Explicit-Formulas Database
Ordinary genus-1 curves over binary fields
Binary Edwards curves EFD / Ordinary genus-1 binary / W coordinates with d1=d2 for binary Edwards curves

W coordinates with d1=d2 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)

W coordinates with d1=d2 [database entry] make the additional assumptions

  d1=d2
and 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: Smallest multiplication counts assuming I=10M, S=0.2M, *param=0M, add=0M, *const=0M:

Summary of all explicit formulas

OperationAssumptionsCostReaddition cost
doubling 1I + 2S + 1*d1
doubling d2overd1plus1=d2/d1+1 1I + 1M + 2S + 1*d2overd1plus1
diffadd 1I + 1M + 2S + 1*d1
diffadd d2overd1plus1=d2/d1+1 1I + 3M + 1S + 1*d2overd1plus1
ladder 2I + 1M + 3S + 2*d1
ladder d2overd1plus1=d2/d1+1 2I + 4M + 3S + 2*d2overd1plus1
scaling 0M

Explicit formulas for addition

Explicit formulas for doubling

The "dbl-2008-blr-1" 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

The "dadd-2008-blr-1" differential-addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

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-1" differential-addition-and-doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

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]: