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A Weil Descent Attack against Elliptic Curve Cryptosystems over Quartic Extension Fields
Seigo ARITA Kazuto MATSUO Kohichi NAGAO Mahoro SHIMURA
Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences
Vol.E89A
No.5
pp.12461254 Publication Date: 2006/05/01
Online ISSN: 17451337
DOI: 10.1093/ietfec/e89a.5.1246
Print ISSN: 09168508 Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications) Category: Keyword: elliptic curve cryptosystems, hyperelliptic curve cryptosystems, Weil descent attack, Scholten form, C_{ab} curves,
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Summary:
This paper proposes a Weil descent attack against elliptic curve cryptosystems over quartic extension fields. The scenario of the attack is as follows: First, one reduces a DLP on a Weierstrass form over the quartic extention of a finite field k to a DLP on a special form, called Scholten form, over the same field. Second, one reduces the DLP on the Scholten form to a DLP on a genus two hyperelliptic curve over the quadratic extension of k. Then, one reduces the DLP on the hyperelliptic curve to one on a C_{ab} model over k. Finally, one obtains the discretelog of original DLP by applying the Gaudry method to the DLP on the C_{ab} model. In order to carry out the scenario, this paper shows that many of elliptic curve discretelog problems over quartic extension fields of odd characteristics are reduced to genus two hyperelliptic curve discretelog problems over quadratic extension fields, and that almost all of the genus two hyperelliptic curve discretelog problems over quadratic extension fields of odd characteristics come under Weil descent attack. This means that many of elliptic curve cryptosystems over quartic extension fields of odd characteristics can be attacked uniformly.

