Explicit-Formulas Database
Genus-1 curves over large-characteristic fields
Edwards curves EFD / Genus-1 large-characteristic / Squared YZ coordinates with square d for Edwards curves

Squared YZ coordinates with square d for Edwards curves

An elliptic curve in Edwards form [more information] has parameters c d and coordinates x y satisfying the following equations:

  x^²+y^²=c^²*(1+d*x^²*y^²)

Squared YZ coordinates with square d [database entry] make the additional assumptions

  c=1
  d=r^²

and represent x y as Y Z satisfying the following equations:

  r*y^²=Y/Z

The following formulas are the outcome of a discussion in 2009 among Daniel J. Bernstein, David Kohel, and Tanja Lange. The core ideas were published by Pierrick Gaudry in 2006.

Best operation counts

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

4M for doubling: 4S.
3M for doubling with Z1=1: 3S.
6M for differential addition: 4M+2S.
5M for differential addition with Z1=1: 3M+2S.
10M for differential addition and doubling: 4M+6S. 4M+6S.
9M for differential addition and doubling with Z1=1: 3M+6S. 3M+6S.
101M for scaling: 1I+1M.

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

3.2M for doubling: 4S.
2.4M for doubling with Z1=1: 3S.
5.6M for differential addition: 4M+2S.
4.6M for differential addition with Z1=1: 3M+2S.
8.8M for differential addition and doubling: 4M+6S. 4M+6S.
7.8M for differential addition and doubling with Z1=1: 3M+6S. 3M+6S.
101M for scaling: 1I+1M.

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

2.68M for doubling: 4S.
2.01M for doubling with Z1=1: 3S.
5.34M for differential addition: 4M+2S.
4.34M for differential addition with Z1=1: 3M+2S.
8.02M for differential addition and doubling: 4M+6S. 4M+6S.
7.02M for differential addition and doubling with Z1=1: 3M+6S. 3M+6S.
101M for scaling: 1I+1M.

Summary of all explicit formulas

Operation	Assumptions	Cost
doubling	Z1=1 and s=(1+r)/(1-r)	3S + 1r + 1s
doubling	s=(1+r)/(1-r)	4S + 1r + 1s
diffadd	Z1=1 and s=(1+r)/(1-r)	3M + 2S + 1*s
diffadd	s=(1+r)/(1-r)	4M + 2S + 1*s
ladder	Z1=1 and s=(1+r)/(1-r)	3M + 6S + 1r + 2s
ladder	Z1=1 and s=(1+r)/(1-r)	3M + 6S + 1r + 2s
ladder	s=(1+r)/(1-r)	4M + 6S + 1r + 2s
ladder	s=(1+r)/(1-r)	4M + 6S + 1r + 2s
scaling		1I + 1M

Explicit formulas for addition

Explicit formulas for doubling

The "mdbl-2006-g" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

Assumptions: Z1=1 and s=(1+r)/(1-r).
Cost: 3S + 1*r + 1*s + 4add + 1*4.
Source: 2006 Gaudry "Variants of the Montgomery form based on Theta functions", page 22/52, with A^²/B^² = (a^²+b^²)/(a^²-b^²) as on page 20/52, replacing incorrect B^²/A^² and b/a on page 22/52 with correct A^²/B^² and a/b; or 2009 Gaudry–Lubicz "The arithmetic of characteristic 2 Kummer surfaces and of elliptic Kummer lines", Section 6.2, replacing incorrect A'/B' = (a^²+b^²)/(a^²-b^²) with correct A'^²/B'^² = (a^²+b^²)/(a^²-b^²), replacing A'^²/B'^² with A^²/B^², and replacing z... with y...; plus notation changes: a/b and A^²/B^² defined as 1/sqrt(r) and (1+r)/(1-r), input x^²/y^² replaced by Z1/Y1, intermediate x'/y' replaced by W/V, output X^²/Y^² replaced by Z3/Y3; plus assumption Z1=1; plus standard simplification.

Explicit formulas:

      W = (1+Y1)^²
      V = s*(W-4*Y1)
      Y3 = (W-V)^²
      Z3 = r*(W+V)^²

The "dbl-2006-g" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

Assumptions: s=(1+r)/(1-r).
Cost: 4S + 1*r + 1*s + 4add.
Source: 2006 Gaudry "Variants of the Montgomery form based on Theta functions", page 22/52, with A^²/B^² = (a^²+b^²)/(a^²-b^²) as on page 20/52, replacing incorrect B^²/A^² and b/a on page 22/52 with correct A^²/B^² and a/b; or 2009 Gaudry–Lubicz "The arithmetic of characteristic 2 Kummer surfaces and of elliptic Kummer lines", Section 6.2, replacing incorrect A'/B' = (a^²+b^²)/(a^²-b^²) with correct A'^²/B'^² = (a^²+b^²)/(a^²-b^²), replacing A'^²/B'^² with A^²/B^², and replacing z... with y...; plus notation changes: a/b and A^²/B^² defined as 1/sqrt(r) and (1+r)/(1-r), input x^²/y^² replaced by Z1/Y1, intermediate x'/y' replaced by W/V, output X^²/Y^² replaced by Z3/Y3.

