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

Inverted coordinates for Edwards curves

An elliptic curve in Edwards form [more information] has parameters c d and coordinates x y satisfying the following equations:
  x2+y2=c2*(1+d*x2*y2)

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

  x=Z/X
  y=Z/Y

This representation was introduced in the Bernstein–Lange paper "Inverted Edwards coordinates" (AAECC 2007). This representation does not cover the points (0,+-c) and (+-c,0).

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 7M + 2*c 7M + 2*c
addition X2=1 8M + 1S + 2*c + 1*d 8M + 1S + 2*c + 1*d
addition Z2=1 8M + 1S + 2*c + 1*d 8M + 1S + 2*c + 1*d
addition Z2=1 9M + 1*c 9M + 1*c
addition 9M + 1S + 2*c + 1*d 9M + 1S + 2*c + 1*d
addition 11M + 1*c 11M + 1*c
doubling ccd2=2*c*c*d and Z1=1 3M + 3S + 1*c
doubling ccd2=2*c*c*d 3M + 4S + 1*ccd2 + 1*c
tripling 9M + 4S + 1*c + 1*d
tripling ccd=c*c*d 7M + 7S + 1*ccd
scaling 1I + 2M

Explicit formulas for addition

The "mmadd-2007-bl" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "xmadd-2007-bl" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "madd-2007-bl" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "madd-20080225-hwcd" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "add-2007-bl" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "add-20080225-hwcd" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

Explicit formulas for doubling

The "mdbl-2007-bl" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "dbl-2007-bl" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

Explicit formulas for tripling

The "tpl-2007-bl" tripling formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "tpl-2007-bl-2" 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

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