数据结构遇上密码学：带预处理的3SUM问题
Data Structures Meet Cryptography: 3SUM with Preprocessing
AI总结：
本文揭示了数据结构与密码学的联系，应用 Fiat--Naor 技术改进 3SUM 预处理上界并反驳旧猜想，同时证明了数据结构下界与抵御预处理攻击的单向函数的等价性。
摘要：
本文展示了数据结构问题与抵御预处理攻击的密码学之间的若干联系。我们的结果涵盖了数据结构上界、密码学应用以及数据结构下界，具体总结如下。首先，我们应用具有密码学起源的 Fiat--Naor inversion 技术，获得了数据结构上界。具体而言，我们的技术为 N 输入 3SUM 问题的预处理版本提供了一组空间为 S 且（在线）时间为 T 的算法，其中 S^3·T = Õ(N^6)。这反驳了一个强猜想（Goldstein et al., WADS 2017），即对于任意常数 δ>0，不存在在 S=N^{2-δ} 和 T=N^{1-δ} 下解决该问题的数据结构。其次，我们证明了一大类（静态）数据结构问题的下界与随机预言机模型中抵御极强预处理攻击的单向函数之间的等价性。具体地，给定随机函数 F: [N] -> [N]（作为预言机访问），我们展示如何将其编译为函数 G^F: [N^2] -> [N^2]，该函数抵御在查询时间 T 内运行且满足 ST=O(N^{2-ε}) 的 S 比特预处理攻击（假设 3SUM 存在相应的数据结构下界）。相比之下，Hellman 的经典结果表明，F 本身更容易被求逆，例如使用 N^{2/3} 比特预处理在 N^{2/3} 时间内完成。我们还证明，从 kSUM 的硬度可得出更强的下界。我们的结果可等价解释为针对高度非均匀、具有大量辅助输入的敌手的安全性，或面对强力后门随机预言机时的安全性。第三，我们给出了 3SUM 及一系列几何问题的非自适应下界，其匹配了静态数据结构问题已知的最佳下界。
英文摘要：
This paper shows several connections between data structure problems and cryptography against preprocessing attacks. Our results span data structure upper bounds, cryptographic applications, and data structure lower bounds, as summarized next. First, we apply Fiat--Naor inversion, a technique with cryptographic origins, to obtain a data structure upper bound. In particular, our technique yields a suite of algorithms with space $S$ and (online) time $T$ for a preprocessing version of the $N$-input 3SUM problem where $S^3\cdot T = \widetilde{O}(N^6)$. This disproves a strong conjecture (Goldstein et al., WADS 2017) that there is no data structure that solves this problem for $S=N^{2-δ}$ and $T = N^{1-δ}$ for any constant $δ>0$. Secondly, we show equivalence between lower bounds for a broad class of (static) data structure problems and one-way functions in the random oracle model that resist a very strong form of preprocessing attack. Concretely, given a random function $F: [N] \to [N]$ (accessed as an oracle) we show how to compile it into a function $G^F: [N^2] \to [N^2]$ which resists $S$-bit preprocessing attacks that run in query time $T$ where $ST=O(N^{2-\varepsilon})$ (assuming a corresponding data structure lower bound on 3SUM). In contrast, a classical result of Hellman tells us that $F$ itself can be more easily inverted, say with $N^{2/3}$-bit preprocessing in $N^{2/3}$ time. We also show that much stronger lower bounds follow from the hardness of kSUM. Our results can be equivalently interpreted as security against adversaries that are very non-uniform, or have large auxiliary input, or as security in the face of a powerfully backdoored random oracle. Thirdly, we give non-adaptive lower bounds for 3SUM and a range of geometric problems which match the best known lower bounds for static data structure problems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
