Explicit-Formulas Database
Genus-1 curves over large-characteristic fields
Short Weierstrass curves EFD / Genus-1 large-characteristic / XYZZ coordinates for short Weierstrass curves

XYZZ coordinates for short Weierstrass curves

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

  y^²=x^³+a*x+b

XYZZ coordinates [database entry] represent x y as X Y ZZ ZZZ satisfying the following equations:

  x=X/ZZ
  y=Y/ZZZ
  ZZ^³=ZZZ^²

Best operation counts

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

14M for addition: 12M+2S.
10M for addition with Z2=1: 8M+2S.
6M for addition with Z1=1 and Z2=1: 4M+2S.
14M for readdition: 12M+2S after 12M+2S.
10M for readdition with Z2=1: 8M+2S after 8M+2S.
6M for readdition with Z1=1 and Z2=1: 4M+2S after 4M+2S.
10M for doubling: 6M+4S.
7M for doubling with Z1=1: 4M+3S.
104M for scaling: 1I+3M+1S.

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

13.6M for addition: 12M+2S.
9.6M for addition with Z2=1: 8M+2S.
5.6M for addition with Z1=1 and Z2=1: 4M+2S.
13.6M for readdition: 12M+2S after 12M+2S.
9.6M for readdition with Z2=1: 8M+2S after 8M+2S.
5.6M for readdition with Z1=1 and Z2=1: 4M+2S after 4M+2S.
9.2M for doubling: 6M+4S.
6.4M for doubling with Z1=1: 4M+3S.
103.8M for scaling: 1I+3M+1S.

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

13.34M for addition: 12M+2S.
9.34M for addition with Z2=1: 8M+2S.
5.34M for addition with Z1=1 and Z2=1: 4M+2S.
13.34M for readdition: 12M+2S after 12M+2S.
9.34M for readdition with Z2=1: 8M+2S after 8M+2S.
5.34M for readdition with Z1=1 and Z2=1: 4M+2S after 4M+2S.
8.68M for doubling: 6M+4S.
6.01M for doubling with Z1=1: 4M+3S.
103.67M for scaling: 1I+3M+1S.

Summary of all explicit formulas

Operation	Assumptions	Cost	Readdition cost
addition	ZZ1=1 and ZZZ1=1 and ZZ2=1 and ZZZ2=1	4M + 2S	4M + 2S
addition	ZZ2=1 and ZZZ2=1	8M + 2S	8M + 2S
addition		12M + 2S	12M + 2S
doubling	ZZ1=1 and ZZZ1=1	4M + 3S
doubling		6M + 4S + 1*a
scaling		1I + 3M + 1S

Explicit formulas for addition

The "mmadd-2008-s" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

Assumptions: ZZ1=1 and ZZZ1=1 and ZZ2=1 and ZZZ2=1.
Cost: 4M + 2S + 6add + 1*2.
Source: 2008 Sutherland.

Explicit formulas:

      P = X2-X1
      R = Y2-Y1
      PP = P^²
      PPP = P*PP
      Q = X1*PP
      X3 = R^²-PPP-2*Q
      Y3 = R*(Q-X3)-Y1*PPP
      ZZ3 = PP
      ZZZ3 = PPP

The "madd-2008-s" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

Assumptions: ZZ2=1 and ZZZ2=1.
Cost: 8M + 2S + 6add + 1*2.
Source: 2008 Sutherland.

Explicit formulas:

      U2 = X2*ZZ1
      S2 = Y2*ZZZ1
      P = U2-X1
      R = S2-Y1
      PP = P^²
      PPP = P*PP
      Q = X1*PP
      X3 = R^²-PPP-2*Q
      Y3 = R*(Q-X3)-Y1*PPP
      ZZ3 = ZZ1*PP
      ZZZ3 = ZZZ1*PPP

The "add-2008-s" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

Cost: 12M + 2S + 6add + 1*2.
Source: 2008 Sutherland.

Explicit formulas:

      U1 = X1*ZZ2
      U2 = X2*ZZ1
      S1 = Y1*ZZZ2
      S2 = Y2*ZZZ1
      P = U2-U1
      R = S2-S1
      PP = P^²
      PPP = P*PP
      Q = U1*PP
      X3 = R^²-PPP-2*Q
      Y3 = R*(Q-X3)-S1*PPP
      ZZ3 = ZZ1*ZZ2*PP
      ZZZ3 = ZZZ1*ZZZ2*PPP

Explicit formulas for doubling

The "mdbl-2008-s-1" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

Assumptions: ZZ1=1 and ZZZ1=1.
Cost: 4M + 3S + 4add + 2*2 + 1*3.
Source: 2008 Sutherland.

Explicit formulas:

      U = 2*Y1
      V = U^²
      W = U*V
      S = X1*V
      M = 3*X1^²+a
      X3 = M^²-2*S
      Y3 = M*(S-X3)-W*Y1
      ZZ3 = V
      ZZZ3 = W

The "dbl-2008-s-1" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

Cost: 6M + 4S + 1*a + 4add + 2*2 + 1*3.
Source: 2008 Sutherland.

Explicit formulas:

      U = 2*Y1
      V = U^²
      W = U*V
      S = X1*V
      M = 3*X1^²+a*ZZ1^²
      X3 = M^²-2*S
      Y3 = M*(S-X3)-W*Y1
      ZZ3 = V*ZZ1
      ZZZ3 = W*ZZZ1

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 + 3M + 1S + 0add.

Explicit formulas:

      A = 1/ZZZ1
      B = (ZZ1*A)^²
      X3 = X1*B
      Y3 = Y1*A
      ZZ3 = 1
      ZZZ3 = 1