2
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Till now all we know is Hasse's theorem, which states that $|\#E(p)-(p+1)| \leq 2\sqrt{p}$, where $\#E(p)$ is the total number of points in $E_p(a,b)$.

Is there any other theorem which defines relation between points on elliptic curve $E_p(a,b)$ ?

I want to find the maximum interval of $x$ co-ordinates between two successive elliptic curve points. For example let us consider the elliptic curve $y^2=x^3+x+1$ mod $5$. This has $8$ points in total. These are $(0,1),(0.4),(2,1),(2,4),(3,1),(3,4),(4,2),(4,3)$. The maximum interval of $x$ co-ordinates in this curve is between $0$ and $2$. I need a generic solution to find the maximum interval of $x$ co-ordinates given $E_p(a,b)$.

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  • $\begingroup$ This question needs some rewording to make it understandable. $\endgroup$ – fkraiem Aug 3 '16 at 4:47
  • $\begingroup$ Have you checked wiki page Counting points on elliptic curves $\endgroup$ – Makif Aug 3 '16 at 7:02
  • $\begingroup$ @fkraiem I have revised my question. Hope it makes understandable $\endgroup$ – Sandy Aug 3 '16 at 7:46
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    $\begingroup$ I want to find the maximum interval of xx co-ordinates between two successive elliptic curve points. I dont know any algorithm, and i dont think it exists, that answers this very specific question which does not require to iterate through all points on the elliptic curve. $\endgroup$ – Makif Aug 3 '16 at 7:54
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    $\begingroup$ Your distance function is not well defined on $\mathbb{F}_p$, since it depends on the choice of representative. For example, $0\equiv 5\bmod 5$ but the distance between 2 and 0 is 2, while the distance between 2 and 5 is 3. So perhaps some context on why this is interesting? $\endgroup$ – CurveEnthusiast Jun 24 '17 at 22:17

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