Explicit-Formulas Database
Ordinary genus-1 curves over binary fields
Short Weierstrass curves EFD / Ordinary genus-1 binary / Lambda coordinates for short Weierstrass curves

# Lambda coordinates for short Weierstrass curves

An elliptic curve in short Weierstrass form [more information] has parameters a2 a6 and coordinates x y satisfying the following equations:
```  y^2+x*y=x^3+a2*x^2+a6
```

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

```  x=X/Z
y/x=(L-X)/Z
```

Source: 2013 Oliveira–López–Aranha–Rodríguez-Henríquez "Lambda coordinates for binary elliptic curves".

## Best operation counts

Smallest multiplication counts assuming I=10M, S=0M, *param=0M, add=0M, *const=0M:
Smallest multiplication counts assuming I=10M, S=0.2M, *param=0M, add=0M, *const=0M:

## Summary of all explicit formulas

addition 11M + 2S 11M + 2S
doubling a21=a2+1 and a226=a22+a6 3M + 4S + 1*a2 + 1*a226 + 1*a21
doubling 4M + 4S + 1*a2

The "add-2013-olar" addition formulas [database entry; Sage verification script; Sage output; three-operand code]:
• Cost: 11M + 2S + 5add.
• Source: 2013 Oliveira–López–Aranha–Rodríguez-Henríquez "Lambda coordinates for binary elliptic curves"; database entry contributed by Chitchanok Chuengsatiansup.
• Explicit formulas:
```      A = L1*Z2
B = L2*Z1
C = A+B
D = X1*Z2
E = X2*Z1
F = D+E
G = F^2
H = C*D
I = C*E
J = C*G
K = J*Z2
X3 = H*I
L3 = (I+G)^2+K*(L1+Z1)
Z3 = K*Z1
```

## Explicit formulas for doubling

The "dbl-2013-olar-2" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:
• Assumptions: a21=a2+1 and a226=a22+a6.
• Cost: 3M + 4S + 1*a2 + 1*a226 + 1*a21 + 8add.
• Source: 2013 Oliveira–López–Aranha–Rodríguez-Henríquez "Lambda coordinates for binary elliptic curves"; database entry contributed by Chitchanok Chuengsatiansup.
• Explicit formulas:
```      A = Z1^2
B = L1+Z1
C = L1*B
D = a2*A
E = C+D
F = L1+X1
G = F^2
H = A^2
X3 = E^2
Z3 = E*A
L3 = G*(G+E+A)+a226*H+X3+a21*Z3
```

The "dbl-2013-olar" doubling formulas [database entry; Sage verification script; Sage output; three-operand code]:
• Cost: 4M + 4S + 1*a2 + 5add.
• Source: 2013 Oliveira–López–Aranha–Rodríguez-Henríquez "Lambda coordinates for binary elliptic curves"; database entry contributed by Chitchanok Chuengsatiansup.
• Explicit formulas:
```      A = Z1^2
B = L1^2
C = L1*Z1
D = a2*A
E = B+C+D
F = X1*Z1
X3 = E^2
Z3 = E*A
L3 = F^2+X3+E*C+Z3
```