Advertisement

A Survey of Local and Global Pairings on Elliptic Curves and Abelian Varieties

  • Joseph H. Silverman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6487)

Abstract

There are many bilinear pairings that naturally appear when one studies elliptic curves, abelian varieties, and related groups. Some of these pairings, notably the Weil and Lichtenbaum–Tate pairings, can be defined over finite fields and have important applications in cryptography. Others, such as the Néron–Tate canonical height pairing and the Cassels–Tate pairing on the Shafarevich–Tate group, are of fundamental importance in number theory and arithmetic geometry, but have seen limited use in cryptography. In this article I will present a survey of some of the pairings that are used to study elliptic curves and abelian varieties. I will attempt to show why they are natural pairings and how they fit into a wider framework.

keywords and Phrases

elliptic curve abelian variety cryptography pairings 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Atiyah, M.F., Wall, C.T.C.: Cohomology of groups. In: Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pp. 94–115. Thompson, Washington (1967)Google Scholar
  2. 2.
    Bombieri, E., Gubler, W.: Heights in Diophantine geometry. New Mathematical Monographs, vol. 4. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar
  3. 3.
    Boneh, D., Franklin, M.: Identity-based encryption from the Weil pairing. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 213–229. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  4. 4.
    Cassels, J.W.S.: Arithmetic on curves of genus 1. IV. Proof of the Hauptvermutung. J. Reine. Angew. Math. 211, 95–112 (1962)zbMATHMathSciNetGoogle Scholar
  5. 5.
    du Sautoy, M.: John Tate wins the Abel Prize (2010), http://www.abelprisen.no/en/prisvinnere/2010/marcus/
  6. 6.
    Hindry, M., Silverman, J.H.: Diophantine Geometry: An Introduction. Graduate Texts in Mathematics, vol. 201. Springer, New York (2000)zbMATHGoogle Scholar
  7. 7.
    Jacobson, M.J., Koblitz, N., Silverman, J.H., Stein, A., Teske, E.: Analysis of the xedni calculus attack. Des. Codes Cryptogr. 20(1), 41–64 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Joux, A.: A one round protocol for tripartite Diffie-Hellman. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 385–393. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  9. 9.
    Koblitz, N.: Elliptic curve cryptosystems. Math. Comp. 48(177), 203–209 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kolyvagin, V.A.: Finiteness of E(ℚ) and III(E,ℚ) for a subclass of Weil curves. Izv. Akad. Nauk SSSR Ser. Mat. 52(3), 522–540 (1988)MathSciNetGoogle Scholar
  11. 11.
    Lang, S.: Fundamentals of Diophantine Geometry. Springer, New York (1983)zbMATHGoogle Scholar
  12. 12.
    Lang, S.: Les formes bilinéaires de Néron et Tate. In: Séminaire Bourbaki, Soc. Math. France, Paris, vol. 8, Exp. No. 274, pp. 435–445 (1995)Google Scholar
  13. 13.
    Menezes, A.J., Okamoto, T., Vanstone, S.A.: Reducing elliptic curve logarithms to logarithms in a finite field. IEEE Trans. Inform. Theory 39(5), 1639–1646 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Miller, V.S.: Use of elliptic curves in cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986)Google Scholar
  15. 15.
    Milne, J.S.: Arithmetic Duality Theorems. Perspectives in Mathematics, vol. 1. Academic Press Inc., Boston (1986)zbMATHGoogle Scholar
  16. 16.
    Néron, A.: Quasi-fonctions et hauteurs sur les variétés abéliennes. Ann. of Math. 82(2), 249–331 (1965)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Rubin, K.: Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication. Invent. Math. 89(3), 527–559 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Serre, J.-P.: Local fields. Graduate Texts in Mathematics, vol. 67. Springer, New York (1979); Translated from the French by Marvin Jay GreenbergzbMATHGoogle Scholar
  19. 19.
    Serre, J.-P.: Galois cohomology, English edn. Springer Monographs in Mathematics. Springer, Berlin (2002); Translated from the French by Patrick Ion and revised by the authorzbMATHGoogle Scholar
  20. 20.
    Silverman, J.H.: The xedni calculus and the elliptic curve discrete logarithm problem. Des. Codes Cryptogr. 20(1), 5–40 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Silverman, J.H.: The Arithmetic of Elliptic Curves, 2nd edn. Graduate Texts in Mathematics, vol. 106. Springer, Dordrecht (2009)zbMATHGoogle Scholar
  22. 22.
    Silverman, J.H., Suzuki, J.: Elliptic curve discrete logarithms and the index calculus. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 110–125. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  23. 23.
    Tate, J.: WC-groups over \({\mathfrak p}\)-adic fields. In: Textes des Conférences; Exposés 152 à 168; 2e éd. corrigée, Exposé 156. Séminaire Bourbaki; 10e année: 1957/1958, vol. 13, Secrétariat mathématique, Paris (1958)Google Scholar
  24. 24.
    Tate, J.: Letter to J.W.S. Cassels (August 1, 1962) (unpublished)Google Scholar
  25. 25.
    Tate, J.: Duality theorems in Galois cohomology over number fields. In: Proc. Internat. Congr. Mathematicians (Stockholm, 1962), pp. 288–295. Inst. Mittag-Leffler, Djursholm (1963)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Joseph H. Silverman
    • 1
  1. 1.Mathematics DepartmentBrown UniversityProvidenceUSA

Personalised recommendations