Supersingular Abelian Varieties in Cryptology

  • Karl Rubin
  • Alice Silverberg
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2442)


For certain security applications, including identity based encryption and short signature schemes, it is useful to have abelian varieties with security parameters that are neither too small nor too large. Supersingular abelian varieties are natural candidates for these applications. This paper determines exactly which values can occur as the security parameters of supersingular abelian varieties (in terms of the dimension of the abelian variety and the size of the finite field), and gives constructions of supersingular abelian varieties that are optimal for use in cryptography.


Elliptic Curve Elliptic Curf Abelian Variety Security Parameter Hyperelliptic Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    L. Adleman, J. DeMarrais and M-D. Huang. A subexponential algorithm for discrete logarithms over the rational subgroup of the Jacobians of large genus hyperelliptic curves over finite fields, in Algorithmic number theory. Lecture Notes in Computer Science, Vol. 877. Springer-Verlag (1994) 28–40.Google Scholar
  2. 2.
    D. Boneh and M. Franklin. Identity based encryption from the Weil pairing, in Advances in Cryptology — Crypto 2001. Lecture Notes in Computer Science, Vol. 2139. Springer-Verlag (2001) 213–229.Google Scholar
  3. 3.
    D. Boneh, B. Lynn and H. Shacham. Short signatures from the Weil pairing, in Advances in Cryptology — Asiacrypt 2001. Lect. Notes in Comp. Sci. 2248 (2001), Springer-Verlag, 514–532.Google Scholar
  4. 4.
    R. Coleman and W. McCallum, Stable reduction of Fermat curves and Jacobi sum Hecke characters. J. Reine Angew. Math. 385 (1988) 41–101.zbMATHMathSciNetGoogle Scholar
  5. 5.
    D. Cox, J. Little and D. O’shea. Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra. Springer-Verlag (1997).Google Scholar
  6. 6.
    G. Frey. Applications of arithmetical geometry to cryptographic constructions, in Finite fields and applications (Augsburg, 1999). Springer-Verlag (2001) 128–161.Google Scholar
  7. 7.
    G. Frey, M. Müller and H-G. Rück. The Tate pairing and the discrete logarithm applied to elliptic curve cryptosystems. IEEE Trans. Inform. Theory 45 (1999) 1717–1719.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    G. Frey and H-G. Rück. A remark concerning m-divisibility and the discrete logarithm in the divisor class group of curves. Math. Comp. 62 (1994) 865–874.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    S. Galbraith. Supersingular curves in cryptography, in Advances in Cryptology — Asiacrypt 2001. Lecture Notes in Computer Science, Vol. 2248. Springer-Verlag (2001) 495–513.CrossRefGoogle Scholar
  10. 10.
    S. Galbraith, F. Hess and N. P. Smart. Extending the GHS Weil descent attack, in Advances in Cryptology — Eurocrypt 2002. Lecture Notes in Computer Science, Vol. 2332. Springer-Verlag (2002) 29–44.CrossRefGoogle Scholar
  11. 11.
    P. Gaudry. A variant of the Adleman-DeMarrais-Huang algorithm and its application to small genera, in Advances in Cryptology — Eurocrypt 2000. Lecture Notes in Computer Science, Vol. 1807. Springer-Verlag (2000) 19–34.CrossRefGoogle Scholar
  12. 12.
    P. Gaudry, F. Hess and N. P. Smart. Constructive and destructive facets of Weil descent on elliptic curves. J. Cryptology 15 (2002) 19–46.CrossRefMathSciNetGoogle Scholar
  13. 13.
    T. Honda. Isogeny classes of abelian varieties over finite fields. J. Math. Soc. Japan 20 (1968) 83–95.zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    B. Huppert and N. Blackburn. Finite groups II. Springer-Verlag (1982).Google Scholar
  15. 15.
    A. Joux. A one round protocol for tripartite Diffie-Hellman, in Algorithmic Number Theory (ANTS-IV), Leiden, The Netherlands, July 2–7, 2000, Lecture Notes in Computer Science, Vol. 1838. Springer-Verlag (2000) 385–394.CrossRefGoogle Scholar
  16. 16.
    A. K. Lenstra and E. R. Verheul. The XTR public key system, in Advances in Cryptology — Crypto 2000. Lecture Notes in Computer Science, Vol. 1880. Springer-Verlag (2000) 1–19.CrossRefGoogle Scholar
  17. 17.
    A. J. Menezes, T. Okamoto and S. A. Vanstone. Reducing elliptic curve logarithms to logarithms in a finite field. IEEE Trans. Inform. Theory 39 (1993) 1639–1646.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    R. Sakai, K. Ohgishi and M. Kasahara, Cryptosystems based on pairing. SCIS2000 (The 2000 Symposium on Cryptography and Information Security), Okinawa, Japan, January 26–28, 2000, C20.Google Scholar
  19. 19.
    G. Shimura. Abelian varieties with complex multiplication and modular functions. Princeton Univ. Press, Princeton, NJ (1998).zbMATHGoogle Scholar
  20. 20.
    J. Silverman. The arithmetic of elliptic curves. Springer-Verlag (1986).Google Scholar
  21. 21.
    J. Tate. Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T. Honda), in Séminaire Bourbaki, 1968/69, Soc. Math. France, Paris (1968) 95–110.Google Scholar
  22. 22.
    E. R. Verheul. Self-blindable credential certificates from the Weil pairing, in Advances in Cryptology — Asiacrypt 2001, Lecture Notes in Computer Science, Vol. 2248. Springer-Verlag (2001) 533–551.CrossRefGoogle Scholar
  23. 23.
    A. Weil. Adeles and algebraic groups. Progress in Math. 23, Birkhäuser, Boston (1982).Google Scholar
  24. 24.
    H. J. Zhu. Group structures of elementary supersingular abelian varieties over finite fields. J. Number Theory 81 (2000) 292–309.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Karl Rubin
    • 1
  • Alice Silverberg
    • 2
  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.Department of MathematicsOhio State UniversityColumbusUSA

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