Explicit-Formulas Database
Genus-1 curves over large-characteristic fields
Twisted Edwards curves EFD / Genus-1 large-characteristic / Projective coordinates for twisted Edwards curves

Projective coordinates for twisted Edwards curves

An elliptic curve in twisted Edwards form [more information] has parameters a d and coordinates x y satisfying the following equations:
  a*x2+y2=1+d*x2*y2

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

  x=X/Z
  y=Y/Z

Best operation counts

Smallest multiplication counts assuming I=100M, S=1M, *param=0M, add=0M, *const=0M: Smallest multiplication counts assuming I=100M, S=0.8M, *param=0M, add=0M, *const=0M: Smallest multiplication counts assuming I=100M, S=0.67M, *param=0M, add=0M, *const=0M:

Summary of all explicit formulas

OperationAssumptionsCostReaddition cost
addition Z1=1 and Z2=1 6M + 1S + 1*a + 1*d 6M + 1S + 1*a + 1*d
addition Z2=1 9M + 1S + 1*a + 1*d 9M + 1S + 1*a + 1*d
addition 10M + 1S + 1*a + 1*d 10M + 1S + 1*a + 1*d
doubling Z1=1 2M + 4S + 1*a
doubling 3M + 4S + 1*a
tripling 9M + 3S + 1*a

Explicit formulas for addition

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

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

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

Explicit formulas for doubling

The "mdbl-2008-bbjlp" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

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

Explicit formulas for tripling

The "tpl-2015-c" tripling formulas [database entry; Sage verification script; Sage output; three-operand code]:

Explicit formulas for differential addition

Explicit formulas for differential addition and doubling

Explicit formulas for scaling