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

XYZ coordinates for Jacobi quartics

An elliptic curve in Jacobi quartic form [more information] has parameters a and coordinates x y satisfying the following equations:

  y^²=x^⁴+2*a*x^²+1

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

  x=X/Z
  y=Y/Z^²

Best operation counts

Smallest multiplication counts assuming I=100M, S=1M, *param=0M, add=0M, *const=0M:

13M for addition: 10M+3S.
11M for addition with Z2=1: 8M+3S.
7M for addition with Z1=1 and Z2=1: 5M+2S.
11M for readdition: 8M+3S after 8M+6S. 9M+2S after 10M+4S. 9M+2S after 10M+4S.
11M for readdition with Z2=1: 8M+3S after 8M+3S. 8M+3S after 8M+6S. 9M+2S after 10M+4S. 9M+2S after 10M+4S.
7M for readdition with Z1=1 and Z2=1: 5M+2S after 5M+2S.
8M for doubling: 2M+6S. 2M+6S.
5M for doubling with Z1=1: 1M+4S.
103M for scaling: 1I+2M+1S.

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

12.4M for addition: 10M+3S.
10.4M for addition with Z2=1: 8M+3S.
6.6M for addition with Z1=1 and Z2=1: 5M+2S.
10.4M for readdition: 8M+3S after 8M+6S.
10.4M for readdition with Z2=1: 8M+3S after 8M+3S. 8M+3S after 8M+6S.
6.6M for readdition with Z1=1 and Z2=1: 5M+2S after 5M+2S.
6.8M for doubling: 2M+6S. 2M+6S.
4.2M for doubling with Z1=1: 1M+4S.
102.8M for scaling: 1I+2M+1S.

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

12.01M for addition: 10M+3S.
10.01M for addition with Z2=1: 8M+3S.
6.34M for addition with Z1=1 and Z2=1: 5M+2S.
10.01M for readdition: 8M+3S after 8M+6S.
10.01M for readdition with Z2=1: 8M+3S after 8M+3S. 8M+3S after 8M+6S.
6.34M for readdition with Z1=1 and Z2=1: 5M+2S after 5M+2S.
6.02M for doubling: 2M+6S. 2M+6S.
3.68M for doubling with Z1=1: 1M+4S.
102.67M for scaling: 1I+2M+1S.

Summary of all explicit formulas

Operation	Assumptions	Cost	Readdition cost
addition	Z1=1 and Z2=1	5M + 2S + 1*a	5M + 2S + 1*a
addition	Z2=1	8M + 3S + 1*a	8M + 3S + 1*a
addition		10M + 3S + 1*a	9M + 3S + 1*a
addition		8M + 6S + 1*a	8M + 3S + 1*a
addition		10M + 4S + 1*a	9M + 2S + 1*a
addition	b=-2*a	10M + 4S + 1*b	9M + 2S + 1*b
addition		19M + 8S + 1*a	18M + 6S + 1*a
addition	Z1=1 and Z2=1	2I + 11M + 5S + 1*a	2I + 11M + 4S + 1*a
doubling	a2=2*a and Z1=1	1M + 4S + 1*a2
doubling	b=4-4*a^²	2M + 6S + 1a + 1b
doubling	a2=2*a	2M + 6S + 1*a2
doubling	a2=2*a	3M + 6S + 2*a2
doubling		1M + 9S + 1*a
doubling		19M + 8S + 1*a
doubling		19M + 8S + 1*a
scaling		1I + 2M + 1S

Explicit formulas for addition

The "mmadd-2002-bj-2" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

Assumptions: Z1=1 and Z2=1.
Cost: 5M + 2S + 1*a + 10add + 1*2.
Cost: 5M + 2S + 1*a + 9add + 1*2 dependent upon the first point.
Source: 2002 Billet–Joye "The Jacobi model of an elliptic curve and side-channel analysis", page 6, plus specialization to Z1=1 and Z2=1.
Strongly unified.

