# Choosing an independent hash function, given hash function value

Supposed we have a function $$h:U\to [m_1]$$. Given this hash function, can we generate without using randomization or a universal hash collection another hash $$h':U \to [m_2]$$, which depends on $$h$$ and the values of $$h'(x)$$ are uniform? i.e., for every value $$y\in [m_2]$$ we have $$\Pr_{x} (h'(x)=y|h(x)=t)=1/m_2$$ ($$x$$ is chosen at random).

I know that if $$U=[2^t]$$, $$m_1=2^{n_1}$$, $$m_2=2^{n_2}$$, $$t>n_1+n_2$$ and $$h(x)$$ is the first $$n_1$$ bits of $$x$$ this problem is easy: we can select $$h'(x)$$ to be the bits $$n_1+1$$ till $$n_2$$ of $$x$$. My question is how to generalized this approche.