Physics, mathematics, and technology

2010 Issue №10

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Overview of effective point counting algorithms for Jacobian of hyperelliptic curve over finite field



Various algorithms for finding of the order of Jacobian, their range of use and efficiency are considered.


1. Colm O hEigeartaigh. A comparison of point counting methods for hyperelliptic curves over prime fields and fields of characteristic 2 // Cryptology ePrint Archive. 2004.

2. Haneda M., Kawazoe M., Takahashi T. Suitable curves for genus-4 HCC over prime fields: point counting formulae for hyperelliptic curves of type.....  // Ibid.

3. Furukawa E., Kawazoe M., Takahashi T. Counting points for hyperelliptic curves of type..... // Ibid. 2002.

4. Haloui S. The minimum and maximum number of rational points on jacobian surfaces over finite fields. URL: http://arxiv. org/abs/1002.3683.2010.

5. Ravnshoj C. R. Generators of Jacobians of genus two curves // Cryptology ePrint Archive. 2008.

6. Ravnshoj C. R. Non-cyclic subgroups of Jacobians of genus two curves // Ibid.

7. Ravnshoj C. R. Non-cyclic subgroups of Jacobians of genus two curves with complex multiplication // Ibid.

8. Dechene I. Arithmetic of generalized Jacobians // Ibid. 2006.

9. Dechene I. On the security of generalized Jacobian cryptosystems // Ibid.

10. Galbraith S. D., Smith B. A. Discrete logarithms in generalized Jacobians // Ibid.

11. Nagao K. Improvement of theriault algorithm of index calculus for Jacobian of hyperelliptic curves of small genus // Ibid. 2004.