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Učenye zapiski Kazanskogo gosudarstvennogo universiteta. Seriâ Fiziko-matematičeskie nauki/Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki

Analysis of the amplitude form of the quantum hash function

Marat Ablayev, Farid Mansurovich Ablayev, Alexander Vasiliev

Published Nov 13, 2023• 2 citations

Abstract

In this article, the properties of quantum hash functions are further explored. Previous findings show that so-called small-bias sets (special subsets of the set of elements of a cyclic group) generate a “phase” quantum hash function. Here, it was proved that they also generate an “amplitude” quantum hash function. Namely, it turned out that constructing small-bias sets while generating amplitude quantum functions yields a well-balanced combination of the cryptographic properties of unidirectionality and collision resistance. As a corollary of the obtained theorem, a general statement about the generation of new amplitude quantum hash functions based on universal hash families and small-bias sets was proved.

DOI: 10.26907/2541-7746.2023.1.5-15

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In the letter we define the notion of a quantum resistant ( -resistant) hash function which consists of a combination of pre-image (one-way) resistance ( ε -resistance) and collision resistance ( δ -resistance) properties. We present examples and discussion that supports the idea of quantum hashing. We present an explicit quantum hash function which is ‘balanced’, one-way resistant and collision resistant and demonstrate how to build a large family of quantum hash functions. Balanced quantum hash functions need a high degree of entanglement between the qubits. We use a phase transformation technique to express quantum hashing constructions, which is an effective way of mapping hash states to coherent states in a superposition of time-bin modes. The phase transformation technique is ready to be implemented with current optical technology.

The integrity of evidence, a core principle of digital forensics, is essential for ensuring the reliability and admissibility of evidence in legal proceedings. Classical cryptographic hash functions have effectively ensured the integrity of digital evidence. However, they cannot be applied to quantum evidence emerging with the advent of the quantum computing era. This is due to fundamental principles of quantum mechanics such as superposition and entanglement. To address the challenge of ensuring quantum evidence integrity, this paper introduces a quantum amplitude hash function as a new paradigm. The function operates by directly hashing the entire amplitude, which represents the totality of information within the quantum evidence state, into a hash qubit. The proposed function is implemented as a quantum circuit of constant depth, ensuring excellent scalability. It is theoretically proven to satisfy key cryptographic properties such as preimage resistance, collision resistance, and sensitivity. Moreover, simulation experiments were conducted to validate the generation of hash qubits for representative quantum algorithms. This work is expected to lay a crucial technical foundation for advancing the field of quantum forensics and preparing for future investigative environments.

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