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
Creative Commons Attribution
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},
      note = {\url{https://eprint.iacr.org/2009/613}},
      url = {https://eprint.iacr.org/2009/613}
}
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