Is there any pattern in points on EC?

I read some where in crypto.stackexchange answer (related to EC-SRP protocol) that there is pattern in points on elliptic curves, i.e. if given some points containing mix of correct and wrong points and all points are less than prime number (p), now my question is:

is it possible to identify the correct points if so what is that logic? Please someone explain (I am not an expert in ecc).

@fgrieu and @CodesInChaos: Thank you both for your answers and I understood your point but if x and y co-ordinates of a point are encrypted using a shared secret e.g. password like in encrypted key exchange, my question is that is there any other attack other than checking x or y value greater than prime p to derive password?

It appears from your answer that after decrypting the point (x, y) by a candidate password the equation allows easily determining if a point of given coordinates x,y valid or not and thus password can be derived after observing few transactions over a period of time, please confirm/clarify with some details?

• The Elliptic Curves used in cryptography have a public equation, e.g. $y^2=x^3+ax+b$ with $x,y$ the coordinate of points on the curve, and $a,b$ public constants, with all arithmetic in $\mathbb Z_p$ for some public constant prime $p$. The equation allows easily determining if a point of given coordinates $x,y$ is on the curve, or not. There are so many points that one can't make a list of all the points, for practical parameters. However it is still possible to determine ("count") how many points there are. $\;$ Is that what you are asking? – fgrieu Mar 30 '15 at 16:19
• Even if you're given a compressed point, i.e. you don't know y, you can still check if x^3+ax+b is a square. This can be avoided by either randomly choosing a point from the curve or its twist (with the correct probabilities) or using something fancy, like elligator. – CodesInChaos Mar 30 '15 at 17:23
• See crypto.stackexchange.com/help/account if you're having trouble accessing your account. – Gilles Mar 31 '15 at 9:36