I would like to have a function $F(x, rnd)$, where $rnd$ is a fresh random value, such that it is hard, given $x_0$ and $x_1$, to distinguish $F(x_0, rnd)$ from $F(x_1, rnd)$, EDIT: but the values should be different, i.e., I should be able to recompute the value from $x_0$ and $rnd$ and see that it is the correct value.
I can do this with an asymmetric encryption function that has the IND-CPA property, but in my case, I do not need decryption. Thus, I thought about using a hash function, and my first simple idea was to simple concatenating the randomness, $H(x || rnd)$, but I'm not sure that this is sufficient.
I have looked into keyed hash functions, but I don't really want to use a key, just a random value. I did not really find a "randomized hash function" that would have a property comparable to IND-CPA.
Do I want something else, like a pseudorandom function, for my case? Can I construct one from a hash function?
Thank You for your help.