## Cryptology ePrint Archive: Report 2007/112

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

*Koh-ichi Nagao*

**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.

**Category / Keywords: **Decomposed Attack, Hyperelliptic curve, Discrete logarithm problem,Weil descent attack

**Date: **received 28 Mar 2007, last revised 10 Feb 2008

**Contact author: **nagao at kanto-gakuin ac jp

**Available format(s): **PDF | BibTeX Citation

**Note: **Many correction of minor errors

**Version: **20080210:174616 (All versions of this report)

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