My scenario is as follows:
Two entities $A_1$, $A_2$ host same versions of replicated data $M$. An entity $A_1$ computes $h_1 = H_1(M)$ and another entity $A_2$ computes $h_2 = H_2(M)$.
Is there any set of functions $H$, $H_1$ and $H_2$ so that Alice, after receiving $h_1$ and $h_2$ can check $H(h_1) = h_2$, but $A_2$ cannot compute $h_2$ using $h_1$?
More precisely, there be two ways to compute $H_2$ with two different keys, a direct way and 2-step way. $A_2$ computes $h_2$ through direct way and Alice compute it through $h_2 = H(H_1(M))$.