Explicit formulas:

      T7 = Y1+X1
      T8 = Y2+X2
      T2 = Y1*Y2
      T7 = T7*T8
      T7 = T7-T2
      T5 = X1*X2
      T6 = T5
      X3 = T7-T6
      T1 = X1+X2
      T3 = T1^²
      T6 = 2*T6
      T3 = T3-T6
      T3 = T3*T6
      T4 = a*T6
      T2 = T2+T4
      T8 = T5^²
      T5 = T8+1
      T2 = T2*T5
      Y3 = T2+T3
      Z3 = 1-T8

The "madd-2002-bj" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

Assumptions: Z2=1.
Cost: 8M + 3S + 1*a + 13add + 1*2.
Cost: 8M + 3S + 1*a + 11add + 1*2 dependent upon the first point.
Source: 2002 Billet–Joye "The Jacobi model of an elliptic curve and side-channel analysis", page 6, plus specialization to Z2=1.
Strongly unified.

Explicit formulas:

      T1 = X1
      T2 = Y1
      T3 = Z1
      T4 = X2
      T5 = Y2
      T7 = T1*T3
      T7 = T2+T7
      T8 = T4
      T8 = T5+T8
      T2 = T2*T5
      T7 = T7*T8
      T7 = T7-T2
      T5 = T1*T4
      T1 = T1+T3
      T8 = T3
      T4 = T4+1
      T6 = T5*T8
      T7 = T7-T6
      T1 = T1*T4
      T1 = T1-T5
      T1 = T1-T8
      T3 = T1^²
      T6 = 2*T6
      T3 = T3-T6
      T3 = T3*T6
      T4 = a*T6
      T2 = T2+T4
      T4 = T8^²
      T8 = T5^²
      T5 = T4+T8
      T2 = T2*T5
      T2 = T2+T3
      T5 = T4-T8
      X3 = T7
      Y3 = T2
      Z3 = T5

The "add-2002-bj-2" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

Cost: 10M + 3S + 1*a + 13add + 1*2.
Cost: 9M + 3S + 1*a + 11add + 1*2 dependent upon the first point.
Source: 2002 Billet–Joye "The Jacobi model of an elliptic curve and side-channel analysis", page 6.
Strongly unified.

Explicit formulas:

      T1 = X1
      T2 = Y1
      T3 = Z1
      T4 = X2
      T5 = Y2
      T6 = Z2
      T7 = T1*T3
      T7 = T2+T7
      T8 = T4*T6
      T8 = T5+T8
      T2 = T2*T5
      T7 = T7*T8
      T7 = T7-T2
      T5 = T1*T4
      T1 = T1+T3
      T8 = T3*T6
      T4 = T4+T6
      T6 = T5*T8
      T7 = T7-T6
      T1 = T1*T4
      T1 = T1-T5
      T1 = T1-T8
      T3 = T1^²
      T6 = 2*T6
      T3 = T3-T6
      T3 = T3*T6
      T4 = a*T6
      T2 = T2+T4
      T4 = T8^²
      T8 = T5^²
      T5 = T4+T8
      T2 = T2*T5
      T2 = T2+T3
      T5 = T4-T8
      X3 = T7
      Y3 = T2
      Z3 = T5

The "add-2007-bl" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

Cost: 8M + 6S + 1*a + 15add + 2*2 + 1*4.
Cost: 8M + 3S + 1*a + 11add + 2*2 + 1*4 dependent upon the first point.
Source: 2007 Bernstein–Lange.
Strongly unified.

Explicit formulas:

      A2 = X2^²
      C2 = Z2^²
      D2 = A2+C2
      B2 = (X2+Z2)^²-D2
      E2 = B2+Y2
      A1 = X1^²
      C1 = Z1^²
      D1 = A1+C1
      B1 = (X1+Z1)^²-D1
      E1 = B1+Y1
      A1A2 = A1*A2
      B1B2 = B1*B2
      C1C2 = C1*C2
      Y1Y2 = Y1*Y2
      F = C1C2+A1A2
      G = 2*B1B2
      X3 = E1*E2-B1B2-Y1Y2
      Y3 = F*(4*Y1Y2+a*G)+(D1*D2-F)*G
      Z3 = 2*(C1C2-A1A2)

The "add-2002-bj-3" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

Cost: 10M + 4S + 1*a + 11add + 1*2.
Cost: 9M + 2S + 1*a + 9add + 1*2 dependent upon the first point.
Source: 2002 Billet–Joye "The Jacobi model of an elliptic curve and side-channel analysis", formula (11), plus separation of X2^²,Z2^² as suggested by 2007 Duquesne.
Strongly unified.

