Explicit-Formulas Database
Ordinary genus-1 curves over binary fields
Short Weierstrass curves EFD / Ordinary genus-1 binary / Affine coordinates for short Weierstrass curves

Affine coordinates for short Weierstrass curves

An elliptic curve in short Weierstrass form [more information] has parameters a2 a6 and coordinates x y satisfying the following equations:
  y2+x*y=x3+a2*x2+a6

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 1I + 2M + 1S 1I + 2M + 1S
doubling 1I + 2M + 1S
scaling 0M

Explicit formulas for addition

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

Explicit formulas for doubling

The "dbl" 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]: