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

2024-03-07

IACR Cryptology ePrint Archive

AI Summary

This work investigates the quantum one-wayness of the single-round sponge construction—underlying SHA-3—under reversible permutations. Addressing the long-standing open problem of rigorously proving quantum one-wayness in the reversible-permutation setting, we first confirm Unruh’s “bidirectional zero-search” conjecture. Methodologically, we introduce Young subgroup theory to construct a symmetry-based analytical framework and establish a tight lower bound within the quantum random oracle model. Leveraging group representation theory and Grover algorithm complexity analysis, we prove that any quantum adversary requires at least Ω(2^{n/2}) quantum queries to invert the sponge construction with non-negligible success probability. This result provides the first rigorous proof of quantum one-wayness for the single-round sponge under reversible permutations, thereby delivering a foundational theoretical guarantee for post-quantum hash function security.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

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 $Omega(2^{n/2})$ many queries -- and this is tight due to Grover's algorithm. At the core of our proof lies a novel"symmetrization 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.

Problem

• Quantum Computing
• Reversible Permutation
• Cryptography Security

Innovation

• Quantum Computing
• Reversible Sponge Hash Structure
• Symmetrization and Yang Subgroup Theory