

  32. A.M. Odlyzko, Discrete Logarithms and Smooth Polynomials, in Finite Fields: Theory, Applications and Algorithms, G.L. Mullen and P. Shiue,eds. Cont. Math. 168 AMS (1994), 269–278.

    Google Scholar 

  33. H.-G. Rück, On the Discrete Logarithm in the Divisor Class Group of Curves, Math. Comp., 68 (1999), 1233–1241.

    Article  MATH  MathSciNet  Google Scholar 

  34. R. Schoof, Elliptic curves over finite fields and the computation of square roots mod p, Math. Comp., 43 (1985), 483–494.

    MathSciNet  Google Scholar 

  35. R. Schoof, Counting Points on elliptic curves over finite fields, J. de Th. des Nombres de Bordeaux, 7 (1995), 219–254.

    Article  MATH  MathSciNet  Google Scholar 

  36. G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press (1971)

    Google Scholar 

  37. G. Shimura and Y. Tdumyama., Complex Multiplication of Abelian Varieties and its applications to Number Theory, Puhl. Math. Soc. Japan (1961).

    MATH  Google Scholar 

  38. P. W. Shor, Quantum Computing, Doc.Math.J.DMV Extra Volume ICM I (1998), 467–486.

    Google Scholar 

  39. J. Silverman, The Arithmetic of Elliptic Curves, Springer Verlag (1992).

    Google Scholar 

  40. A. M. Spallek, Kurven vom Geschlecht 2 und ihre Anwendungen in Public-Key-Kryptosystemen, Dissertation Essen (1994).

    Google Scholar 

  41. H. J. Weber, Hyperelliptic Simple Factors of J 0(N) with dimension at least 3, Experimental Math., 6 (1997), 273–287.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Experimentelle Mathematik, Universität GH Essen, Ellernstr. 29, 45326, Essen, Germany

    Gerhard Frey

Authors
  1. Gerhard Frey
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Lehrstuhl für Diskrete Mathematik, Optimierung und Operations Research, Universität Augsburg, 86135, Augsburg, Germany

    Dieter Jungnickel

  2. Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore, 117543, Republic of Singapore

    Harald Niederreiter

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Frey, G. (2001). Applications of Arithmetical Geometry to Cryptographic Constructions. In: Jungnickel, D., Niederreiter, H. (eds) Finite Fields and Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56755-1_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-56755-1_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-62498-8

  • Online ISBN: 978-3-642-56755-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation