Can elliptic curve point common factors be detected? Consider the two EC points $X=abG$ and $Y=bG$. Is it possible for someone to examine $X$ and $Y$ to determine if there is a common factor, as long as $a$ and $b$ are randomly chosen numbers between 0 and the size of the finite field? Even if the common factor $b$ cannot be determined, can it be determined if there is, or if there is likely to be a common factor? Assume the ed25519 curve is being used.
Is it possible for someone to examine $X$ and $Y$ to determine if there is a common factor, as long as $a$ and $b$ are randomly chosen numbers between 0 and the size of the finite field?


(Actually, you mean the size of the group)

Technically, the answer is "yes, it is easy to determine that".  The easiest subcase is if $G$ (and hence $X$ and $Y$) belong to the large prime subgroup of x25519; then the rule is "unless Y is the point-at-infinity and X is not, the answer is "yes".

That is, there will always be an $a$ such that $X = aY$.

It's a little trickier if $G$ generates a larger subgroup, but it is still easy (it might take a little computation on the attacker's part).

Of course, the fact that it's always true (whether or not you consciously generated such a relationship between $X$ and $Y$) means that the attacker can't deduce anything from that information.