Paper 2013/487
Classification of Elliptic/hyperelliptic Curves with Weak Coverings against GHS Attack under an Isogeny Condition
Tsutomu Iijima and 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}. Such curves C_{0}/k_{d} can be attacked by the GHS attack and index calculus algorithms. In this paper, we will classify all elliptic curves and hyperelliptic curves C_{0}/k_{d} of genus 2, 3 which possess (2,...,2) covering C/k of \Bbb{P}^1 under the isogeny condition (i.e. g(C)=d \cdot g(C_{0})) in odd characteristic case. Our main approach is analysis of ramification points and representation of the extension of Gal(k_{d}/k) acting on the covering group cov(C/\Bbb{P}^1). Consequently, all explicit defining equations of such curves C_0/k_d and existential conditions of a model of C over k are provided.
Metadata
- Available format(s)
- Category
- Public-key cryptography
- Publication info
- Preprint. MINOR revision.
- Keywords
- Weil descent attackGHS attackElliptic curve cryptosystemsHyperelliptic curve cryptosystemsIndex calculusGalois representation
- Contact author(s)
- tiijima @ jt3 so-net ne jp
- History
- 2015-02-18: last of 3 revisions
- 2013-08-14: received
- See all versions
- Short URL
- https://ia.cr/2013/487
- License
-
CC BY