Explicit-Formulas Database
Genus-1 curves over large-characteristic fields
Short Weierstrass curves EFD / Genus-1 large-characteristic / Projective coordinates with a4=-1 for short Weierstrass curves

Projective coordinates with a4=-1 for short Weierstrass curves

An elliptic curve in short Weierstrass form [more information] has parameters a b and coordinates x y satisfying the following equations:
  y2=x3+a*x+b

Projective coordinates with a4=-1 [database entry] make the additional assumptions

  a=-1
and 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 5M + 2S 5M + 2S
addition Z2=1 9M + 2S 9M + 2S
addition 12M + 2S 12M + 2S
addition 13M + 3S 13M + 3S
addition 11M + 6S + 1*a 11M + 6S + 1*a
addition 12M + 5S + 1*a 12M + 5S + 1*a
addition 10M + 4S + 13 10M + 4S + 13
addition 16M + 3S + 33 16M + 3S + 33
doubling Z1=1 3M + 5S
doubling 5M + 6S + 1*a
doubling 6M + 5S + 1*a
doubling 6M + 5S + 13 + 1*a
scaling 1I + 2M

Explicit formulas for addition

The "mmadd-1998-cmo" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "madd-1998-cmo" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "add-1998-cmo-2" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "add-2002-bj-2" 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-2002-bj" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "add-1986-cc" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "add-1998-cmo" 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]:

The "dbl-1998-cmo-2" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "dbl-1998-cmo" 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

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