Special properties of the points addition law of non-cyclic Edwards curves / Bessalov, / Abramov. (2022)
Ukrainian

English  Cybernetics and Systems Analysis   /     Issue (2022, 58 (6))

Bessalov A.V., Abramov S.V.
Special properties of the points addition law of non-cyclic Edwards curves

The authors analyze the special properties of two classes of quadratic and twisted Edwards curves over a prime field, which take into account their non-cyclic structure and the incompleteness of the point addition law. Both classes of curves contain singular points of 2nd and 4th orders with respect to one infinite coordinate, which generate points with uncertainty 0/0 in one of the coordinates of the sum, called fuzzy points. Five theorems are formulated and proved, which allow resolving these uncertainties and establishing the conditions whereby the point addition law in these classes of curves is complete. © 2023, Springer Science+Business Media, LLC, part of Springer Nature.

Keywords: complete Edwards curve, curve order, fuzzy point, generalized Edwards curve, point order, point wheel, quadratic Edwards curve, quadratic non-residue, quadratic residue, singular point, twisted Edwards curve, Complete edward curve, Curve order, Fuzzy point, Generalized edward curve, Point order, Point wheel, Quadratic edward curve, Quadratic non-residue, Quadratic residues, Singular points, Twisted edward curve, Algebra


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Cite:
Bessalov A.V., Abramov S.V. (2022). Special properties of the points addition law of non-cyclic Edwards curves. Cybernetics and Systems Analysis, 58 (6), 3–14. doi: https://doi.org/10.1007/s10559-023-00518-w http://jnas.nbuv.gov.ua/article/UJRN-0001368544 [In Ukrainian].


 

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