Distinguishing two sets of pseudorandom values when their keys differ by one Suppose we use a pseudorandom function $PRF$ and a random key $k$ to generate a set of pseudorandom values:

$\forall i, 1\leq i \leq n:  w_i=PRF(k,i)$

Now, consider instead of picking a fresh key, we increment the key by one:$k'=k+1$. Then, we compute a set of pseudorandom values:

$\forall i, 1\leq i \leq n:  q_i=PRF(k',i)$



Question: Are the set of $w_i$ values are computationally indistinguishable from the set of $q_i$ values?
Question: Are the set of $w_i$ values are computationally indistinguishable from the set of $q_i$ values?


No, that is not guaranteed by standard security notions.

AES for example is considered a secure PRP (and thus PRF up to a bound), but has related-key attacks. Granted, the known attacks have more complex key-dependence between a large number of keys, but still, in theory an otherwise secure PRF may be insecure if you see outputs for such related keys. They are only required to be secure with a randomly chosen key.