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A Subexponential Algorithm for Evaluating Large Degree Isogenies

  • David Jao
  • Vladimir Soukharev
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6197)

Abstract

An isogeny between elliptic curves is an algebraic morphism which is a group homomorphism. Many applications in cryptography require evaluating large degree isogenies between elliptic curves efficiently. For ordinary curves of the same endomorphism ring, the previous best known algorithm has a worst case running time which is exponential in the length of the input. In this paper we show this problem can be solved in subexponential time under reasonable heuristics. Our approach is based on factoring the ideal corresponding to the kernel of the isogeny, modulo principal ideals, into a product of smaller prime ideals for which the isogenies can be computed directly. Combined with previous work of Bostan et al., our algorithm yields equations for large degree isogenies in quasi-optimal time given only the starting curve and the kernel.

Keywords

Prime Ideal Elliptic Curve Elliptic Curf Endomorphism Ring Prime Degree 
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.

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References

  1. 1.
    Bisson, G., Sutherland, A.: Computing the endomorphism ring of an ordinary elliptic curve over a finite field. Journal of Number Theory (to appear 2009)Google Scholar
  2. 2.
    Blake, I.F., Seroussi, G., Smart, N.P.: Elliptic curves in cryptography. London Mathematical Society Lecture Note Series, vol. 265. Cambridge University Press, Cambridge (2000); Reprint of the 1999 original (1999)Google Scholar
  3. 3.
    Bostan, A., Morain, F., Salvy, B., Schost, É.: Fast algorithms for computing isogenies between elliptic curves. Math. Comp. 77(263), 1755–1778 (2008)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Bröker, R., Charles, D., Lauter, K.: Evaluating large degree isogenies and applications to pairing based cryptography. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 100–112. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Bröker, R., Lauter, K., Sutherland, A.: Modular polynomials via isogeny volcanoes (2010)Google Scholar
  6. 6.
    Buchmann, J., Vollmer, U.: Binary quadratic forms. Algorithms and Computation in Mathematics, vol. 20. Springer, Berlin (2007); An algorithmic approachMATHGoogle Scholar
  7. 7.
  8. 8.
  9. 9.
    Cohen, H., Frey, G., Avanzi, R., Doche, C., Lange, T., Nguyen, K., Vercauteren, F. (eds.): Handbook of elliptic and hyperelliptic curve cryptography. Discrete Mathematics and its Applications. Chapman & Hall/CRC (2006)Google Scholar
  10. 10.
    Cohen, H.: A course in computational algebraic number theory. Graduate Texts in Mathematics, vol. 138. Springer, Berlin (1993)MATHGoogle Scholar
  11. 11.
    Couveignes, J.-M., Morain, F.: Schoof’s algorithm and isogeny cycles. In: Huang, M.-D.A., Adleman, L.M. (eds.) ANTS 1994. LNCS, vol. 877, pp. 43–58. Springer, Heidelberg (1994)Google Scholar
  12. 12.
    Cox, D.A.: Primes of the form x 2 + ny 2. A Wiley-Interscience Publication, John Wiley & Sons Inc., New York (1989); Fermat, class field theory and complex multiplicationGoogle Scholar
  13. 13.
    Enge, A.: Computing modular polynomials in quasi-linear time. Math. Comp. 78(267), 1809–1824 (2009)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Fouquet, M., Morain, F.: Isogeny volcanoes and the SEA algorithm. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 276–291. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  15. 15.
    Freeman, D., Scott, M., Teske, E.: A taxonomy of pairing-friendly elliptic curves. J. Cryptology (to appear 2010)Google Scholar
  16. 16.
    Galbraith, S.D.: Constructing isogenies between elliptic curves over finite fields. LMS J. Comput. Math. 2, 118–138 (1999) (electronic)MATHMathSciNetGoogle Scholar
  17. 17.
    Galbraith, S.D., Hess, F., Smart, N.P.: Extending the GHS Weil descent attack. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 29–44. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  18. 18.
    Hafner, J., McCurley, K.: A rigorous subexponential algorithm for computation of class groups. J. Amer. Math. Soc. 2(4), 837–850 (1989)MATHMathSciNetGoogle Scholar
  19. 19.
    Hardy, K., Muskat, J.B., Williams, K.S.: A deterministic algorithm for solving n = fu 2 + gv 2 in coprime integers u and v. Math. Comp. 55(191), 327–343 (1990)MATHMathSciNetGoogle Scholar
  20. 20.
    Jao, D., Miller, S.D., Venkatesan, R.: Do all elliptic curves of the same order have the same difficulty of discrete log? In: Roy, B. (ed.) ASIACRYPT 2005. LNCS, vol. 3788, pp. 21–40. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  21. 21.
    Kohel, D.: Endomorphism rings of elliptic curves over finite fields. PhD thesis, University of California, Berkeley (1996)Google Scholar
  22. 22.
    MAGMA Computational Algebra System, http://magma.maths.usyd.edu.au/
  23. 23.
    Menezes, A., Teske, E., Weng, A.: Weak fields for ECC. In: Okamoto, T. (ed.) CT-RSA 2004. LNCS, vol. 2964, pp. 366–386. Springer, Heidelberg (2004)Google Scholar
  24. 24.
    Schönhage, A.: Fast reduction and composition of binary quadratic forms. In: ISSAC 1991: Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation, pp. 128–133. ACM, New York (1991)CrossRefGoogle Scholar
  25. 25.
    Schoof, R.: Counting points on elliptic curves over finite fields. J. Théor. Nombres Bordeaux 7(1), 219–254 (1995); Les Dix-huitièmes Journées Arithmétiques (Bordeaux, 1993)MATHMathSciNetGoogle Scholar
  26. 26.
    Seysen, M.: A probabilistic factorization algorithm with quadratic forms of negative discriminant. Math. Comp. 48(178), 757–780 (1987)MATHMathSciNetGoogle Scholar
  27. 27.
    Silverman, J.: The arithmetic of elliptic curves. Graduate Texts in Mathematics, vol. 106. Springer, New York (1992); Corrected reprint of the 1986 original (1986)Google Scholar
  28. 28.
    Sutherland, A.:Smoothrelation, http://math.mit.edu/~drew/smoothrelation_v1.tar
  29. 29.
    Tate, J.: Endomorphisms of abelian varieties over finite fields. Invent. Math. 2, 134–144 (1966)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Teske, E.: An elliptic curve trapdoor system. J. Cryptology 19(1), 115–133 (2006)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • David Jao
    • 1
  • Vladimir Soukharev
    • 1
  1. 1.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada

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