It seems to me that the Elliptic Curve Addition Algorithm is undefined for the case $P+P$ if $P=(0,0)$.
Are there any alternative computation methods for this case?
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Is this on a Weierstrass curve $y^2=x^3+ax^2+bx$? – Angina Seng Sep 15 '19 at 15:11
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yes, sorry for not mentioning this – jvdh Sep 15 '19 at 15:12
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Thanks, @anomaly, I think my confusion stemmed from the fact that I overlooked that $\mathcal{O} = (0,0)$ – jvdh Sep 15 '19 at 15:15
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1For a curve in short Weierstrass form, any point $P$ with $y$-coordinate $0$ is a 2-torsion point. So $P+P=\infty$. – Viktor Vaughn Sep 15 '19 at 15:20
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@anomaly I don't understand your comment. To say that the point is at infinity means that it does not appear in the affine picture. That's why it's "at infinity." – Viktor Vaughn Sep 15 '19 at 15:23
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1@André3000: And this is why I don't post before breakfast. – anomaly Sep 15 '19 at 15:32
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The point at $\infty$ is $(\infty,\infty)$ not $(0,0)$. To find $P+P$ we need the tangent at $P$ that is the line $(x_P+ct,y_P+dt)$ ($(c,d) \ne (0,0)$) such that $f'(0) = 0$ where $f(t) = (y_P+dt)^2-(x_P+ct)^3-a(x_P+ct)^2-b(x_P+ct)$ then as usual we pick the 3rd root of $f(t)=-c^3 t^2(t-T), T = 1+f(1)/c^3$ then $P+P = (x_P+cT,-y_P-dT)$. Here with $P = (0,0)$ the tangent line is $c=0,d=1$ and $T=\infty$,$P+P = (\infty,\infty)$ ($c^3T$ is non-zero that's why $cT = \infty$ even if $c=0$ .... hard to justify without rational functions on the curve or projective curves stuffs) – reuns Sep 15 '19 at 15:33