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Non-Cyclic Subgroups of Jacobians of Genus Two Curves
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Abstract: | Let E be an elliptic curve defined over a finite field. Balasubramanian and Koblitz have proved that if the l-th roots of unity m_l is not contained in the ground field, then a field extension of the ground field contains m_l if and only if the l-torsion points of E are rational over the same field extension. We generalize this result to Jacobians of genus two curves. In particular, we show that the Weil- and the Tate-pairing are non-degenerate over the same field extension of the ground field. From this generalization we get a complete description of the l-torsion subgroups of Jacobians of supersingular genus two curves. In particular, we show that for l>3, the l-torsion points are rational over a field extension of degree at most 24. |
BibTeX
@misc{eprint-2008-17706, title={Non-Cyclic Subgroups of Jacobians of Genus Two Curves}, booktitle={IACR Eprint archive}, keywords={Jacobians, hyperelliptic genus two curves, pairings, embedding degree, supersingular curves}, url={http://eprint.iacr.org/2008/029}, note={ cr@imf.au.dk 13900 received 22 Jan 2008}, author={Christian Robenhagen Ravnshoj}, year=2008 }