Academic Editor: J. Rodellar
Copyright © 2011 Lin You et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Hyperelliptic curves have been widely studied for cryptographic applications, and some special
hyperelliptic curves are often considered to be used in practical cryptosystems. Computing Jacobian group orders is an important operation in constructing hyperelliptic curve cryptosystems,
and the most common method used for the computation of Jacobian group orders is by computing
the zeta functions or the characteristic polynomials of the related hyperelliptic curves. For the
hyperelliptic curve : over the field with being a power of an odd prime p, Duursma and Sakurai obtained its characteristic polynomial for , and . In
this paper, we determine the characteristic polynomials of over the finite field for , 2 and , . We also give some computational data which show that many of those curves
have large prime factors in their Jacobian group orders, which are both practical and vital for the
constructions of efficient and secure hyperelliptic curve cryptosystems.