[Resource Topic] 2002/181: Counting Points for Hyperelliptic Curves of type $y^2=x^5+ax$ over Finite Prime Fields

Welcome to the resource topic for 2002/181

Counting Points for Hyperelliptic Curves of type y^2=x^5+ax over Finite Prime Fields

Authors: Eisaku Furukawa, Mitsuru Kawazoe, Tetsuya Takahashi


Counting rational points on Jacobian varieties of hyperelliptic curves
over finite fields is very important for constructing
hyperelliptic curve cryptosystems (HCC),
but known algorithms for general curves over given large prime
fields need very long running times.
In this article, we propose an extremely fast point counting algorithm for
hyperelliptic curves of type y^2=x^5+ax over given large
prime fields \Fp, e.g. 80-bit fields.
For these curves, we also determine the necessary condition
to be suitable for HCC, that is, to satisfy that the order
of the Jacobian group is of the form l\cdot c where l
is a prime number greater than about 2^{160} and
c is a very small integer.
We show some examples of suitable curves for HCC obtained by
using our algorithm.
We also treat curves of type y^2=x^5+a where a is not
square in \Fp.

ePrint: https://eprint.iacr.org/2002/181

See all topics related to this paper.

Feel free to post resources that are related to this paper below.

Example resources include: implementations, explanation materials, talks, slides, links to previous discussions on other websites.

For more information, see the rules for Resource Topics .