UDC 621.391.15:519.7
SPECIAL PROPERTIES OF THE POINTS ADDITION LAW
OF NON-CYCLIC EDWARDS CURVES
Abstract. The authors analyze the speclial properties of two classes of quadratic and twisted Edwards
curves over a prime field, related to their non-cyclic structure and the incompleteness of the points addition law.
Both classes of curves contain special 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 ambiguous points.
Five theorems are formulated and proved that allow resolving these uncertainties and proving the conditions
whereby the points addition law in these classes of curves is complete.
Keywords: generalized Edwards curve, complete Edwards curve, twisted Edwards curve, quadratic Edwards curve,
curve order, point order, special point, ambiguous point, wheel of points, quadratic residue, quadratic non-residue.
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