security for PRF when key is changed often Here's the situation. Suppose I have a secure PRF $F$ and use $F$ in the following way:

for equal size message $m_i$ in $\{m_j\}^{q}_{j=1}$( $q=poly(n)$, $n$ is security parameter ), randomly select $k_i \xleftarrow{\$}K$ and compute $c_i=F(k_i, m_i)$, where $K$ is key space. All previous computation results $\{(k_j, m_j, c_j)\}^{i-1}_{j=1}$ are revealed to adversary $A$ when using ($k_i$, $m_i$) to evaluate $F$. The main point is that the key is changed and some previous keys revealed, which is different to traditional pseudo-randomness definition of $PRF$. My security goal is adversary $A$ cannot distinguish $c_i$ from a random string $r_i$ using any $PPT$ distinguisher $D$. Intuitively, I think $F$ is secure in this setting. Is there any formal security model for this situation? and under what situation is it secure or insecure?
Yes, this is secure since the keys are chosen randomly. Remember that even in the "traditional" PRF game, the adversary can generate keys at random, so the output of F on past keys does not help him (since he can generate such outputs anyway).