Skip to main content

Index Calculus in Class Groups of Non-hyperelliptic Curves of Genus Three


We study an index calculus algorithm to solve the discrete logarithm problem (DLP) in degree 0 class groups of non-hyperelliptic curves of genus 3 over finite fields. We present a heuristic analysis of the algorithm which indicates that the DLP in degree 0 class groups of non-hyperelliptic curves of genus 3 can be solved in an expected time of \(\tilde{O}(q)\) . This heuristic result relies on one heuristic assumption which is studied experimentally.

We also present experimental data which show that a variant of the algorithm is faster than the Rho method even for small group sizes, and we address practical limitations of the algorithm.


  1. [1]

    F. Abu Salem, K. Khuri-Makdisi, Fast Jacobian group operations for C 3,4 curves over a large finite field. LMS J. Comput. Math. 10, 307–328 (2007)

    MathSciNet  Google Scholar 

  2. [2]

    F. Bahr, M. Böhm, J. Franke, T. Kleinjung, Factorization of RSA-200 by GNFS, May 2005. Unpublished electronic mail

  3. [3]

    A. Basiri, A. Enge, J.-C. Faugère, N. Gürel, Implementing the arithmetic of C 3,4-curves, in Algorithmic Number Theory—ANTS VI. Lecture Notes in Comput. Sci. (Springer, Berlin, 2004), pp. 87–101

    Google Scholar 

  4. [4]

    A. Basiri, A. Enge, J.-C. Faugère, N. Gürel, The arithmetic of Jacobian groups of superelliptic cubics, Math. Comput. 74(249), 389–410 (2005)

    MATH  Google Scholar 

  5. [5]

    M. Bauer, E. Teske, A. Weng, Point counting on Picard curves in large characteristic, Math. Comput. 74(252), 1983–2005 (2005)

    MATH  Article  MathSciNet  Google Scholar 

  6. [6]

    S. Cavallar, Strategies in filtering in the number field sieve, in Algorithmic Number Theory — ANTS-IV, ed. by W. Bosma. Lecture Notes in Comput. Sci., vol. 1838 (Springer, Berlin, 2000), pp. 209–231

    Chapter  Google Scholar 

  7. [7]

    F. Chung, L. Lu, The diameter of random sparse graphs, Adv. Appl. Math. 26, 257–279 (2001)

    MATH  Article  MathSciNet  Google Scholar 

  8. [8]

    T. Cormen, C. Leiserson, R. Rivest, C. Stein, Introduction to algorithms, 2nd edn. (MIT Press/McGraw–Hill, Cambridge/New York, 2001)

    MATH  Google Scholar 

  9. [9]

    C. Diem, An index calculus algorithm for plane curves of small degree, in Algorithmic Number Theory—ANTS VII, ed. by F. Hess, S. Pauli, M. Pohst. Lecture Notes in Comput. Sci., vol. 4076 (Springer, Berlin, 2006), pp. 543–557

    Chapter  Google Scholar 

  10. [10]

    A. Enge, P. Gaudry, A general framework for subexponential discrete logarithm algorithms, Acta Arith. 102(1), 83–103 (2002)

    MATH  MathSciNet  Article  Google Scholar 

  11. [11]

    S. Flon, R. Oyono, Fast arithmetic on Jacobians of Picard curves, in Advances in Cryptology—PKC 2004, ed. by F. Bao et al. Lecture Notes in Comput. Sci., vol. 2947 (Springer, Berlin, 2004), pp. 55–68

    Google Scholar 

  12. [12]

    S. Flon, R. Oyono, C. Ritzenthaler, Fast addition on non-hyperelliptic genus 3 curves. IACR Eprint report 2004/118, available at, 2004

  13. [13]

    P. Gaudry, E. Thomé, N. Thériault, C. Diem, A double large prime variation for small genus hyperelliptic index calculus, Math. Comput. 76(257), 475–492 (2007)

    MATH  Article  Google Scholar 

  14. [14]

    R. Hartshorne, Algebraic Geometry. Grad. Texts in Math., vol. 52 (Springer, Berlin, 1977)

    MATH  Google Scholar 

  15. [15]

    F. Heß, Computing Riemann-Roch spaces in algebraic function fields and related topics, J. Symb. Comput. 33(4), 425–445 (2002)

    MATH  Article  Google Scholar 

  16. [16]

    S. Janson, T. Luczak, A. Rucinski, Random Graphs (Wiley, New York, 2000)

    MATH  Google Scholar 

  17. [17]

    A. Joux, R. Lercier, Discrete logarithms in GF(p)—130 digits. Electronic mail to the NMBRTHRY mailing list. Available at, June 2005

  18. [18]

    K. Koyke, A. Weng, Construction of CM-Picard curves, Math. Comput. 74(249), 499–518 (2005)

    Google Scholar 

  19. [19]

    V.K. Murty, J. Scherk, Effective versions of the Chebotarev density theorem for function fields, C. R. Acad. Sci. Paris Sér. I Math. 319, 523–528 (1994)

    MATH  MathSciNet  Google Scholar 

  20. [20]

    J. Pila, Frobenius maps of Abelian varieties and finding roots of unity in finite fields, Math. Comput. 55(192), 745–763 (1990)

    MATH  Article  MathSciNet  Google Scholar 

  21. [21]

    J.H. Silverman, The Arithmetic of Elliptic Curves. Grad. Texts in Math., vol. 106 (Springer, Berlin, 1986)

    MATH  Google Scholar 

  22. [22]

    H. Stichtenoth, Über die automorphismengruppe eines algebraischen funktionenkörpers von primzahlcharakteristik. I. Eine abschätzung der ordnung der automorphismengruppe, Arch. Math. 24, 527–544 (1973)

    MATH  Article  MathSciNet  Google Scholar 

  23. [23]

    H. Stichtenoth, Algebraic Function Fields and Codes. Universitext (Springer, Berlin, 1993)

    MATH  Google Scholar 

  24. [24]

    E. Thomé, Computation of discrete logarithms in \(\mathbb{F}_{2^{607}}\) , in Advances in Cryptology—ASIACRYPT 2001, ed. by C. Boyd, E. Dawson. Lecture Notes in Comput. Sci., vol. 2248 (Springer, Berlin, 2001), pp. 107–124

    Chapter  Google Scholar 

  25. [25]

    P.C. van Oorschot, M.J. Wiener, Parallel collision search with cryptanalytic applications, J. Cryptol. 12, 1–28 (1999)

    MATH  Article  Google Scholar 

  26. [26]

    A. Weng, A low-memory algorithm for point counting on Picard curves, Des. Codes Cryptogr. 38, 383–393 (2005)

    Article  MathSciNet  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Claus Diem.

Additional information

Communicated by Arjen K. Lenstra

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Diem, C., Thomé, E. Index Calculus in Class Groups of Non-hyperelliptic Curves of Genus Three. J Cryptol 21, 593–611 (2008).

Download citation


  • Index calculus
  • Non-hyperelliptic curves
  • Class groups
  • Jacobians