Paper 2009/613
Classification of Elliptic/hyperelliptic Curves with Weak Coverings against GHS Attack without Isogeny Condition
Tsutomu Iijima, Fumiyuki Momose, and Jinhui Chao
Abstract
The GHS attack is known as a method to map the discrete logarithm problem(DLP) in the Jacobian of a curve C_{0} defined over the d degree extension k_{d} of a finite field k to the DLP in the Jacobian of a new curve C over k which is a covering curve of C_{0}. Recently, classification and density analysis were shown for all elliptic and hyperelliptic curves C_{0}/k_d of genus 2, 3 which possess (2, \ldots ,2) covering C/k of {\mathbb{P}^{1}} under the isogeny condition (i.e. when g(C)=d \cdot g(C_{0})). In this paper, we show a complete classification of small genus hyperelliptic curves C_0/k_d which possesses (2,..,2) covering C over k without the isogeny condition. Our main approach is to use representation of the extension of Gal(k_{d}/k) acting on cov(C/\mathbb{P}^{1}). Explicit defining equations of such curves and the existence of a model of C over k are also presented.
Metadata
- Available format(s)
-
PDF
- Category
- Public-key cryptography
- Publication info
- Published elsewhere. Unknown status
- Keywords
- Weil descent attackGHS attackElliptic curve cryptosystemsHyperelliptic curve cryptosystemsIndex calculusGalois representation
- Contact author(s)
- tiijima @ jt3 so-net ne jp
- History
- 2013-11-22: last of 2 revisions
- 2009-12-14: received
- See all versions
- Short URL
- https://ia.cr/2009/613
- License
-
CC BY
BibTeX
@misc{cryptoeprint:2009/613,
author = {Tsutomu Iijima and Fumiyuki Momose and Jinhui Chao},
title = {Classification of Elliptic/hyperelliptic Curves with Weak Coverings against {GHS} Attack without Isogeny Condition},
howpublished = {Cryptology {ePrint} Archive, Paper 2009/613},
year = {2009},
url = {https://eprint.iacr.org/2009/613}
}