Explicit formulas:

      V = s*(Z1-Y1)^²
      W = (Z1+Y1)^²
      Y3 = (W-V)^²
      Z3 = r*(W+V)^²

Explicit formulas for tripling

Explicit formulas for differential addition

The "mdadd-2006-g" differential-addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

Assumptions: Z1=1 and s=(1+r)/(1-r).
Cost: 3M + 2S + 1*s + 6add.
Source: 2006 Gaudry "Variants of the Montgomery form based on Theta functions", page 23/52, with A^²/B^² = (a^²+b^²)/(a^²-b^²) as on page 20/52, replacing incorrect B^²/A^² on page 23/52 with correct A^²/B^² and a/b; or 2009 Gaudry–Lubicz "The arithmetic of characteristic 2 Kummer surfaces and of elliptic Kummer lines", Section 6.2, replacing incorrect A'/B' = (a^²+b^²)/(a^²-b^²) with correct A'^²/B'^² = (a^²+b^²)/(a^²-b^²), replacing A'^²/B'^² with A^²/B^², and replacing z... with y...; plus notation changes: a/b and A^²/B^² defined as 1/sqrt(r) and (1+r)/(1-r), input x^²/y^² etc. replaced by Z2/Y2 and Z3/Y3 and Z1/Y1, intermediate x'/y' replaced by W/V, output X^²/Y^² replaced by Z5/Y5; plus assumption Z1=1.

Explicit formulas:

      V = s*(Z2-Y2)*(Z3-Y3)
      W = (Z2+Y2)*(Z3+Y3)
      Y5 = (W-V)^²
      Z5 = Y1*(W+V)^²

The "dadd-2006-g" differential-addition formulas [database entry; Sage verification script; Sage output; three-operand code]:

Assumptions: s=(1+r)/(1-r).
Cost: 4M + 2S + 1*s + 6add.
Source: 2006 Gaudry "Variants of the Montgomery form based on Theta functions", page 23/52, with A^²/B^² = (a^²+b^²)/(a^²-b^²) as on page 20/52, replacing incorrect B^²/A^² on page 23/52 with correct A^²/B^² and a/b; or 2009 Gaudry–Lubicz "The arithmetic of characteristic 2 Kummer surfaces and of elliptic Kummer lines", Section 6.2, replacing incorrect A'/B' = (a^²+b^²)/(a^²-b^²) with correct A'^²/B'^² = (a^²+b^²)/(a^²-b^²), replacing A'^²/B'^² with A^²/B^², and replacing z... with y...; plus notation changes: a/b and A^²/B^² defined as 1/sqrt(r) and (1+r)/(1-r), input x^²/y^² etc. replaced by Z2/Y2 and Z3/Y3 and Z1/Y1, intermediate x'/y' replaced by W/V, output X^²/Y^² replaced by Z5/Y5.

Explicit formulas:

      V = s*(Z2-Y2)*(Z3-Y3)
      W = (Z2+Y2)*(Z3+Y3)
      Y5 = Z1*(W-V)^²
      Z5 = Y1*(W+V)^²

Explicit formulas for differential addition and doubling

The "mladd-2006-g-2" differential-addition-and-doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

Assumptions: Z1=1 and s=(1+r)/(1-r).
Cost: 3M + 6S + 1*r + 2*s + 8add.
Source: 2006 Gaudry "Variants of the Montgomery form based on Theta functions", pages 22/52 and 23/52, with A^²/B^² = (a^²+b^²)/(a^²-b^²) as on page 20/52, replacing incorrect B^²/A^² on pages 22/52 and 23/52 with correct A^²/B^² and a/b; or 2009 Gaudry–Lubicz "The arithmetic of characteristic 2 Kummer surfaces and of elliptic Kummer lines", Section 6.2, replacing incorrect A'/B' = (a^²+b^²)/(a^²-b^²) with correct A'^²/B'^² = (a^²+b^²)/(a^²-b^²), replacing A'^²/B'^² with A^²/B^², and replacing z... with y...; plus notation changes: a/b and A^²/B^² defined as 1/sqrt(r) and (1+r)/(1-r), input x^²/y^² etc. replaced by Z2/Y2 and Z3/Y3 and Z1/Y1, intermediate x'/y' replaced by W/V, output X^²/Y^² replaced by Z5/Y5; plus common-subexpression elimination; plus assumption Z1=1.

