For elliptic-curve cryptography, does a 256-bit key imply that $x$ and $y$ are each 256-bits or 128-bits? In the wikipedia article, the claim is made that "256-bit elliptic curve public key should provide comparable security to a 3072-bit RSA public key". Since, in ECC, the public-key consists of a point $(x, y)$, when they say a "256-bit public key", does that imply both $x$ and $y$ are each 256-bits? Or that, together, they are 256 bits (meaning that each would be 128 bits)?
In ECC, the xxx-bit curve implies the modulus length. Also, this implies the length of each coordinate in $(x,y)$. Therefore, $x$ and $y$ are each 256-bit