Quantum Time–Space Tradeoff for Finding Multiple Collision Pairs

IF 0.8 Q3 COMPUTER SCIENCE, THEORY & METHODS
Yassine Hamoudi, F. Magniez
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引用次数: 15

Abstract

We study the problem of finding K collision pairs in a random function f : [N] → [N] by using a quantum computer. We prove that the number of queries to the function in the quantum random oracle model must increase significantly when the size of the available memory is limited. Namely, we demonstrate that any algorithm using S qubits of memory must perform a number T of queries that satisfies the tradeoff T3 S ≥ Ω (K3 N). Classically, the same question has only been settled recently by Dinur [22], who showed that the Parallel Collision Search algorithm of van Oorschot and Wiener [36] achieves the optimal time–space tradeoff of T2 S = Θ (K2 N). Our result limits the extent to which quantum computing may decrease this tradeoff. Our method is based on a novel application of Zhandry’s recording query technique [42] for proving lower bounds in the exponentially small success probability regime. As a second application, we give a simpler proof of the time–space tradeoff T2 S ≥ Ω (N3) for sorting N numbers on a quantum computer, which was first obtained by Klauck, Špalek, and de Wolf [30].
寻找多个碰撞对的量子时空权衡
我们研究了在随机函数f:[N]中寻找K个碰撞对的问题→ [N] 通过使用量子计算机。我们证明,当可用内存的大小有限时,对量子随机预言机模型中函数的查询数量必须显著增加。也就是说,我们证明了任何使用存储器的S个量子位的算法都必须执行满足折衷T3 S≥Ω(K3 N)的T个查询。传统上,Dinur[22]最近才解决了同样的问题,他表明van Oorshot和Wiener[36]的并行碰撞搜索算法实现了T2 S=θ(K2 N)的最佳时间-空间折衷。我们的结果限制了量子计算可以在多大程度上减少这种权衡。我们的方法基于Zhandry的记录查询技术[42]的一个新应用,用于证明指数小成功概率情况下的下界。作为第二个应用,我们给出了在量子计算机上排序N个数的时间-空间折衷T2 S≥Ω(N3)的一个更简单的证明,这是Klauck、Špalek和de Wolf[30]首次获得的。
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来源期刊
ACM Transactions on Computation Theory
ACM Transactions on Computation Theory COMPUTER SCIENCE, THEORY & METHODS-
CiteScore
2.30
自引率
0.00%
发文量
10
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