Explicit-Formulas Database
Genus-1 curves over large-characteristic fields
Jacobi quartics EFD / Genus-1 large-characteristic / XXYZZ coordinates for Jacobi quartics

XXYZZ coordinates for Jacobi quartics

An elliptic curve in Jacobi quartic form [more information] has parameters a and coordinates x y satisfying the following equations:
  y2=x4+2*a*x2+1

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

  x=X/Z
  y=Y/ZZ
  XX=X2
  ZZ=Z2

This representation was introduced by Hisil, Carter, and Dawson in the paper "New formulae for efficient elliptic curve arithmetic" at Indocrypt 2007.

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 Z2=1 and k=a-1 6M + 3S + 1*k 6M + 3S + 1*k
addition k=a-1 7M + 4S + 1*k 7M + 3S + 1*k
doubling Z1=1 6S + 1*a
doubling Z1=1 1M + 5S
doubling 2M + 5S + 1*a
doubling 3M + 4S
doubling 1M + 8S + 1*a
doubling a2=2*a 3M + 8S + 1*a2 + 1*a
tripling 8M + 6S + 1*a
tripling b=a2-1 4M + 11S + 1*a + 1*b
scaling 1I + 2M + 2S

Explicit formulas for addition

The "madd-20080225-hwcd" 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-20090311-hwcd" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

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

The "dbl-20090311-hwcd" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

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

The "dbl-2007-fw-2" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

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

Explicit formulas for tripling

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

The "tpl-2007-hcd-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]: