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Decomposed Attack for the Jacobian of a Hyperelliptic Curve over an Extension Field

http://www.firstlight.cn2008/9/8

[作者] Koh-ichi Nagao

[单位] Dept. of Engineering, Kanto Gakuin Univ.

[摘要] We study the solution of the discrete logarithm problem for the Jacobian of a curve of genus g defined over an extension field Fqn, by decomposed attack, considering a external elements B0 given by points of…

[关键词] Decomposed Attack Hyperelliptic curve Discrete logarithm problem

Abstract. We study the solution of the discrete logarithm problem for

the Jacobian of a curve of genus g defined over an extension field Fqn, by

decomposed attack, considering a external elements B0 given by points

of the curve whose x-coordinates are defined in Fq. In the decomposed

attack, an element of the group which is written by a sum of some elements

of external elements is called (potentially) decomposed and the

set of the terms, that appear in the sum, is called decomposed factor. In

order for the running of the decomposed attack, a test for the (potential)

decomposeness and the computation of the decomposed factor are

needed. Here, we show that the test to determine if an element of the

Jacobian (i.e., reduced divisor) is written by an ng sum of the elements

of the external elements and the computation of decomposed factor are

reduced to the problem of solving some multivariable polynomial system

of equations by using the Riemann-Roch theorem. In particular, in the

case of a hyperelliptic curve, we construct a concrete system of equations,

which satisfies these properties and consists of (n2¡n)g quadratic

equations. Moreover, in the case of (g; n) = (1; 3); (2; 2) and (3; 2), we

give examples of the concrete computation of the decomposed factors by

using the computer algebra system Magma.

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原文发布时间:2008/9/8

引用本文:

Koh-ichi Nagao.Decomposed Attack for the Jacobian of a Hyperelliptic Curve over an Extension Fieldhttp://bsnc.firstlight.cn/View.aspx?infoid=502756&cb=zhangyingyingxg
发布时间:2008/9/8.检索时间:2024/12/18

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