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Lobachevskii Journal of Mathematics

Cryptographic Properties of the Quantum Hashing Based on Expander Graphs

I. Zinnatullin

Published Feb 1, 2023• 4 citations
Publisher

Abstract

Quantum hashing is a useful technique in different computational and cryptographic scenarios in the quantum world. A set of random parameters is required to construct a quantum hashing scheme. For instance, random walks on expander graphs (expanders) are known to be efficient randomness generators in many areas of computer science. We analyze a scheme based on expanders. Collision resistance and preimage resistance of this scheme are considered. We show that quantum hashing based on expanders is collision-resistant (in quantum sense), and the considered scheme’s accessible information is $$O(1)$$ .

DOI
10.1134/s1995080223020397

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Quantum hashing is a useful technique in different computational and cryptographic scenarios in the quantum world. A set of random parameters is required to construct a quantum hashing scheme. For instance, random walks on expander graphs (expanders) are known to be efficient randomness generators in many areas of computer science. We analyze a scheme based on expanders. Collision resistance and preimage resistance of this scheme are considered. We show that quantum hashing based on expanders is collision-resistant (in quantum sense), and the considered scheme’s accessible information is O(1)O(1)O(1) .
In the paper we define a notion of quantum resistant ((δ, є)-resistant) hash function which combine together a notion of pre-image (one-way) resistance (δ-resistance) property and the notion of collision resistance (є-resistance) properties. We present a discussion that supports the idea of quantum hashing oriented for cryptographical purposes. We propose a quantum setting of a classical digital signature scheme do demonstrate a theoretical possibilities and restrictions of (δ, є)-hashing. The assumption we use is that a set of qubits (quantum hash) we generate, send, and receive during the execution of a protocol can be stored for a certain (a large enough) amount of time; next, the scheme requires the high degree of entanglement between the qubits which makes such a quantum hash. These properties make quantum hash cryptographically efficient.
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The rapid advancement of quantum computing presents a fundamental challenge to modern cryptographic security, particularly in the domain of hash functions that ensure data integrity, authentication, and blockchain security. Traditional crypto graphic hash functions such as SHA-256, SHA-3, and BLAKE2 rely on computational hardness assumptions that become obsolete in the presence of large-scale quantum computers. Shor’s algorithm can efficiently break RSA and ECC-based cryptosys tems, while Grover’s algorithm reduces the security of traditional hash functions by square root complexity, significantly weakening their preimage and collision resistance. This quantum threat necessitates the development of post-quantum secure hashing techniques that remain resilient against both classical and quantum adversaries. This paper proposes Quantum Hashing, a novel cryptographic framework that integrates quantum entanglement, lattice-based cryptography, and hybrid quantum classical hashing to construct post-quantum secure hash functions. We introduce a formal model for Quantum Collision Resistance (QCR) and provide entropy-based ran domness enhancement to ensure unpredictable hash outputs. Unlike classical hashing approaches, our framework leverages the hardness of lattice problems (e.g., Shortest Vector Problem, Learning with Errors) to withstand quantum attacks while incorpo rating Quantum Key Distribution (QKD) mechanisms to enhance entropy and key management. Furthermore, we evaluate the security of Quantum Hashing under various attack models, comparing its resistance against Grover’s search and collision attacks. We benchmark its performance against NIST Post-Quantum Cryptography (PQC) final ists, including CRYSTALS-DILITHIUM, SPHINCS+, and Falcon, demonstrating that our approach offers superior resilience while maintaining computational feasibility. Additionally, we present an implementation of Quantum Hashing using Qiskit, show casing its practical applicability in quantum circuits and quantum-secure blockchain architectures. Our findings highlight that Quantum Hashing provides a scalable, entropy-efficient, and post-quantum resilient cryptographic primitive suitable for next-generation cryptographic applications. This work paves the way for secure post-quantum digital signatures, blockchain consensus mechanisms, and zero-knowledge proof systems that require tamper-resistant hashing in a quantum computing era.