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I had a lecture last week which dealt with the Frobenius Endomorphism on elliptic curves. The lecturer showed an example at the end of the lecture, when almost out of time and I don't quite understand it. We had the elliptic curve $E: Y^2+XY=X^3+X$ over $\mathbb{F}_2$. He then wrote the following: $[2](X,Y)=(X,Y)+(X,Y)=(X^2+\frac{1}{X^2},1+\frac{1}{X^2}+Y^2+\frac{Y^2}{X^4})=\text{Frob}(X+\frac{1}{X},1+\frac{1}{X}+Y+\frac{Y}{X^2})$

Now I can understand the last equal sign, since it seems the Frobenius endomorphism just raises everything to the (in this case) second power. However, I don't really know how we deduce the other parts. It might be that the teacher skipped quite a few steps, so I don't quite get it. Anybody any tips (wikipedia isn't much help).

user106421
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  • So you problem is not with the Frobenius part, but with the double-point formula? – awllower Nov 11 '13 at 09:41
  • I suppose, I just don't really understand where it comes from. – user106421 Nov 11 '13 at 10:04
  • So a question arises: what does it mean to add two points on an elliptic curve. I think it suffices to check the definitions and do some calculations. :) – awllower Nov 11 '13 at 10:11
  • I need to know this for a homework exercise (which uses a different curve, and I want to do it myself, hence the question about this example). In this exercise I am asked to find the dual to the frobenius endomorphism, so is that just sending $X\to X+\frac{1}{X}$ and $Y$ similarly according to above? – user106421 Nov 11 '13 at 10:14
  • Also I'm trying to work the addition out but I am really not getting $X^2+\frac{1}{X^2}$ as $X$-coordinate. – user106421 Nov 11 '13 at 11:28
  • Could it be that your teacher meant $E:Y^2+XY=X^3+X^2$? – Willem Beek Dec 23 '15 at 14:47

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