# Cryptography/Prime Curve/Chudnovsky Coordinates

### Introduction

edit**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).

### Point Doubling (5M + 6S or 5M + 4S)

editLet (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.

### Point Addition (11M + 3S)

editLet (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)

### Mixed Addition (with affine point) (8M + 3S)

editLet (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 occurrence of "Z2" by "1" (and thereby dropping three field multiplications).

### Mixed Addition (with jacobian point) (11M + 3S)

editSee Jacobian Coordinates for further details.