Explicit formulas:

      A = Z2-Y2
      B = Z2+Y2
      V2 = s*A^²
      W2 = B^²
      Y4 = (W2-V2)^²
      Z4 = r*(W2+V2)^²
      V = s*A*(Z3-Y3)
      W = B*(Z3+Y3)
      Y5 = (W-V)^²
      Z5 = Y1*(W+V)^²

The "mladd-2006-g" differential-addition-and-doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

Assumptions: Z1=1 and s=(1+r)/(1-r).
Cost: 3M + 6S + 1*r + 2*s + 10add.
Source: 2006 Gaudry "Variants of the Montgomery form based on Theta functions", pages 22/52 and 23/52, with A^²/B^² = (a^²+b^²)/(a^²-b^²) as on page 20/52, replacing incorrect B^²/A^² on pages 22/52 and 23/52 with correct A^²/B^² and a/b; or 2009 Gaudry–Lubicz "The arithmetic of characteristic 2 Kummer surfaces and of elliptic Kummer lines", Section 6.2, replacing incorrect A'/B' = (a^²+b^²)/(a^²-b^²) with correct A'^²/B'^² = (a^²+b^²)/(a^²-b^²), replacing A'^²/B'^² with A^²/B^², and replacing z... with y...; plus notation changes: a/b and A^²/B^² defined as 1/sqrt(r) and (1+r)/(1-r), input x^²/y^² etc. replaced by Z2/Y2 and Z3/Y3 and Z1/Y1, intermediate x'/y' replaced by W/V, output X^²/Y^² replaced by Z5/Y5; plus assumption Z1=1.

Explicit formulas:

      V2 = s*(Z2-Y2)^²
      W2 = (Z2+Y2)^²
      Y4 = (W2-V2)^²
      Z4 = r*(W2+V2)^²
      V = s*(Z2-Y2)*(Z3-Y3)
      W = (Z2+Y2)*(Z3+Y3)
      Y5 = (W-V)^²
      Z5 = Y1*(W+V)^²

The "ladd-2006-g-2" differential-addition-and-doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

Assumptions: s=(1+r)/(1-r).
Cost: 4M + 6S + 1*r + 2*s + 8add.
Source: 2006 Gaudry "Variants of the Montgomery form based on Theta functions", pages 22/52 and 23/52, with A^²/B^² = (a^²+b^²)/(a^²-b^²) as on page 20/52, replacing incorrect B^²/A^² on pages 22/52 and 23/52 with correct A^²/B^² and a/b; or 2009 Gaudry–Lubicz "The arithmetic of characteristic 2 Kummer surfaces and of elliptic Kummer lines", Section 6.2, replacing incorrect A'/B' = (a^²+b^²)/(a^²-b^²) with correct A'^²/B'^² = (a^²+b^²)/(a^²-b^²), replacing A'^²/B'^² with A^²/B^², and replacing z... with y...; plus notation changes: a/b and A^²/B^² defined as 1/sqrt(r) and (1+r)/(1-r), input x^²/y^² etc. replaced by Z2/Y2 and Z3/Y3 and Z1/Y1, intermediate x'/y' replaced by W/V, output X^²/Y^² replaced by Z5/Y5; plus common-subexpression elimination.

Explicit formulas:

      A = Z2-Y2
      B = Z2+Y2
      V2 = s*A^²
      W2 = B^²
      Y4 = (W2-V2)^²
      Z4 = r*(W2+V2)^²
      V = s*A*(Z3-Y3)
      W = B*(Z3+Y3)
      Y5 = Z1*(W-V)^²
      Z5 = Y1*(W+V)^²

The "ladd-2006-g" differential-addition-and-doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:

Assumptions: s=(1+r)/(1-r).
Cost: 4M + 6S + 1*r + 2*s + 10add.
Source: 2006 Gaudry "Variants of the Montgomery form based on Theta functions", pages 22/52 and 23/52, with A^²/B^² = (a^²+b^²)/(a^²-b^²) as on page 20/52, replacing incorrect B^²/A^² on pages 22/52 and 23/52 with correct A^²/B^² and a/b; or 2009 Gaudry–Lubicz "The arithmetic of characteristic 2 Kummer surfaces and of elliptic Kummer lines", Section 6.2, replacing incorrect A'/B' = (a^²+b^²)/(a^²-b^²) with correct A'^²/B'^² = (a^²+b^²)/(a^²-b^²), replacing A'^²/B'^² with A^²/B^², and replacing z... with y...; plus notation changes: a/b and A^²/B^² defined as 1/sqrt(r) and (1+r)/(1-r), input x^²/y^² etc. replaced by Z2/Y2 and Z3/Y3 and Z1/Y1, intermediate x'/y' replaced by W/V, output X^²/Y^² replaced by Z5/Y5.

Explicit formulas:

      V2 = s*(Z2-Y2)^²
      W2 = (Z2+Y2)^²
      Y4 = (W2-V2)^²
      Z4 = r*(W2+V2)^²
      V = s*(Z2-Y2)*(Z3-Y3)
      W = (Z2+Y2)*(Z3+Y3)
      Y5 = Z1*(W-V)^²
      Z5 = Y1*(W+V)^²

Explicit formulas for scaling

The "scale" scaling formulas [database entry; Sage verification script; Sage output; three-operand code]:

Cost: 1I + 1M + 0add.
Explicit formulas:
```
      Y3 = Y1/Z1
      Z3 = 1
```