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Paper: Inverted Edwards coordinates

Authors:
Daniel J. Bernstein
Tanja Lange
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URL: http://eprint.iacr.org/2007/410
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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.
BibTeX
@misc{eprint-2007-13690,
  title={Inverted Edwards coordinates},
  booktitle={IACR Eprint archive},
  keywords={public-key cryptography / Elliptic curves, addition, doubling, explicit  formulas, Edwards coordinates, inverted Edwards coordinates,  side-channel countermeasures, unified addition formulas,  strongly unified addition formulas.},
  url={http://eprint.iacr.org/2007/410},
  note={ tanja@hyperelliptic.org 13811 received 25 Oct 2007},
  author={Daniel J. Bernstein and Tanja Lange},
  year=2007
}