Journal of the Ramanujan Mathematical Society

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An Addition Algorithm on the Jacobian Varieties of Curves


Affiliations
1 Institute of Information Security, Graduate School of Information Security, 2-14-1, Tsuruya-Cho, Kanagawa-Ku, Yokohama 221-0835, Japan
2 Research and Development Initiative, Chuo University, 1-13-27, Kasuga Bunkyo-Ku, Tokyo 112-8551, Japan
3 Department of Mathematics, Chuo University, 1-13-27, Kasuga Bunkyo-Ku, Tokyo 112-8551, Japan
     

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The present paper computes addition on Jacobian varieties of curves using its singular plane model. Given a nonsingular curve C, let C1 be its singular (especially plane) CA model. After making clear the relationship between generalized Jacobian variety of C1 and Jacobian variety of C, we show that Arita’s algorithm for non singular CA curves also works for generalized Jacobian variety of singular CA curve C1. This means that we can compute addition on Jacobian variety of any curve (with at least one rational point) using its singular CA models.
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  • An Addition Algorithm on the Jacobian Varieties of Curves

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Authors

S. Arita
Institute of Information Security, Graduate School of Information Security, 2-14-1, Tsuruya-Cho, Kanagawa-Ku, Yokohama 221-0835, Japan
S. Miura
Research and Development Initiative, Chuo University, 1-13-27, Kasuga Bunkyo-Ku, Tokyo 112-8551, Japan
T. Sekiguchi
Department of Mathematics, Chuo University, 1-13-27, Kasuga Bunkyo-Ku, Tokyo 112-8551, Japan

Abstract


The present paper computes addition on Jacobian varieties of curves using its singular plane model. Given a nonsingular curve C, let C1 be its singular (especially plane) CA model. After making clear the relationship between generalized Jacobian variety of C1 and Jacobian variety of C, we show that Arita’s algorithm for non singular CA curves also works for generalized Jacobian variety of singular CA curve C1. This means that we can compute addition on Jacobian variety of any curve (with at least one rational point) using its singular CA models.