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

Lopez-Dahab 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

Lopez-Dahab coordinates [database entry] represent x y as X Y Z satisfying the following equations:

  x=X/Z
  y=Y/Z2

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 Z1=1 and Z2=1 5M + 3S + 1*a2 5M + 3S + 1*a2
addition Z2=1 8M + 5S + 1*a2 8M + 5S + 1*a2
addition 13M + 4S 13M + 3S
doubling Z1=1 1M + 3S + 1*a2 + 1*a6
doubling 3M + 5S + 1*a2 + 1*a6
doubling 4M + 4S + 1*a2

Explicit formulas for addition

The "mmadd-2005-dl" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "madd-2005-dl" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "add-2005-dl" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

Explicit formulas for doubling

The "mdbl-2005-dl" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "dbl-2005-dl" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "dbl-2005-l" 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