Inverted Edwards coordinates

Daniel J.  Bernstein; Tanja Lange

Paper 2007/410 Original: eprint.iacr.org/2007/410 ↗

Inverted Edwards coordinates

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.

Metadata

Available format(s): PDF
Category: Public-key cryptography
Publication info: Preprint.
History: 2007-10-26: last updated; 2007-10-26: published
Short URL: https://ia.cr/2007/410
License: https://creativecommons.org/licenses/by/4.0/

BibTeX

@misc{cryptoeprint:2007/410,
      author = {Daniel J.  Bernstein and Tanja Lange},
      title = {Inverted Edwards coordinates},
      howpublished = {Cryptology {ePrint} Archive, Paper 2007/410},
      year = {2007},
      url = {https://eprint.iacr.org/2007/410}
}