y^^{2}=x^^{3}+a*x+b
Projective coordinates with a4=-1 [database entry] make the additional assumptions
a=-1and 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 | 5M + 2S | 5M + 2S |
addition | Z2=1 | 9M + 2S | 9M + 2S |
addition | 12M + 2S | 12M + 2S | |
addition | 13M + 3S | 13M + 3S | |
addition | 11M + 6S + 1*a | 11M + 6S + 1*a | |
addition | 12M + 5S + 1*a | 12M + 5S + 1*a | |
addition | 10M + 4S + 1^^{3} | 10M + 4S + 1^^{3} | |
addition | 16M + 3S + 3^^{3} | 16M + 3S + 3^^{3} | |
doubling | Z1=1 | 3M + 5S | |
doubling | 5M + 6S + 1*a | ||
doubling | 6M + 5S + 1*a | ||
doubling | 6M + 5S + 1^^{3} + 1*a | ||
scaling | 1I + 2M |
u = Y2-Y1 uu = u^^{2} v = X2-X1 vv = v^^{2} vvv = v*vv R = vv*X1 A = uu-vvv-2*R X3 = v*A Y3 = u*(R-A)-vvv*Y1 Z3 = vvv
u = Y2*Z1-Y1 uu = u^^{2} v = X2*Z1-X1 vv = v^^{2} vvv = v*vv R = vv*X1 A = uu*Z1-vvv-2*R X3 = v*A Y3 = u*(R-A)-vvv*Y1 Z3 = vvv*Z1
Y1Z2 = Y1*Z2 X1Z2 = X1*Z2 Z1Z2 = Z1*Z2 u = Y2*Z1-Y1Z2 uu = u^^{2} v = X2*Z1-X1Z2 vv = v^^{2} vvv = v*vv R = vv*X1Z2 A = uu*Z1Z2-vvv-2*R X3 = v*A Y3 = u*(R-A)-vvv*Y1Z2 Z3 = vvv*Z1Z2
U1 = X1*Z2 U2 = X2*Z1 S1 = Y1*Z2 S2 = Y2*Z1 ZZ = Z1*Z2 T = U1+U2 M = S1+S2 R = (T-ZZ)*(T+ZZ)-U1*U2 F = ZZ*M L = M*F G = T*L W = R^^{2}-G X3 = 2*F*W Y3 = R*(G-2*W)-L^^{2} Z3 = 2*F*F^^{2}
U1 = X1*Z2 U2 = X2*Z1 S1 = Y1*Z2 S2 = Y2*Z1 ZZ = Z1*Z2 T = U1+U2 TT = T^^{2} M = S1+S2 R = TT-U1*U2+a*ZZ^^{2} F = ZZ*M L = M*F LL = L^^{2} G = (T+L)^^{2}-TT-LL W = 2*R^^{2}-G X3 = 2*F*W Y3 = R*(G-2*W)-2*LL Z3 = 4*F*F^^{2}
U1 = X1*Z2 U2 = X2*Z1 S1 = Y1*Z2 S2 = Y2*Z1 ZZ = Z1*Z2 T = U1+U2 M = S1+S2 R = T^^{2}-U1*U2+a*ZZ^^{2} F = ZZ*M L = M*F G = T*L W = R^^{2}-G X3 = 2*F*W Y3 = R*(G-2*W)-L^^{2} Z3 = 2*F*F^^{2}
U1 = X1*Z2 U2 = X2*Z1 S1 = Y1*Z2 S2 = Y2*Z1 W = Z1*Z2 P = U2-U1 R = S2-S1 X3 = P*(-(U1+U2)*P^^{2}+W*R^^{2}) Y3 = (R*(-2*W*R^^{2}+3*(U1+U2)*P^^{2})-P^^{3}*(S1+S2))/2 Z3 = W*P^^{3}
u = Y2*Z1-Y1*Z2 v = X2*Z1-X1*Z2 A = u^^{2}*Z1*Z2-v^^{3}-2*v^^{2}*X1*Z2 X3 = v*A Y3 = u*(v^^{2}*X1*Z2-A)-v^^{3}*Y1*Z2 Z3 = v^^{3}*Z1*Z2
XX = X1^^{2} w = a+3*XX Y1Y1 = Y1^^{2} R = 2*Y1Y1 sss = 4*Y1*R RR = R^^{2} B = (X1+R)^^{2}-XX-RR h = w^^{2}-2*B X3 = 2*h*Y1 Y3 = w*(B-h)-2*RR Z3 = sss
XX = X1^^{2} ZZ = Z1^^{2} w = a*ZZ+3*XX s = 2*Y1*Z1 ss = s^^{2} sss = s*ss R = Y1*s RR = R^^{2} B = (X1+R)^^{2}-XX-RR h = w^^{2}-2*B X3 = h*s Y3 = w*(B-h)-2*RR Z3 = sss
w = a*Z1^^{2}+3*X1^^{2} s = Y1*Z1 ss = s^^{2} sss = s*ss R = Y1*s B = X1*R h = w^^{2}-8*B X3 = 2*h*s Y3 = w*(4*B-h)-8*R^^{2} Z3 = 8*sss
w = a*Z1^^{2}+3*X1^^{2} s = Y1*Z1 B = X1*Y1*s h = w^^{2}-8*B X3 = 2*h*s Y3 = w*(4*B-h)-8*Y1^^{2}*s^^{2} Z3 = 8*s^^{3}
A = 1/Z1 X3 = A*X1 Y3 = A*Y1 Z3 = 1