Journal of Cryptology

, Volume 21, Issue 4, pp 593–611 | Cite as

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

  • Claus Diem
  • Emmanuel Thomé


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.


Index calculus Non-hyperelliptic curves Class groups Jacobians 


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  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) MathSciNetGoogle Scholar
  2. [2]
    F. Bahr, M. Böhm, J. Franke, T. Kleinjung, Factorization of RSA-200 by GNFS, May 2005. Unpublished electronic mail Google Scholar
  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) zbMATHGoogle Scholar
  5. [5]
    M. Bauer, E. Teske, A. Weng, Point counting on Picard curves in large characteristic, Math. Comput. 74(252), 1983–2005 (2005) zbMATHCrossRefMathSciNetGoogle 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 CrossRefGoogle Scholar
  7. [7]
    F. Chung, L. Lu, The diameter of random sparse graphs, Adv. Appl. Math. 26, 257–279 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    T. Cormen, C. Leiserson, R. Rivest, C. Stein, Introduction to algorithms, 2nd edn. (MIT Press/McGraw–Hill, Cambridge/New York, 2001) zbMATHGoogle 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 CrossRefGoogle Scholar
  10. [10]
    A. Enge, P. Gaudry, A general framework for subexponential discrete logarithm algorithms, Acta Arith. 102(1), 83–103 (2002) zbMATHMathSciNetCrossRefGoogle 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) zbMATHCrossRefGoogle Scholar
  14. [14]
    R. Hartshorne, Algebraic Geometry. Grad. Texts in Math., vol. 52 (Springer, Berlin, 1977) zbMATHGoogle Scholar
  15. [15]
    F. Heß, Computing Riemann-Roch spaces in algebraic function fields and related topics, J. Symb. Comput. 33(4), 425–445 (2002) zbMATHCrossRefGoogle Scholar
  16. [16]
    S. Janson, T. Luczak, A. Rucinski, Random Graphs (Wiley, New York, 2000) zbMATHGoogle 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) zbMATHMathSciNetGoogle 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) zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    J.H. Silverman, The Arithmetic of Elliptic Curves. Grad. Texts in Math., vol. 106 (Springer, Berlin, 1986) zbMATHGoogle 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) zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    H. Stichtenoth, Algebraic Function Fields and Codes. Universitext (Springer, Berlin, 1993) zbMATHGoogle 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 CrossRefGoogle Scholar
  25. [25]
    P.C. van Oorschot, M.J. Wiener, Parallel collision search with cryptanalytic applications, J. Cryptol. 12, 1–28 (1999) zbMATHCrossRefGoogle Scholar
  26. [26]
    A. Weng, A low-memory algorithm for point counting on Picard curves, Des. Codes Cryptogr. 38, 383–393 (2005) CrossRefMathSciNetGoogle Scholar

Copyright information

© International Association for Cryptologic Research 2007

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität LeipzigLeipzigGermany
  2. 2.INRIA Lorraine, CACAO—bât. AVillers-lès-NancyFrance

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