There is an encryption scheme using elliptic curves given by @tylo explained here: @tylo's answer on ElGamal with elliptic curves and here: ElGamal with elliptic curves I. The encryption idea is to multiply (in the field) the message $m$ with the $x$-coordinate of a specific point on the elliptic curve (which is assumed to be hard to compute without breaking discrete log).
My question is why this is a secure encryption -- basically, only about half the $x$-coordinates are in use on an elliptic curve, so I guess it's a bit hard to say that the $x$-coordinates are really random. Is it assumed that the $x$-coordinate of a random elliptic curve point is pseudo-random? (There is no mentioned hashing)
Could anyone please supply a link to further reading or a brief explanation on the security of this scheme? Another question (also appearing in the mentioned questions) is if using addition in the field (instead of multiplication) is also secure.
EDIT: Following the discussion in the comments, I have a concern regarding the security of the scheme -- let's suppose that there are only two possible messages, $m_0$ and $m_1$. Then, the encryption is done by multiplying, in the field, the chosen message with the $x$-coordinate of an unknown point on the curve. A challenger receiving the cipher-text $c=x\cdot m_i$ can then compute $x_0=c/m_0$ and $x_1=c/m_1$. If only one of these is an $x$-coordinate of a point on the curve, then the other cannot be the chosen message. Can someone tell me what I missed in this "attack"?