Explicit formulas:

      A2 = X2^²
      B2 = X2*Z2
      C2 = Z2^²
      A1 = X1^²
      B1 = X1*Z1
      C1 = Z1^²
      A1A2 = A1*A2
      B1B2 = B1*B2
      C1C2 = C1*C2
      Y1Y2 = Y1*Y2
      E = C1C2+A1A2
      F = (A1+C1)*(A2+C2)-E
      G = 2*B1B2
      X3 = (B1+Y1)*(B2+Y2)-B1B2-Y1Y2
      Y3 = E*(Y1Y2+a*G)+G*F
      Z3 = C1C2-A1A2

The "add-2007-d" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

Assumptions: b=-2*a.
Cost: 10M + 4S + 1*b + 12add + 1*2.
Cost: 9M + 2S + 1*b + 10add + 1*2 dependent upon the first point.
Source: 2007 Duquesne, page 103, Table 1.
Strongly unified.

Explicit formulas:

      T1 = X1^²
      T2 = X2^²
      T3 = X1*Z1
      T4 = X2*Z2
      T5 = Z1^²
      T6 = Z2^²
      T7 = Y1
      T8 = Y2
      T9 = T7*T8
      T7 = T7+T3
      T8 = T8+T4
      T3 = T3*T4
      T7 = T7*T8
      T7 = T7-T9
      T7 = T7-T3
      T4 = T1*T2
      T8 = T5*T6
      T1 = T1+T5
      T2 = T2+T6
      T5 = T1*T2
      T5 = T5-T4
      T5 = T5-T8
      T1 = T8-T4
      T2 = T8+T4
      T6 = b*T3
      T6 = T9-T6
      T6 = T6*T2
      T3 = 2*T3
      T3 = T5*T3
      T8 = T6+T3
      X3 = T7
      Y3 = T8
      Z3 = T1

The "add-2002-bj" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

Cost: 19M + 8S + 1*a + 6add + 2*2.
Cost: 18M + 6S + 1*a + 6add + 2*2 dependent upon the first point.
Source: 2002 Billet–Joye "The Jacobi model of an elliptic curve and side-channel analysis", formula (11).
Strongly unified.

Explicit formulas:

      X3 = X1*Z1*Y2+Y1*X2*Z2
      Y3 = ((Z1*Z2)^²+(X1*X2)^²)*(Y1*Y2+2*a*X1*X2*Z1*Z2)+2*X1*X2*Z1*Z2*(X1^²*Z2^²+Z1^²*X2^²)
      Z3 = (Z1*Z2)^²-(X1*X2)^²

The "mmadd-2002-bj" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

Assumptions: Z1=1 and Z2=1.
Cost: 2I + 11M + 5S + 1*a + 6add + 2*2.
Cost: 2I + 11M + 4S + 1*a + 6add + 2*2 dependent upon the first point.
Source: 2002 Billet–Joye "The Jacobi model of an elliptic curve and side-channel analysis", formula (11), plus specialization to affine.
Strongly unified.

Explicit formulas:

      r = 1-(X1*X2)^²
      X3 = (X1*Y2+Y1*X2)/r
      Y3 = ((1+(X1*X2)^²)*(Y1*Y2+2*a*X1*X2)+2*X1*X2*(X1^²+X2^²))/r^²
      Z3 = 1

Explicit formulas for doubling

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

Assumptions: a2=2*a and Z1=1.
Cost: 1M + 4S + 1*a2 + 9add + 1*4.
Source: 2007 Feng–Wu, first JQN2 doubling formula, plus correction of obvious typos, plus epsilon=1, plus delta=-a, plus Z1=1, plus common-subexpression elimination.

Explicit formulas:

      U1 = (X1+Y1)^²
      U2 = Y1^²
      S1 = X1^²
      V1 = S1^²
      aS1 = a2*S1
      T = U2-V1-aS1
      X3 = U1-U2-S1
      Y3 = (T+V1)*(U2+aS1)+4*V1
      Z3 = T-V1

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

Assumptions: b=4-4*a^².
Cost: 2M + 6S + 1*a + 1*b + 5add + 1*2.
Source: 2007 Hisil–Carter–Dawson.

