基于伪随机性的单元清晰性与集合相交性的时空折衷
Time-Space Tradeoffs for Element Distinctness and Set Intersection via Pseudorandomness
摘要：
在元素区分问题中，给定一个来自$[poly(n)]$的整数数组$a_1,\dots, a_n$，并负责决定$\{a_i\}$是否相互区分。Beame，Clifford和Machmouchi（FOCS 2013）给出了一个在空间$S(n)$和时间$T(n)$中运行的低空间算法，其中$T(n) \le \widetilde{O}(n^{3/2}/S(n)^{1/2})$，假设一个随机预言（即，对多项式多个随机比特的随机访问）。Chen，Jin，Williams和Wu最近的一项突破（SODA 2022）展示了如何在$S(n) = polylog(n)$和$T(n) = \widetilde{O}(n^{3/2})$体系中消除随机预言假设。他们设计了第一个真正的$polylog(n)$ -空间，$\widetilde{O}(n^{3/2})$ -时间算法，通过构造一小族具有一定伪随机属性的哈希函数$\mathcal{H} \subseteq \{h | h:[poly(n)]\to [n]\}$。在本文中，我们给出了一个显着简化的分析，伪随机哈希家庭的Chen等人。我们的分析清楚地确定了关键的伪随机属性所需的欺骗的伪随机算法，使我们能够探索这种建设的全部潜力。作为我们的主要结果，我们显示了一个时间-空间权衡的元素清晰没有随机预言。也就是说，对于每个$S(n),T(n)$，使得$T\approx \widetilde{O}(n^{3/2}/S(n)^{1/2})$，我们的算法可以解决问题的空间$S(n)$和时间$T(n)$。我们的算法也适用于一个相关的问题集相交，由于Dinur的匹配下限，这种权衡是紧密的（Eurocrypt 2020）。作为两个额外的贡献，我们显示了一个更一般的伪随机属性的哈希家庭，并略有改善的种子长度采样的伪随机哈希函数。
英文摘要：
In the Element Distinctness problem, one is given an array $a_1,\dots, a_n$ of integers from $[poly(n)]$ and is tasked to decide if $\{a_i\}$ are mutually distinct. Beame, Clifford and Machmouchi (FOCS 2013) gave a low-space algorithm for this problem running in space $S(n)$ and time $T(n)$ where $T(n) \le \widetilde{O}(n^{3/2}/S(n)^{1/2})$, assuming a random oracle (i.e., random access to polynomially many random bits). A recent breakthrough by Chen, Jin, Williams and Wu (SODA 2022) showed how to remove the random oracle assumption in the regime $S(n) = polylog(n)$ and $T(n) = \widetilde{O}(n^{3/2})$. They designed the first truly $polylog(n)$-space, $\widetilde{O}(n^{3/2})$-time algorithm by constructing a small family of hash functions $\mathcal{H} \subseteq \{h | h:[poly(n)]\to [n]\}$ with a certain pseudorandom property. In this paper, we give a significantly simplified analysis of the pseudorandom hash family by Chen et al. Our analysis clearly identifies the key pseudorandom property required to fool the BCM algorithm, allowing us to explore the full potential of this construction. As our main result, we show a time-space tradeoff for Element Distinctness without random oracle. Namely, for every $S(n),T(n)$ such that $T\approx \widetilde{O}(n^{3/2}/S(n)^{1/2})$, our algorithm can solve the problem in space $S(n)$ and time $T(n)$. Our algorithm also works for a related problem Set Intersection, for which this tradeoff is tight due to a matching lower bound by Dinur (Eurocrypt 2020). As two additional contributions, we show a more general pseudorandom property of the hash family, and slightly improve the seed length to sample the pseudorandom hash function.
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。
