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MD4 is Not One-Way. The attack described in the 2008 paper is a theoretical attack with complexity $2^{102}$, which is better than the brute force complexity of $2^{128}$. In later theoretical results reported here the complexity dropped to $2^{94.98}$ and here it dropped even further to $2^{69.4}$ for secondary pre-image attacks. Similarly, The MD2 Hash ...


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There are different birthday bounds when we draw independent uniform random integers less then $d$  (for some large $d$, including $d=2^{32}$ of the question) and watch for collision(s): In crypto, we often consider the bound of $\sqrt d$  ($65536$ for $d=2^{32}$) draws, at which there is a fair probability of collision: $p\approx1-1/\sqrt ...



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