Explicit-Formulas Database
Ordinary genus-1 curves over binary fields
Binary Edwards curves EFD / Ordinary genus-1 binary / Affine coordinates for binary Edwards curves

Affine 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*(x2+y2)=(x+x2)*(y+y2)

Affine coordinates [database entry] represent x y as X Y satisfying the following equations:

  x=X
  y=Y

Best operation counts

Smallest multiplication counts assuming I=10M, S=0M, *param=0M, add=0M, *const=0M: Smallest multiplication counts assuming I=10M, S=0.2M, *param=0M, add=0M, *const=0M:

Summary of all explicit formulas

OperationAssumptionsCostReaddition cost
addition 2I + 8M + 2S + 2*d1 + 1*d2 2I + 7M + 2S + 2*d1 + 1*d2
doubling d2d1=d2/d1 1I + 2M + 4S + 1*d2 + 1*d2d1
doubling 4I + 4M + 10S + 64 + 4*d2
scaling 0M

Explicit formulas for addition

The "add-2008-blr" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

Explicit formulas for doubling

The "dbl-2008-blr+cse" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

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

Explicit formulas for differential addition and doubling

Explicit formulas for scaling

The "copy" scaling formulas [database entry; Sage verification script; Sage output; three-operand code]: