Modified version of a CPA-secure encryption scheme The following encryption scheme: choose a random string $r \leftarrow \{0,1\}^n$ and compute the ciphertext $c= \thinspace &lt;c_0, c_1&gt;= \thinspace &lt;r, F_k(r) \oplus m&gt;$, assuming $F$ a PRF; is CPA-secure, as we have a randomized/non-deterministic encryption process. Why, if applying a modified encryption process, i.e. computing the ciphertext $c'= \thinspace &lt;c_0, c_1'&gt;= \thinspace &lt;r, F_r(k) \oplus m&gt;$, this  is no more CPA-secure? I mean, this modified encryption process is also non-deterministic. Could it be because on each encryption invocation the input to the PRF $F$ is the fixed value $k$? I'm not sure if this makes sense.
Ask a CPA-query with a known $m$ and get back $c_0,c_1'$. Compute $c_1'' = c_1' \oplus m$. Then, compute $F^{-1}_r(c_1'')$ and this will be $k$. Now you know the key.

Of course, this attack assumes that you can invert $F$, but nothing in the definition says you cannot (and in practice you often can).