A Weil Descent Attack against Elliptic Curve Cryptosystems over Quartic Extension Fields

Seigo ARITA  Kazuto MATSUO  Koh-ichi NAGAO  Mahoro SHIMURA  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E89-A   No.5   pp.1246-1254
Publication Date: 2006/05/01
Online ISSN: 1745-1337
DOI: 10.1093/ietfec/e89-a.5.1246
Print ISSN: 0916-8508
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,  Cab 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 Cab model over k. Finally, one obtains the discrete-log of original DLP by applying the Gaudry method to the DLP on the Cab model. In order to carry out the scenario, this paper shows that many of elliptic curve discrete-log problems over quartic extension fields of odd characteristics are reduced to genus two hyperelliptic curve discrete-log problems over quadratic extension fields, and that almost all of the genus two hyperelliptic curve discrete-log 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.