Explicit formulas:

      T0 = X1*Z1
      X3 = T0*Y1
      X3 = X3+X3
      T0 = T0^²
      T1 = 2*a*T0
      T2 = Y1^²
      Z3 = T2-T1
      Y3 = T0^²
      Y3 = b*Y3
      T0 = T2^²
      Y3 = Y3+T0
      T0 = X1^²
      T0 = T0^²
      Z3 = Z3-T0
      Z3 = Z3-T0

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

Assumptions: a2=2*a.
Cost: 2M + 6S + 1*a2 + 9add + 1*2.
Source: 2007 Feng–Wu, first JQN2 doubling formula, plus correction of obvious typos, plus epsilon=1, plus delta=-a, plus common-subexpression elimination.

Explicit formulas:

      XZ1 = X1*Z1
      U1 = (XZ1+Y1)^²
      U2 = Y1^²
      V1 = (X1^²)^²
      S1 = XZ1^²
      aS1 = a2*S1
      T = U2-V1-aS1
      X3 = U1-U2-S1
      Y3 = (T+V1)*(U2+aS1)+(2*S1)^²
      Z3 = T-V1

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

Assumptions: a2=2*a.
Cost: 3M + 6S + 2*a2 + 9add + 1*4.
Source: 2007 Feng–Wu, first JQN2 doubling formula, plus correction of obvious typos, plus epsilon=1, plus substitution delta=-a.

Explicit formulas:

      U1 = (X1*Z1+Y1)^²
      U2 = Y1^²
      V1 = (X1^²)^²
      S1 = (X1*Z1)^²
      T = U2-V1-a2*S1
      X3 = U1-U2-S1
      Y3 = (T+V1)*(U2+a2*S1)+4*S1^²
      Z3 = T-V1

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

Cost: 1M + 9S + 1*a + 10add + 2*2 + 1*4.
Source: 2007 Bernstein–Lange.

Explicit formulas:

      XX = X1^²
      XXXX = XX^²
      YY = Y1^²
      ZZ = Z1^²
      ZZZZ = ZZ^²
      M = XX+ZZ
      XZ = (X1+Z1)^²-M
      XZXZ = XZ^²
      X3 = (Y1+XZ)^²-YY-XZXZ
      Y3 = (ZZZZ+XXXX)*(4*YY+2*a*XZXZ)+XZXZ^²
      Z3 = 2*(ZZZZ-XXXX)

The "dbl-2002-bj-2" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

Cost: 19M + 8S + 1*a + 6add + 2*2.
Source: 2002 Billet–Joye "The Jacobi model of an elliptic curve and side-channel analysis", formula (11), plus specialization to doubling.

Explicit formulas:

      X3 = X1*Z1*Y1+Y1*X1*Z1
      Y3 = ((Z1*Z1)^²+(X1*X1)^²)*(Y1*Y1+2*a*X1*X1*Z1*Z1)+2*X1*X1*Z1*Z1*(X1^²*Z1^²+Z1^²*X1^²)
      Z3 = (Z1*Z1)^²-(X1*X1)^²

The "dbl-2002-bj" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

Cost: 19M + 8S + 1*a + 6add + 2*2.
Source: 2002 Billet–Joye "The Jacobi model of an elliptic curve and side-channel analysis", formula (11).

Explicit formulas:

      X2 = X1
      Y2 = Y1
      Z2 = Z1
      X3 = X1*Z1*Y2+Y1*X2*Z2
      Y3 = ((Z1*Z2)^²+(X1*X2)^²)*(Y1*Y2+2*a*X1*X2*Z1*Z2)+2*X1*X2*Z1*Z2*(X1^²*Z2^²+Z1^²*X2^²)
      Z3 = (Z1*Z2)^²-(X1*X2)^²

Explicit formulas for tripling

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]:

Cost: 1I + 2M + 1S + 0add.

Explicit formulas:

      A = 1/Z1
      X3 = X1*A
      Y3 = Y1*A^²
      Z3 = 1