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**2024/414**

**Title:**

Quantum One-Wayness of the Single-Round Sponge with Invertible Permutations

**Authors:**
Joseph Carolan, Alexander Poremba

**Abstract:**

Sponge hashing is a novel 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 short digest which consists of a subset of the final output bits. While much is known about the post-quantum security of the sponge construction in the case when the block function is modeled as a random function or 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 $\Omega(2^{n/2})$ many queries---and this is tight due to Grover's algorithm. At the core of our proof lies a novel `

symmetrization argumentâ€™â€™ 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.

**ePrint:**
https://eprint.iacr.org/2024/414

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