y^^{2}+x*y=x^^{3}+a2*x^^{2}+a6
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 | Z1=1 and Z2=1 | 7M + 1S + 1*a2 | 7M + 1S + 1*a2 |
addition | Z2=1 | 11M + 1S + 1*a2 | 11M + 1S + 1*a2 |
addition | Z2=1 | 11M + 2S + 1*a2 | 11M + 2S + 1*a2 |
addition | 14M + 1S + 1*a2 | 14M + 1S + 1*a2 | |
addition | 15M + 2S + 1*a2 | 15M + 2S + 1*a2 | |
addition | 15M + 2S + 1^^{3} + 1*a2 | 15M + 2S + 1^^{3} + 1*a2 | |
doubling | Z1=1 | 5M + 3S + 1*a2 | |
doubling | 7M + 3S + 1*a2 | ||
doubling | 7M + 4S + 1*a2 | ||
scaling | 1I + 2M |
A = Y1+Y2
B = X1+X2
AB = A+B
C = B^^{2}
Z3 = B*C
F = A*AB+a2*C+Z3
X3 = B*F
Y3 = C*(A*X1+B*Y1)+AB*F
A = Y1+Z1*Y2
B = X1+Z1*X2
AB = A+B
C = B^^{2}
E = B*C
F = (A*AB+a2*C)*Z1+E
X3 = B*F
Y3 = C*(A*X1+B*Y1)+AB*F
Z3 = E*Z1
A = Y1+Z1*Y2 B = X1+Z1*X2 C = B^^{2} E = B*C F = (A^^{2}+A*B+a2*C)*Z1+E X3 = B*F Y3 = C*(A*X1+Y1*B)+(A+B)*F Z3 = E*Z1
Y1Z2 = Y1*Z2
X1Z2 = X1*Z2
A = Y1Z2+Z1*Y2
B = X1Z2+Z1*X2
AB = A+B
C = B^^{2}
D = Z1*Z2
E = B*C
F = (A*AB+a2*C)*D+E
X3 = B*F
Y3 = C*(A*X1Z2+B*Y1Z2)+AB*F
Z3 = E*D
A = Y1*Z2+Z1*Y2 B = X1*Z2+Z1*X2 C = B^^{2} D = Z1*Z2 E = B*C F = (A^^{2}+A*B+a2*C)*D+E X3 = B*F Y3 = C*(A*X1+Y1*B)*Z2+(A+B)*F Z3 = E*D
A = Y1*Z2+Z1*Y2 B = X1*Z2+Z1*X2 C = B^^{2} D = Z1*Z2 E = (A^^{2}+A*B+a2*C)*D+B*C X3 = B*E Y3 = C*(A*X1+Y1*B)*Z2+(A+B)*E Z3 = B^^{3}*D
A = X1^^{2} B = A+Y1 E = (B^^{2}+B*X1+a2*A) X3 = X1*E Y3 = (B+X1)*E+A^^{2}*X1 Z3 = X1*A
A = X1^^{2} B = A+Y1*Z1 C = X1*Z1 BC = B+C D = C^^{2} E = B*BC+a2*D X3 = C*E Y3 = BC*E+A^^{2}*C Z3 = C*D
A = X1^^{2} B = A+Y1*Z1 C = X1*Z1 D = C^^{2} E = (B^^{2}+B*C+a2*D) X3 = C*E Y3 = (B+C)*E+A^^{2}*C Z3 = C*D
A = 1/Z1 X3 = A*X1 Y3 = A*Y1 Z3 = 1