# Cryptography/Prime Curve/Standard Projective Coordinates

### Introduction

[edit | edit source]**Standard Projective Coordinates** are used to represent elliptic curve points on prime curves y^2 = x^3 + ax + b. Their usage might give a speed benefit over *Affine Coordinates* when the cost for field inversions is significantly higher than field multiplications. In *Standard Projective Coordinates* the triple (X, Y, Z) represents the affine point (X / Z, Y / Z).

### Point Doubling (7M + 5S or 7M + 3S)

[edit | edit source]Let (X, Y, Z) be a point (unequal to the *point at infinity*) represented in *Standard Projective Coordinates*. Then its double (X', Y', Z') can be calculated by

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

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

### Point Addition (12M + 2S)

[edit | edit source]Let (X1, Y1, Z1) and (X2, Y2, Z2) be two points (both unequal to the *point at infinity*) represented in *Standard Projective Coordinates*. Then the sum (X3, Y3, Z3) can be calculated by

U1 = Y2*Z1 U2 = Y1*Z2 V1 = X2*Z1 V2 = X1*Z2 if (V1 == V2) if (U1 != U2) return POINT_AT_INFINITY else return POINT_DOUBLE(X1, Y1, Z1) U = U1 - U2 V = V1 - V2 W = Z1*Z2 A = U^2*W - V^3 - 2*V^2*V2 X3 = V*A Y3 = U*(V^2*V2 - A) - V^3*U2 Z3 = V^3*W return (X3, Y3, Z3)

### Mixed Addition (with affine point) (9M + 2S)

[edit | edit source]Let (X1, Y1, Z1) be a point represented in *Standard Projective 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 standard projective point addition by replacing each occurrence of "Z2" by "1" (and thereby dropping three field multiplications).