Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 437541, 25 pages
doi:10.1155/2011/437541
Research Article

Computing the Characteristic Polynomials of a Class of Hyperelliptic Curves for Cryptographic Applications

1College of Communication Engineering, Hangzhou Dianzi University, Hangzhou 310018, China
2College of Engineering and Science, Clemson University, Clemson, SC 29631, USA
3School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Received 12 November 2010; Accepted 21 March 2011

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.

Abstract

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 𝐶 𝑞 : 𝑣 2 = 𝑢 𝑝 + 𝑎 𝑢 + 𝑏 over the field 𝔽 𝑞 with 𝑞 being a power of an odd prime p, Duursma and Sakurai obtained its characteristic polynomial for 𝑞 = 𝑝 , 𝑎 = 1 , and 𝑏 𝔽 𝑝 . In this paper, we determine the characteristic polynomials of 𝐶 𝑞 over the finite field 𝔽 𝑝 𝑛 for 𝑛 = 1 , 2 and 𝑎 , 𝑏 F 𝑝 𝑛 . 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.