带可逆置换的单轮Sponge的量子单向性
Quantum One-Wayness of the Single-Round Sponge with Invertible Permutations
AI总结：
该研究证明了Unruh提出的双侧零搜索猜想，通过基于杨子群理论的对称化论证得出随机2n比特置换零对搜索的查询下界，并将其推广至一般双侧搜索问题，最终证明了带可逆置换的单轮Sponge在量子随机谕言机模型下的量子单向性，解决了SHA-3相关的基础性开放问题。
摘要：
Sponge哈希是一类应用广泛的密码哈希算法，是当前国际哈希函数标准SHA-3的底层基础。简而言之，Sponge函数接收任意长度的比特流作为输入，通过简单的迭代过程进行处理：它反复将输入的每个块输入到所谓的块函数中，随后通过对最终输出比特再次迭代块函数来生成摘要。尽管当块函数被建模为随机函数或单向置换时，人们对Sponge构造的后量子安全性已有较多了解，但更准确地刻画SHA-3底层构造的可逆置换情形，迄今仍是一个基础性的开放问题。\n在本研究中，我们在攻克这一障碍方面取得了新进展，并得出了若干结果。首先，我们证明了Unruh（eprint' 2021）提出的“双侧零搜索”猜想，并表明在随机2n比特置换中寻找零对至少需要Ω(2^{n/2})次查询——而由于Grover算法的存在，该下界是紧的。我们证明的核心是一种新颖的“对称化论证”，它运用了来自杨子群理论的洞见。其次，我们研究了双侧搜索问题的更一般变体，并得出了类似的查询下界。作为应用，我们在量子随机谕言机模型下证明了带可逆置换的单轮Sponge的量子单向性。
英文摘要：
Sponge hashing is a widely used class of cryptographic hash algorithms which underlies the current international hash function standard SHA-3. In a nutshell, a sponge function takes as input a bit-stream of any length and processes it via a simple iterative procedure: it repeatedly feeds each block of the input into a so-called block function, and then produces a digest by once again iterating the block function on the final output bits. While much is known about the post-quantum security of the sponge construction when the block function is modeled as a random function or one-way permutation, the case of invertible permutations, which more accurately models the construction underlying SHA-3, has so far remained a fundamental open problem. In this work, we make new progress towards overcoming this barrier and show several results. First, we prove the "double-sided zero-search" conjecture proposed by Unruh (eprint' 2021) and show that finding zero-pairs in a random $2n$-bit permutation requires at least $Ω(2^{n/2})$ many queries -- and this is tight due to Grover's algorithm. At the core of our proof lies a novel "symmetrization argument" which uses insights from the theory of Young subgroups. Second, we consider more general variants of the double-sided search problem and show similar query lower bounds for them. As an application, we prove the quantum one-wayness of the single-round sponge with invertible permutations in the quantum random oracle model.
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