Explicit-Formulas Database
Genus-1 curves over large-characteristic fields
Short Weierstrass curves EFD / Genus-1 large-characteristic / Modified Jacobian coordinates for short Weierstrass curves

Modified Jacobian coordinates 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

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

  x=X/Z2
  y=Y/Z3
  T=a*Z4

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 3M + 4S + 1*a 3M + 4S + 1*a
addition Z2=1 7M + 6S + 1*a 7M + 6S + 1*a
addition 11M + 7S + 1*a 10M + 6S + 1*a
addition 12M + 6S + 1*a 11M + 5S + 1*a
doubling Z1=1 2M + 5S
doubling 3M + 5S
doubling 4M + 4S

Explicit formulas for addition

The "mmadd-2009-bl" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "madd-2009-bl" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "add-2009-bl" 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]:

Explicit formulas for doubling

The "mdbl-2009-bl" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "dbl-2009-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]:

Explicit formulas for tripling

Explicit formulas for differential addition

Explicit formulas for differential addition and doubling

Explicit formulas for scaling