Cryptology ePrint Archive: Report 2007/410

Inverted Edwards coordinates

Daniel J. Bernstein and 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.

Category / 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.

Date: received 25 Oct 2007

Contact author: tanja at hyperelliptic org

Available format(s): PDF | BibTeX Citation

Version: 20071026:095603 (All versions of this report)

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