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Paper: On the Decomposition of an Element of Jacobian of a Hyperelliptic Curve

Authors:
Koh-ichi Nagao
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URL: http://eprint.iacr.org/2007/112
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Abstract: In this manuscript, if a reduced divisor $D_0$ of hyperelliptic curve of genus $g$ over an extension field $F_{q^n}$ is written by a linear sum of $ng$ lements of $F_{q^n}$-rational points of the hyperelliptic curve whose $x$-coordinates are in the base field $F_q$, $D_0$ is noted by a decomposed divisor and the set of such $F_{q^n}$-rational points is noted by the decomposed factor of $D_0$. We propose an algorithm which checks whether a reduced divisor is decomposed or not, and compute the decomposed factor, if it is decomposed. This algorithm needs a process for solving equations system of degree $2$, $(n^2-n)g$ variables, and $(n^2-n)g$ equations over $F_q$. Further, for the cases $(g,n)=(1,3),(2,2),$ and $(3,2)$, the concrete computations of decomposed factors are done by computer experiments.
BibTeX
@misc{eprint-2007-13394,
  title={On the Decomposition of an Element of Jacobian of a Hyperelliptic Curve},
  booktitle={IACR Eprint archive},
  keywords={ndex calculus attack, Jacobian, Hyperelliptic curve, DLP, Weil descent attack},
  url={http://eprint.iacr.org/2007/112},
  note={ nagao@kanto-gakuin.ac.jp 13662 received 28 Mar 2007, last revised 29 May 2007},
  author={Koh-ichi Nagao},
  year=2007
}