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I want to know why the distribution the points in Elliptic curves over a finite field $\mathbb{F}_p$ where $p$ is prime is uniform. That means the number of points in elliptic curve $E$ with $x$-coordinate in the interval $[0,p/3][p/3,2*p/3][2*p/3,p]$ is roughly equal.

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Do you have a citation for this result? –  Gerry Myerson Dec 21 '12 at 16:41
    
In my estimation, the 'why' is obvious -- there's seemingly nothing special about any interval -- just difficult to prove (if if is true). –  Hurkyl Dec 21 '12 at 17:39
    
No I don't have citation for this result but Experimently if we take arbitrary elliptic curve we confirm this result more than this, Pollard use the x coordinate to split E(fp) to 3 sets roughly equal –  user53917 Dec 21 '12 at 19:57
    
Wait --- so you are asking why something is true, when you don't even know whether it is true? Maybe you should rewrite your question to reflect the actual state of play. Be sure to include a bibliographic citation and a clear statement of what Pollard did. Also, have you thought of writing to Pollard? –  Gerry Myerson Dec 22 '12 at 6:46

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