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Erschienen in:
2007 | OriginalPaper | Buchkapitel
Inverted Edwards Coordinates
verfasst von : Daniel J. Bernstein, Tanja Lange
Erschienen in: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Verlag: Springer Berlin Heidelberg
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Edwards curves have attracted great interest for several reasons. When curve parameters are chosen properly, the addition formulas use only 10
M
+ 1
S
. The formulas are
strongly unified
, i.e., work without change for doublings; even better, they are
complete
, i.e., work without change for all inputs. Dedicated doubling formulas use only 3
M
+ 4
S
, and dedicated tripling formulas use only 9
M
+ 4
S
.
This paper introduces
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 9
M
+ 1
S
. The formulas are not complete but still are strongly unified. Dedicated doubling formulas use only 3
M
+ 4
S
, and dedicated tripling formulas use only 9
M
+ 4
S
. Inverted Edwards coordinates thus save 1
M
for each addition, without slowing down doubling or tripling.