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

Lambda 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

Lambda coordinates [database entry] represent x y as X L Z satisfying the following equations:

  x=X/Z
  y/x=(L-X)/Z

Source: 2013 Oliveira–López–Aranha–Rodríguez-Henríquez "Lambda coordinates for binary elliptic curves".

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 11M + 2S 11M + 2S
doubling a21=a2+1 and a226=a22+a6 3M + 4S + 1*a2 + 1*a226 + 1*a21
doubling 4M + 4S + 1*a2

Explicit formulas for addition

The "add-2013-olar" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

Explicit formulas for doubling

The "dbl-2013-olar-2" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "dbl-2013-olar" 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