基于Cuckoo Hashing的硬度保持约简
Hardness-Preserving Reductions via Cuckoo Hashing
摘要：
这项工作的重点是\emph{hardness-preserving}变换的有点有限的伪随机函数的家庭（PRF）到更通用的特性。考虑伪随机函数的\emph{domain extension}问题：给定一个以某个域$U$的元素作为输入的PRF，我们希望得到一个更大域上的PRF。我们能在不显著影响系统安全性的情况下完成这项工作吗？一种方法是首先将较大的域散列成较小的域，然后应用原始的PRF。然而，这样的减少易受“生日攻击”的影响：在$\sqrt{\size{U}}$查询所得到的PRF之后，很可能发生冲突（\ie两个不同的输入具有相同的散列值）。因此，针对进行该数量的查询的攻击者，所得到的PRF是\emph{insecure}。在这项工作中，我们将展示如何超越上述生日攻击障碍，用\textit{cuckoo hashing}的变体替换上述简单的哈希方法，这是一种哈希范式，通过使用两个哈希函数和两个表来解决表中的冲突，巧妙地将每个元素分配给两个表中的一个。我们使用这种方法来获得：（i）域扩展方法，需要{ \em只是两个调用}的原始PRF，可以承受许多查询的原始域的大小，并具有区分的概率是指数小的非加密工作量;和（ii）{ \em安全保护}减少从非自适应的自适应PRF。
英文摘要：
The focus of this work is \emph{hardness-preserving} transformations of somewhat limited pseudorandom functions families (PRFs) into ones with more versatile characteristics. Consider the problem of \emph{domain extension} of pseudorandom functions: given a PRF that takes as input elements of some domain $U$, we would like to come up with a PRF over a larger domain. Can we do it with little work and without significantly impacting the security of the system? One approach is to first hash the larger domain into the smaller one and then apply the original PRF. Such a reduction, however, is vulnerable to a "birthday attack": after $\sqrt{\size{U}}$ queries to the resulting PRF, a collision (\ie two distinct inputs having the same hash value) is very likely to occur. As a consequence, the resulting PRF is \emph{insecure} against an attacker making this number of queries. In this work we show how to go beyond the aforementioned birthday attack barrier by replacing the above simple hashing approach with a variant of \textit{cuckoo hashing}, a hashing paradigm that resolves collisions in a table by using two hash functions and two tables, cleverly assigning each element to one of the two tables. We use this approach to obtain: (i) a domain extension method that requires {\em just two calls} to the original PRF, can withstand as many queries as the original domain size, and has a distinguishing probability that is exponentially small in the amount of non-cryptographic work; and (ii) a {\em security-preserving} reduction from non-adaptive to adaptive PRFs.
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