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.
Distribution the points in Elliptic curves over a finite field F_p where p is prime.
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0Do you have a citation for this result? – 2012-12-21
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0In my estimation, the 'why' is obvious -- there's seemingly nothing special about any interval -- just difficult to prove (if if is true). – 2012-12-21
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1No 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 – 2012-12-21
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0Wait --- 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? – 2012-12-22
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0Look at Sato–Tate conjecture (if $E$ has no CM). – 2018-11-23