Exchanging Keyspace and Message space in PRF Suppose there is a secure PRF $F: K \times X \to Y$. Then is $F': X \times K \to Y$, defined by $F'(x,k):=F(k,x)$, a secure PRF?


In my opinion it should be because exchanging the keyspace with the message space shouldn't effect the security. As we were choosing a random key from the key space, similarly we can choose a key from message space randomly and can use to compute $F'$. However, I am not sure about my intuition.
In general this is not the case. Consider a PRF F which ignores the first bit of the key. Then you can distinguish F'(k,.)=F(.,k) from a random function (and thus break its security as a PRF) by querying it on inputs x and x', where with x' we denote x with the first bit flipped. Note that the outputs of F'(k,.) will collide on inputs x and x' as F(x,k)=F(x',k).