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

Extended coordinates for Jacobi intersections

An elliptic curve in Jacobi intersection form [more information] has parameters a and coordinates s c d satisfying the following equations:
  s2+c2=1
  a*s2+d2=1

Extended coordinates [database entry] represent s c d as S C D Z SC DZ satisfying the following equations:

  s=S/Z
  c=C/Z
  d=D/Z
  SC=S*C
  DZ=D*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 Z2=1 10M + 1S + 2*a 10M + 1S + 2*a
addition Z1=1 and Z2=1 10M + 2S + 1*a 10M + 1S + 1*a
addition 11M + 1S + 2*a 11M + 1S + 2*a
addition Z2=1 13M + 2S + 1*a 12M + 2S + 1*a
addition S2=1 13M + 2S + 1*a 12M + 2S + 1*a
addition 15M + 2S + 1*a 13M + 2S + 1*a
addition 16M + 2S + 1*a 14M + 2S + 1*a
addition 22M + 2S + 1*a 20M + 2S + 1*a
doubling 2M + 5S + 1*a
doubling Z1=1 4M + 4S
doubling 5M + 4S
doubling 6M + 3S
doubling 7M + 3S
doubling 14M + 9S
tripling b=a-1 and b2=2*b and b3=3*b and bb2=2*b*b 6M + 10S + 1*bb2 + 1*b2 + 2*a + 1*b3
tripling b=a-1 and b2=2*b and bb2=2*b*b and b3=3*b 6M + 10S + 1*bb2 + 1*b2 + 2*a + 1*b3
tripling b=a-1 9M + 7S + 3*b
tripling b=a-1 9M + 7S + 5*b
scaling 1I + 4M

Explicit formulas for addition

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

The "mmadd-2001-ls" 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]:

The "madd-2001-ls" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "smadd-2001-ls" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "add-2001-ls" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "add-1986-cc-2" 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]:

Explicit formulas for doubling

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

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-2001-ls" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "dbl-1986-cc-2" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

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

Explicit formulas for tripling

The "tpl-2007-hcd-4" tripling formulas [database entry; Sage verification script; Sage output; three-operand code]:

The "tpl-2007-hcd-3" 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]:

The "tpl-2007-hcd" 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]: