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

Projective coordinates for twisted Hessian curves

An elliptic curve in twisted Hessian form [more information] has parameters a d and coordinates x y satisfying the following equations:
  a*x3+y3+1=d*x*y

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 12M + 1*a 12M + 1*a
doubling minustwo=-2 7M + 1S + 1*minustwo + 1*d
doubling i2=-1 and minustwo=-2 and 2d=2*d 8M + 1*i + 1*minustwo + 1*2d
doubling 6M + 3S + 1*a
doubling 3M + 33 + 1*a
tripling d*recipd=1 8M + 6S + 1*a + 1*recipd

Explicit formulas for addition

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

Explicit formulas for doubling

The "dbl-2012-c" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "dbl-2009-bkl-3" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "dbl-2009-bkl-2" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

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

Explicit formulas for tripling

The "tpl-2009-bkl" 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