I don't think that x coordinates on secp256k1 are uniformly distributed.
You can actually see visually that they're not. (I know this is not over Zp, but you get the idea)
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if you work in Finite Field F(P) around a half of x coordinate between 1-2^256 lie on the curve y**2=x**3 + 7 (mod P) and there are perfectly distributed (even we wish for) . because if not ECSDA will be have a bias and it can be broken..
You can actually see visually that they're not. (I know this is not over Zp, but you get the idea)
[/quote]
if you work in Finite Field F(P) around a half of x coordinate between 1-2^256 lie on the curve y**2=x**3 + 7 (mod P) and there are perfectly distributed (even we wish for) . because if not ECSDA will be have a bias and it can be broken..