Cryptography/Prime Curve/Chudnovsky Coordinates

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[edit] Introduction

Chudnovsky Coordinates are used to represent elliptic curve points on prime curves y^2 = x^3 + ax + b. They give a speed benefit over Affine Coordinates when the cost for field inversions is significantly higher than field multiplications. In Chudnovsky Coordinates the quintuple (X, Y, Z, Z^2, Z^3) represents the affine point (X / Z^2, Y / Z^3).

[edit] Point Doubling (5M + 6S or 5M + 4S)

Let (X, Y, Z, Z^2, Z^3) be a point (unequal to the point at infinity) represented in Chudnovsky Coordinates. Then its double (X', Y', Z', Z'^2, Z'^3) can be calculated by

if (Y == 0)
  return POINT_AT_INFINITY
S = 4*X*Y^2
M = 3*X^2 + a*(Z^2)^2
X' = M^2 - 2*S
Y' = M*(S - X') - 8*Y^4
Z' = 2*Y*Z
Z'^2 = Z'^2
Z'^3 = Z'^2 * Z'
return (X', Y', Z', Z'^2, Z'^3)

Note: if a = -3, then M can also be calculated as M = 3*(X + Z^2)*(X - Z^2), saving 2 field squarings.

[edit] Point Addition (11M + 3S)

Let (X1, Y1, Z1, Z1^2, Z1^3) and (X2, Y2, Z2, Z2^2, Z2^3) be two points (both unequal to the point at infinity) represented in Chudnovsky Coordinates. Then the sum (X3, Y3, Z3, Z3^2, Z3^3) can be calculated by

U1 = X1*Z2^2
U2 = X2*Z1^2
S1 = Y1*Z2^3
S2 = Y2*Z1^3
if (U1 == U2)
  if (S1 != S2)
    return POINT_AT_INFINITY
  else 
    return POINT_DOUBLE(X1, Y1, Z1, Z1^2, Z1^3)
H = U2 - U1
R = S2 - S1
X3 = R^2 - H^3 - 2*U1*H^2
Y3 = R*(U1*H^2 - X3) - S1*H^3
Z3 = H*Z1*Z2
Z3^2 = Z3^2
Z3^3 = Z3^2 * Z3
return (X3, Y3, Z3)

[edit] Mixed Addition (with affine point) (8M + 3S)

Let (X1, Y1, Z1, Z1^2, Z1^3) be a point represented in Chudnovsky Coordinates and (X2, Y2) a point in Affine Coordinates (both unequal to the point at infinity). A formula to add those points can be readily derived from the regular chudnovsky point addition by replacing each occurance of "Z2" by "1" (and thereby dropping three field multiplications).

[edit] Mixed Addition (with jacobian point) (11M + 3S)

See Jacobian Coordinates for further details.

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