Cuckoo hashing is a powerful primitive that enables storing items using small space with efficient querying. At a high level, cuckoo hashing maps $n$ items into $b$ entries storing at most $\ell$ items such that each item is placed into one of $k$ randomly chosen entries. Additionally, there is an overflow stash that can store at most $s$ items. Many cryptographic primitives rely upon cuckoo hashing to privately embed and query data where it is integral to ensure small failure probability when constructing cuckoo hashing tables as it directly relates to the privacy guarantees. As our main result, we present a more query-efficient cuckoo hashing construction using more hash functions. For construction failure probability $\epsilon$, the query overhead of our scheme is $O(1 + \sqrt{\log(1/\epsilon)/\log n})$. Our scheme has quadratically smaller query overhead than prior works for any target failure probability $\epsilon$. We also prove lower bounds matching our construction. Our improvements come from a new understanding of the locality of cuckoo hashing failures for small sets of items. We also initiate the study of robust cuckoo hashing where the input set may be chosen with knowledge of the hash functions. We present a cuckoo hashing scheme using more hash functions with query overhead $\tilde{O}(\log \lambda)$ that is robust against poly$(\lambda)$ adversaries. Furthermore, we present lower bounds showing that this construction is tight and that extending previous approaches of large stashes or entries cannot obtain robustness except with $\Omega(n)$ query overhead. As applications of our results, we obtain improved constructions for batch codes and PIR. In particular, we present the most efficient explicit batch code and blackbox reduction from single-query PIR to batch PIR.

翻译：布谷鸟哈希是一种强大的基础工具，能够以较小空间存储条目并实现高效查询。在高层抽象中，布谷鸟哈希将 $n$ 个条目映射到 $b$ 个存储位置中（每个位置最多容纳 $\ell$ 个条目），使得每个条目被放置到随机选择的 $k$ 个位置之一。此外，系统包含一个最多存储 $s$ 个条目的溢出暂存区。许多密码学原语依赖布谷鸟哈希进行数据私有嵌入与查询，此时确保哈希表构建的失败概率极小至关重要——该概率直接关联隐私保障强度。作为主要成果，我们提出了一种通过增加哈希函数数量实现更高查询效率的布谷鸟哈希构造方案。对于构造失败概率 $\epsilon$，该方案的查询开销为 $O(1 + \sqrt{\log(1/\epsilon)/\log n})$。对于任意目标失败概率 $\epsilon$，本方案的查询开销较以往工作呈二次方缩减。我们同时证明了与构造相匹配的下界。这些改进源于对小型条目集合的布谷鸟哈希失败局部性的新认识。我们还率先开展了鲁棒布谷鸟哈希的研究——在此场景中，输入集可能由知晓哈希函数的对手选择。我们提出一种采用更多哈希函数的布谷鸟哈希方案，其查询开销为 $\tilde{O}(\log \lambda)$，且能抵抗 $\text{poly}(\lambda)$ 规模的敌手。进一步地，我们给出的下界表明该构造已达到最优，并且以往扩增暂存区或条目容量的方法若未引入 $\Omega(n)$ 查询开销则无法实现鲁棒性。作为应用，我们获得了批处理码和PIR的改进构造。其中特别地，我们提出了最高效的显式批处理码，以及从单查询PIR到批处理PIR的黑盒归约。