Consider the Miyaguchi–Preneel construction:
$H_0 = E(0,m_0) \oplus m_0$ (0 here means a vector filled with zeros)
$H_1 = E(H_0,m_1) \oplus H_0 \oplus m_1$
where $E(K,M)$ is a block cipher (for example AES), $m_0, m_1$ are messages. What's the best way to find messages $m_0$, $m_1$, such that $H_1$ will have a given prefix? $prefix(H_1, len(P)) = P$? Is there a faster way than birthday paradox?