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*x^³+y^³+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:

12M for addition: 12M.
12M for readdition: 12M after 12M.
8M for doubling: 7M+1S. 8M.
14M for tripling: 8M+6S.

Smallest multiplication counts assuming I=100M, S=0.8M, *param=0M, add=0M, *const=0M:

12M for addition: 12M.
12M for readdition: 12M after 12M.
7.8M for doubling: 7M+1S.
12.8M for tripling: 8M+6S.

Smallest multiplication counts assuming I=100M, S=0.67M, *param=0M, add=0M, *const=0M:

12M for addition: 12M.
12M for readdition: 12M after 12M.
7.67M for doubling: 7M+1S.
12.02M for tripling: 8M+6S.

Summary of all explicit formulas

Operation	Assumptions	Cost	Readdition cost
addition		12M + 1*a	12M + 1*a
doubling	minustwo=-2	7M + 1S + 1minustwo + 1d
doubling	i^²=-1 and minustwo=-2 and 2d=2*d	8M + 1i + 1minustwo + 1*2d
doubling		6M + 3S + 1*a
doubling		3M + 3^³ + 1*a
tripling	d*recipd=1	8M + 6S + 1a + 1recipd

Explicit formulas for addition

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

Cost: 12M + 1*a + 3add.
Source: 2009 Bernstein–Kohel–Lange.
Strongly unified.

Explicit formulas:

      A = X1*Z2
      B = Z1*Z2
      C = Y1*X2
      D = Y1*Y2
      E = Z1*Y2
      F = a*X1*X2
      X3 = A*B-C*D
      Y3 = D*E-F*A
      Z3 = F*C-B*E

Explicit formulas for doubling

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

Assumptions: minustwo=-2.
Cost: 7M + 1S + 1*minustwo + 1*d + 7add + 1*2 + 1*3.
Source: 2012 Chuengsatiansup.

Explicit formulas:

      P = Y1*Z1
      2P = 2*P
      S = Y1+Z1
      A = S^²-P
      C = (A-2P)*S
      D = A*(Z1-Y1)
      E = 3*C-d*X1*2P
      X3 = minustwo*X1*D
      Y3 = (D-E)*Z1
      Z3 = (D+E)*Y1

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

Assumptions: i^²=-1 and minustwo=-2 and 2d=2*d.
Cost: 8M + 1*i + 1*minustwo + 1*2d + 9add + 1*3.
Source: 2009 Bernstein–Kohel–Lange.

Explicit formulas:

      iZ = i*Z1
      A = (Y1-iZ)*(Y1+iZ)
      B = Y1*Z1
      C = (A-B)*(Y1+Z1)
      D = (A+B)*(Z1-Y1)
      E = 3*C-2d*X1*B
      X3 = minustwo*X1*D
      Y3 = (D-E)*Z1
      Z3 = (D+E)*Y1

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

Cost: 6M + 3S + 1*a + 3add.
Source: 2009 Bernstein–Kohel–Lange.

Explicit formulas:

      A = X1^²
      B = Y1^²
      C = Z1^²
      D = A*X1
      E = B*Y1
      F = C*Z1
      G = a*D
      X3 = X1*(E-F)
      Y3 = Z1*(G-E)
      Z3 = Y1*(F-G)

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

Cost: 3M + 3^³ + 1*a + 3add.
Source: 2009 Bernstein–Kohel–Lange.

Explicit formulas:

      D = X1^³
      E = Y1^³
      F = Z1^³
      G = a*D
      X3 = X1*(E-F)
      Y3 = Z1*(G-E)
      Z3 = Y1*(F-G)

Explicit formulas for tripling

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

Assumptions: d*recipd=1.
Cost: 8M + 6S + 1*a + 1*recipd + 12add + 2*2.
Source: 2009 Bernstein–Kohel–Lange.

Explicit formulas:

      U = a*X1*X1^²
      V = Y1*Y1^²
      W = Z1*Z1^²
      A = (U-V)^²
      B = (U-W)^²
      C = (V-W)^²
      D = A+C
      E = A+B
      X3 = recipd*(U+V+W)*(B+D)
      Y3 = 2*U*C-V*(C-E)
      Z3 = 2*V*B-U*(B-D)

Projective coordinates for twisted Hessian curves

Best operation counts

Summary of all explicit formulas

Explicit formulas for addition

Explicit formulas for doubling

Explicit formulas for tripling

Explicit formulas for differential addition

Explicit formulas for differential addition and doubling

Explicit formulas for scaling