ref: 34f5a8333090c5090e1d0b6ddbaf900d64340eea
dir: /sys/src/libsec/port/jacobian.mp/
# Elliptic curve group operations in jacobian coordinates: # x=X/Z^2 # x=Y/Z^3 jacobian_new(x,y,z, X,Y,Z) { X = x; Y = y; Z = z; } jacobian_inf(X,Y,Z) { X,Y,Z = jacobian_new(0,1,0); } jacobian_affine(p, X,Y,Z) mod(p) { if(Z != 0) { ZZ = Z^2; ZZZ = ZZ*Z; X = X / ZZ; Y = Y / ZZZ; Z = 1; } } jacobian_dbl(p,a, X1,Y1,Z1, X3,Y3,Z3) mod(p) { if(Y1 == 0) { X3,Y3,Z3 = jacobian_inf(); } else { XX = X1^2; YY = Y1^2; YYYY = YY^2; ZZ = Z1^2; S = 2*((X1+YY)^2-XX-YYYY); M = 3*XX+a*ZZ^2; Z3 = (Y1+Z1)^2-YY-ZZ; X3 = M^2-2*S; Y3 = M*(S-X3)-8*YYYY; } } jacobian_add(p,a, X1,Y1,Z1, X2,Y2,Z2, X3,Y3,Z3) mod(p) { Z1Z1 = Z1^2; Z2Z2 = Z2^2; U1 = X1*Z2Z2; U2 = X2*Z1Z1; S1 = Y1*Z2*Z2Z2; S2 = Y2*Z1*Z1Z1; if(U1 == U2) { if(S1 != S2) { X3,Y3,Z3 = jacobian_inf(); } else { X3,Y3,Z3 = jacobian_dbl(p,a, X1,Y1,Z1); } } else { H = U2-U1; I = (2*H)^2; J = H*I; r = 2*(S2-S1); V = U1*I; X3 = r^2-J-2*V; Y3 = r*(V-X3)-2*S1*J; Z3 = ((Z1+Z2)^2-Z1Z1-Z2Z2)*H; } }