# Complex Numbers on Elliptic Curves & Usage in Tate Pairing

I'm working with understanding the internals of the Tate Pairing. I was going through an example of the curve $E: y^2 = x^3 + 3x$ over $\mathbb{F_{11}}$. The author is showing the computation of $e(P,Q)$, where $P=(1,9)$ and $Q = \phi (P) = (10,9i)$ using Miller's Algorithm. Through this algorithm, the author chose a point $Q' = (6,6)$ and then computes $S = Q + Q' = (8+7i, 10+6i)$. This is where I'm having an issue, I can't seem to determine how the complex addition of points occurs in elliptic curves, and I'm looking for an explanation in the usage for the Tate pairing.

Paper being referenced : http://www.win.tue.nl/~bdeweger/downloads/MT%20Martijn%20Maas.pdf , Section 4.3.1

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He's working over a quadratic extension, and the formulas work exactly the same as over any finite field. –  Watson Ladd Jun 1 '13 at 19:40
Watson is correct. Here's the example you mention spelled out in Sage. –  Samuel Neves Jun 1 '13 at 20:06
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