verifying points on an elliptic curve

Question

I have an assignment to write a program to verify the points

P = (1+T+T², 1+T), Q = (T²,T), and R = (1+T+T², 1+T²) are in the field E(F₈) on the curve:

E:Y² +XY+Y = X³ +TX+(T+1)

on the field:

F ₈= F₂[T]/(T³+T+1)

And I can not for the life of my figure out how to make both sides equal to each other. Ive done the math by hand and with the program but it never comes out right. Can someone please explain how I can verify that these points are on the curve?

Answer 1

I presume your $X^2$ should be $X^3$.

I'll just look at $Q$, that's the easy one. Substituting $X=T^2$ and $Y=T$ into $$Y^2+XY+Y+X^3+TX+T+1$$ gives $$T^2+T^3+T+T^6+T^3+T+1=T^6+T^2+1=(T^3+T+1)^2=0$$ in $\Bbb F_8$. So $Q$ is on the curve. Throughout I am using the fact that $1=-1$ in the field.