a*x^3+y^3+1=d*x*y
Projective coordinates [database entry] represent x y as X Y Z satisfying the following equations:
x=X/Z y=Y/Z
Operation | Assumptions | Cost | Readdition cost |
---|---|---|---|
addition | 12M + 1*a | 12M + 1*a | |
doubling | minustwo=-2 | 7M + 1S + 1*minustwo + 1*d | |
doubling | i^2=-1 and minustwo=-2 and 2d=2*d | 8M + 1*i + 1*minustwo + 1*2d | |
doubling | 6M + 3S + 1*a | ||
doubling | 3M + 3^3 + 1*a | ||
tripling | d*recipd=1 | 8M + 6S + 1*a + 1*recipd |
A = X1*Z2 B = Z1*Z2 C = Y1*X2 D = Y1*Y2 E = Z1*Y2 F = a*X1*X2 X3 = A*B-C*D Y3 = D*E-F*A Z3 = F*C-B*E
P = Y1*Z1
2P = 2*P
S = Y1+Z1
A = S^2-P
C = (A-2P)*S
D = A*(Z1-Y1)
E = 3*C-d*X1*2P
X3 = minustwo*X1*D
Y3 = (D-E)*Z1
Z3 = (D+E)*Y1
iZ = i*Z1 A = (Y1-iZ)*(Y1+iZ) B = Y1*Z1 C = (A-B)*(Y1+Z1) D = (A+B)*(Z1-Y1) E = 3*C-2d*X1*B X3 = minustwo*X1*D Y3 = (D-E)*Z1 Z3 = (D+E)*Y1
A = X1^2 B = Y1^2 C = Z1^2 D = A*X1 E = B*Y1 F = C*Z1 G = a*D X3 = X1*(E-F) Y3 = Z1*(G-E) Z3 = Y1*(F-G)
D = X1^3 E = Y1^3 F = Z1^3 G = a*D X3 = X1*(E-F) Y3 = Z1*(G-E) Z3 = Y1*(F-G)
U = a*X1*X1^2 V = Y1*Y1^2 W = Z1*Z1^2 A = (U-V)^2 B = (U-W)^2 C = (V-W)^2 D = A+C E = A+B X3 = recipd*(U+V+W)*(B+D) Y3 = 2*U*C-V*(C-E) Z3 = 2*V*B-U*(B-D)