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Title:
Inverted Edwards coordinates
Authors: Daniel J. Bernstein, Tanja Lange
Abstract:Edwards curves have attracted great interest for several reasons. When curve parameters are chosen properly, the addition formulas use only 10M+1S. The formulas are {\it strongly unified}, i.e., work without change for doublings; even better, they are {\it complete}, i.e., work without change for all inputs. Dedicated doubling formulas use only 3M+4S, and dedicated tripling formulas use only 9M+4S. This paper introduces {\it inverted Edwards coordinates}. Inverted Edwards coordinates (X_1:Y_1:Z_1) represent the affine point (Z_1/X_1,Z_1/Y_1) on an Edwards curve; for comparison, standard Edwards coordinates (X_1:Y_1:Z_1) represent the affine point (X_1/Z_1,Y_1/Z_1). This paper presents addition formulas for inverted Edwards coordinates using only 9M+1S. The formulas are not complete but still are strongly unified. Dedicated doubling formulas use only 3M+4S, and dedicated tripling formulas use only 9M+4S. Inverted Edwards coordinates thus save 1M for each addition, without slowing down doubling or tripling.
ePrint: https://eprint.iacr.org/2007/410
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