Explicit-Formulas Database

Ordinary genus-1 curves over binary fields

Binary Edwards curves
EFD / Ordinary genus-1 binary / W coordinates for binary Edwards curves
# W coordinates for binary Edwards curves

An elliptic curve in binary Edwards form
[more information]
has parameters
d1
d2
and coordinates
x
y
satisfying the following equations:
d1*(x+y)+d2*(x^^{2}+y^^{2})=(x+x^^{2})*(y+y^^{2})

W coordinates
[database entry]
represent
x
y
as
w
satisfying the following equations:

x+y=w

## Best operation counts

Smallest multiplication counts assuming I=10M, S=0M, *param=0M, add=0M, *const=0M:
- 11M for doubling: 1I+1M+2S.
- 13M for differential addition: 1I+3M+1S.
- 24M for differential addition and doubling: 2I+4M+3S.
- 0M for scaling: 0M.

Smallest multiplication counts assuming I=10M, S=0.2M, *param=0M, add=0M, *const=0M:
- 11.4M for doubling: 1I+1M+2S.
- 13.2M for differential addition: 1I+3M+1S.
- 24.6M for differential addition and doubling: 2I+4M+3S.
- 0M for scaling: 0M.

## Summary of all explicit formulas

## Explicit formulas for addition

## Explicit formulas for doubling

The "dbl-2008-blr" doubling formulas
[database entry;
Sage verification script;
Sage output;
three-operand code]:
## Explicit formulas for tripling

## Explicit formulas for differential addition

The "dadd-2008-blr" differential-addition formulas
[database entry;
Sage verification script;
Sage output;
three-operand code]:
## Explicit formulas for differential addition and doubling

The "ladd-2008-blr" differential-addition-and-doubling formulas
[database entry;
Sage verification script;
Sage output;
three-operand code]:
## Explicit formulas for scaling

The "copy" scaling formulas
[database entry;
Sage verification script;
Sage output;
three-operand code]:
- Cost: 0add.
- Explicit formulas:
w3 = w1