Bilinear Form: Weil pairing

Question

I am doing an example on Weil pairings, and for that purpose I follow the thesis of Alex Edward Aftuck, The Weil Pairing on Elliptic Curves and Its Cryptographic Applications.

By following his thesis, on page #39 he selects 4 points on the elliptic curve $Y^2=X^3+2X^2-3X$ and calculates six divisors from it as shown in the figure:

Using divisors, he calculated four rational functions as but in the last step, he calculates the Weil pairing between $P_1 $ and $P_3$. The sums of points are $P_3+S=(-2.496,-2.047) $ and $P_1-S=(20.798,-98.990) $.

My question is: how can I can put the points into the functions $f_{p_1} $ and $f_{p_3} $? All of the functions have three variables $X,Y,Z$, but we have only two points. What is the value of $Z$, and how can I calculate functions using these points? Please help me as I'm really confused.

Answer 1

See page 17 of the thesis you linked to. There he mentions affine coordinates and homogenization. For your points, $Z=1$. So just set $Z=1$ in your functions above.

To find the poles, we must remember that as a rational function, $Y = Y/1$. Homogenizing, we have $f(X, Y, Z) = Y/Z$. Then to find the poles of $f$, we must analyze $Z$ as a polynomial. For all affine points on $E$, $Z = 1$, so $Z = 0$ only at $\mathcal O$.

Here's more about projective coordinates.

Keep in mind that in general, the sum of two points in projective coordinates having $Z=1$ is not a point with $Z=1$. In this case, the author already did the sums $P_3 + S$ and $P_1 - S$ in affine coordinates.