Explicit Formulas for Real Hyperelliptic Curves of Genus 2 in Affine Representation

  • Stefan Erickson
  • Michael J. JacobsonJr.
  • Ning Shang
  • Shuo Shen
  • Andreas Stein
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4547)


In this paper, we present for the first time efficient explicit formulas for arithmetic in the degree 0 divisor class group of a real hyperelliptic curve. Hereby, we consider real hyperelliptic curves of genus 2 given in affine coordinates for which the underlying finite field has characteristic > 3. These formulas are much faster than the optimized generic algorithms for real hyperelliptic curves and the cryptographic protocols in the real setting perform almost as well as those in the imaginary case. We provide the idea for the improvements and the correctness together with a comprehensive analysis of the number of field operations. Finally, we perform a direct comparison of cryptographic protocols using explicit formulas for real hyperelliptic curves with the corresponding protocols presented in the imaginary model.


hyperelliptic curve reduced divisor infrastructure and distance Cantor’s algorithm explicit formulas efficient implementation cryptographic key exchange 


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  1. 1.
    Avanzi, R.M.: Aspects of hyperelliptic curves over large prime fields in software implementations. In: Joye, M., Quisquater, J.-J. (eds.) CHES 2004. LNCS, vol. 3156, pp. 148–162. Springer, Heidelberg (2004)Google Scholar
  2. 2.
    Cohen, H., Frey, G. (eds.): Handbook of Elliptic and Hyperelliptic Curve Cryptography. Discrete Mathematics and Its Applications, vol. 34. Chapman & Hall/CRC, Sydney, Australia (2005)Google Scholar
  3. 3.
    Enge, A.: How to distinguish hyperelliptic curves in even characteristic. In: Alster, K., Urbanowicz, J., Williams, H.C., (eds.) Public-Key Cryptography and Computational Number Theory, pp. 49–58, De Gruyter, Berlin (2001)Google Scholar
  4. 4.
    Gaudry, P.: On breaking the discrete log on hyperelliptic curves. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 19–34. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  5. 5.
    Gaudry, P., Thomé, E., Thériault, N., Diem, C.: A double large prime variation for small genus hyperelliptic index calculus. Mathematics of Computation 76, 475–492 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Jacobson Jr., M.J., Menezes, A.J., Stein, A.: Hyperelliptic curves and cryptography. In: High Primes and Misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams. Fields Institute Communications Series, vol. 41, pp. 255–282. American Mathematical Society (2004)Google Scholar
  7. 7.
    Jacobson Jr., M.J., Scheidler, R., Stein, A.: Cryptographic protocols on real and imaginary hyperelliptic curves. Accepted to Advances in Mathematics of Communications pending revisions (2007)Google Scholar
  8. 8.
    Jacobson Jr., M.J., Scheidler, R., Stein, A.: Fast Arithmetic on Hyperelliptic Curves Via Continued Fraction Expansions. To appear in Advances in Coding Theory and Cryptology. In: Shaaska, T., Huffman, W.C., Joyner, D., Ustimenko, V. (eds.) Series on Coding, Theory and Cryptology, vol. 2, World Scientific Publishing (2007) Google Scholar
  9. 9.
    Koblitz, N.: Hyperelliptic cryptosystems. Journal of Cryptology 1, 139–150 (1988)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Lange, T.: Formulae for arithmetic on genus 2 hyperelliptic curves. Applicable Algebra in Engineering, Communication, and Computing 15, 295–328 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Menezes, A.J., Wu, Y., Zuccherato, R.J.: An elementary introduction to hyperelliptic curves. Technical Report CORR 96-19, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, 1996. In: Koblitz, N. (ed.) Algebraic Aspects of Cryptography, Springer, Heidelberg (1998)Google Scholar
  12. 12.
    Müller, V., Stein, A., Thiel, C.: Computing discrete logarithms in real quadratic congruence function fields of large genus. Mathematics of Computation 68, 807–822 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Mumford, D.: Tata Lectures on Theta I, II. Birkhäuser, Boston (1983/84)Google Scholar
  14. 14.
    National Institute of Standards and Technology (NIST). Recommendation on key establishment schemes. NIST Special Publication 800-56 (January 2003)Google Scholar
  15. 15.
    Paulus, S., Rück, H.-G.: Real and imaginary quadratic representations of hyperelliptic function fields. Mathematics of Computation 68, 1233–1241 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Pelzl, J., Wollinger, T., Paar, C.: Low cost security: explicit formulae for genus-4 hyperelliptic curves. In: Matsui, M., Zuccherato, R.J. (eds.) SAC 2003. LNCS, vol. 3006, pp. 1–16. Springer, Heidelberg (2003)Google Scholar
  17. 17.
    Scheidler, R.: Cryptography in quadratic function fields. Designs, Codes and Cryptography 22, 239–264 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Scheidler, R., Stein, A., Williams, H.C.: Key-exchange in real quadratic congruence function fields. Designs, Codes and Cryptography 7, 153–174 (1996)zbMATHMathSciNetGoogle Scholar
  19. 19.
    V. Shoup. NTL: A library for doing number theory. Software (2001) See
  20. 20.
    Stein, A.: Sharp upper bounds for arithmetics in hyperelliptic function fields. Journal of the Ramanujan Mathematical Society 9-16(2), 1–86 (2001)Google Scholar
  21. 21.
    Wollinger, T., Pelzl, J., Paar, C.: Cantor versus Harley: optimization and analysis of explicit formulae for hyperelliptic curve cryptosystems. IEEE Transactions on Computers 54, 861–872 (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Stefan Erickson
    • 1
  • Michael J. JacobsonJr.
    • 2
  • Ning Shang
    • 3
  • Shuo Shen
    • 3
  • Andreas Stein
    • 4
  1. 1.Department of Mathematics and Computer Science, Colorado College, 14 E. Cache La Poudre, Colorado Spgs., CO. 80903USA
  2. 2.Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4Canada
  3. 3.Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067USA
  4. 4.Department of Mathematics, University of Wyoming 1000 E. University Avenue, Laramie, WY 82071-3036USA

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