Abstract
Any non-singular plane cubic with a rational point is an elliptic curve, and is therefore birationally equivalent to a curve in Weierstraß form. Such a birational equivalence can be found using generic techniques, but they are computationally quite inefficient.
As early as 1928, Nagell proposed a much simpler procedure to construct that birational equivalence in the particular case of plane cubics, which is implemented in computer algebra packages to this day. However, the procedure fails in even characteristic. We show how the algorithm can be modified to work in any characteristic.
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Tibouchi, M. (2012). A Nagell Algorithm in Any Characteristic. In: Naccache, D. (eds) Cryptography and Security: From Theory to Applications. Lecture Notes in Computer Science, vol 6805. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28368-0_30
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DOI: https://doi.org/10.1007/978-3-642-28368-0_30
Publisher Name: Springer, Berlin, Heidelberg
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