A New Sampling Protocol and Applications to Basing Cryptographic Primitives on the Hardness of NP

We investigate the question of what languages can be decided efficiently with the help of a recursive collision-finding oracle. Such an oracle can be used to break collision-resistant hash functions or, more generally, statistically hiding commitments. The oracle we consider, $\Sam_d$ where $d$ is the recursion depth, is based on the identically-named oracle defined in the work of Haitner et al. (FOCS '07). Our main result is a constant-round public-coin protocol $\AMSam$'' that allows an efficient verifier to emulate a $\Sam_d$ oracle for any constant depth $d = O(1)$ with the help of a $\BPP^\NP$ prover. $\AMSam$ allows us to conclude that if $L$ is decidable by a $k$-adaptive randomized oracle algorithm with access to a $\Sam_{O(1)}$ oracle, then $L \in \AM[k] \cap \coAM[k]$. The above yields the following corollary: assume there exists an $O(1)$-adaptive reduction that bases constant-round statistically hiding commitment on $\NP$-hardness, then $\NP \subseteq \coAM$ and the polynomial hierarchy collapses. The same result holds for any primitive that can be broken by $\Sam_{O(1)}$ including collision-resistant hash functions and $O(1)$-round oblivious transfer where security holds statistically for one of the parties. We also obtain non-trivial (though weaker) consequences for $k$-adaptive reductions for any $k = \poly(n)$. Prior to our work, most results in this research direction either applied only to non-adaptive reductions (\citeauthor{BogdanovT06}, SIAM J. of Comp. '06 and \citeauthor{AkaviaGGM06}, FOCS '06) or to one-way permutations (\citeauthor{Brassard79} FOCS '79). The main technical tool we use to prove the above is a new constant-round public-coin protocol ($\SWS$), which we believe to be of interest in its own right, that guarantees the following: given an efficient function $f$ on $n$ bits, let $D$ be the output distribution $D = f(U_n)$, then $\SWS$ allows an efficient verifier Arthur to use an all-powerful prover Merlin's help to sample a random $y \getsr D$ along with a good multiplicative approximation of the probability $p_y = \Pr_{y' \getsr D}[y' = y]$. The crucial feature of $\SWS$ is that it extends even to distributions of the form $D = f(U_\cs)$, where $U_\cs$ is the uniform distribution on an efficiently decidable subset $\cs \subseteq \zo^n$ (such $D$ are called efficiently samplable with \emph{post-selection}), as long as the verifier is also given a good approximation of the value $|\cs|$.