Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript reviewed the quantum hashing functions. We have the following comments.

(1) In Page 2, “qubits (unitary transformations) and gates” may be “qubits and gates(unitary transformations”.

(2) The manuscript discusses the quantum hash method in depth. Starting from the quantum basic theoretical model, this manuscript introduces the hash condition, finite group, multi quadit state hash and implementation in devices in turn. The content of the full text is comprehensive and well-designed, which is of great help for learning quantum hash. It is suggested to publish.

Author Response

We thank the anonymous reviewer for useful comments. We do our best for fixing the pointed issues.

Comment 1: (1) In Page 2, “qubits (unitary transformations) and gates” may be “qubits and gates(unitary transformations”.

Responce 1: We fixed the issue.

Reviewer 2 Report

Comments and Suggestions for Authors

Referee Report on "Theory and Applications of Quantum Hashing"

This manuscript presents a thorough and timely review of quantum hashing, covering its theoretical foundations, practical constructions, circuit representations, and even experimental implementations. The authors have provided a comprehensive account that will be valuable both for newcomers and experts in quantum cryptography and quantum algorithms. The article is well-organized. Its strengths include the clear definitions of δ-one-way and ε-collision resistance, insightful examples illustrating trade-offs, and practical considerations for implementation on various quantum architectures.

Before recommendation for acceptance, some matters need to be addressed:

1. To appeal to a wider community, the authors should expand the introductions on recent trends on quantum cryptography such as:
   - "The quantum internet: A synergy of quantum information technologies and 6G networks." IET Quantum Communication 4.4 (2023): 147-166.
   - "Artificial intelligence and quantum cryptography." Journal of Analytical Science and Technology 15.1 (2024): 4.
2. Captions in figures 2 and 4, 9 10, 12, 13, 14, 15, 16 should be expanded and include a full description of the content.
3. Figures 10, 11 should be revised. The labels are unclear and there is no x axis.
4. In page 21, how does the quantum hashing technique contribute to demonstrating computational advantages of quantum models over classical models across different computational frameworks such as automata, branching programs, and online algorithms?
5. General statements:
   The notation for quantum states sometimes alternates between |ψ(x)⟩ and |ψx⟩, and the formatting for superscripts and indices is occasionally inconsistent (e.g., bj vs. b_j). A more uniform use of notation would improve readability and precision, especially for pedagogical purposes.
6. There are a few minor grammar and phrasing issues throughout the text (e.g., "has been shown" vs. "we have shown", "probability of extracting initial message" is better as "probability of recovering the original message"). A light language edit would improve polish and clarity.

The paper should be revised and reconsidered after another revision.

Author Response

We thank the anonymous reviewer for useful comments. We do our best for fixing the pointed issues.

Comment 1: To appeal to a wider community, the authors should expand the introductions on recent trends on quantum cryptography such as:

- "The quantum internet: A synergy of quantum information technologies and 6G networks." IET Quantum Communication 4.4 (2023): 147-166.
- "Artificial intelligence and quantum cryptography." Journal of Analytical Science and Technology 15.1 (2024): 4.

Responce 1: We added discussion that includes citation of theses papers to intoduction.

Comment 2: Captions in figures 2 and 4, 9 10, 12, 13, 14, 15, 16 should be expanded and include a full description of the content.
Responce 2: We updated captions of the mentioned figures.

Comment 3: Figures 10, 11 should be revised. The labels are unclear and there is no x axis.
Responce 3: We updated captions of the mentioned figures and added detailed discriptions for axises

Comment 4: In page 21, how does the quantum hashing technique contribute to demonstrating computational advantages of quantum models over classical models across different computational frameworks such as automata, branching programs, and online algorithms?
Responce 4: We added description on each of the mentioned items

Comment 5:  General statements: The notation for quantum states sometimes alternates between |ψ(x)⟩ and |ψx⟩, and the formatting for superscripts and indices is occasionally inconsistent (e.g., bj vs. b_j). A more uniform use of notation would improve readability and precision, especially for pedagogical purposes.

Responce 5: We update the notation everywhere were we found the issue.

Comment 6: There are a few minor grammar and phrasing issues throughout the text (e.g., "has been shown" vs. "we have shown", "probability of extracting initial message" is better as "probability of recovering the original message"). A light language edit would improve polish and clarity.

Responce 6: We fixed the mentioned issues. We do our best for improving English of the paper

Reviewer 3 Report

Comments and Suggestions for Authors

The manuscrpit is good, but in order to be published the following references must be added:

Vladlen Statiev et al, Quantum Rep. 7 (2), 16 (2025), https://doi.org/10.3390/quantum7020016;

Alexey Pakhomchik, Valerii Vinokur, Cryptoalgorithms in the World of Ideal Quantum Annealers, The Role of Cybersecurity in the Industry 5.0 Era, 2025, ISBN = {978-0-85014-484-0},
url = {https://doi.org/10.5772/intechopen.114620},
DOI = {10.5772/intechopen.114620},
publisher = {IntechOpen},
year = {2025},
month = {Feb};

V. Slepnev, et al, Fluxonium-based superconducting qubit magnetometer: Optimization of phase estimation algorithms; Physical Review B 110 (21), 214423 (2024);

N. Kirsanov et al, Loss Control-Based Key Distribution under Quantum Protection, Entropy 26 (6), 437 (2024).

Author Response

We thank the anonymous reviewer for pointing out article [V. Slepnev, et al, Fluxonium-based superconducting qubit magnetometer: Optimization of phase estimation algorithms; Physical Review B 110 (21), 214423 (2024);], which is dedicated to the physical implementation of the QPE algorithm, and refer to it in section 6. However, other articles suggested by the reviewer ([Vladlen Statiev et al, Quantum Rep. 7 (2), 16 (2025), https://doi.org/10.3390/quantum7020016;], [Alexey Pakhomchik, Valerii Vinokur, Cryptoalgorithms in the World of Ideal Quantum Annealers, The Role of Cybersecurity in the Industry 5.0 Era,], [N. Kirsanov et al, Loss Control-Based Key Distribution under Quantum Protection, Entropy 26 (6), 437 (2024).]) are dedicated to the implementation of quantum key distribution, which is a large and important topic, but unfortunately outside the scope of our review.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have undertaken a thorough and detailed revision. I appreciate their efforts in addressing the previous feedback. There are, however, a few minor issues that should be addressed before publication:

1. Equations throughout the manuscript are generally not numbered. Please ensure all equations are properly numbered for clarity and reference.

2. The equation referenced in line 113 should be explicitly cited in the text.

3. While most figures in the paper are clear and well-presented, Figure 11 requires revision. Specifically, the numbers on top of the blue bars should be minimized—retain only a few key values (e.g., 4042, 1053, 828) to avoid clutter and improve readability.

Subject to these minor revisions, the paper should be suitable for publication.

Author Response

Thank you for your useful comments

Comment 1: Equations throughout the manuscript are generally not numbered. Please ensure all equations are properly numbered for clarity and reference.

Reply 1: We have numbered all the equations, and have also used subequation numbering for formulae 16a, 16b, 23a, 23b.

Comment 2: The equation referenced in line 113 should be explicitly cited in the text.

Reply 2: Unfortunately, it is not clear which equation is meant. If we are not mistaken, it appears that line 113 contains the basic gate definitions, so there is no reference in this line. However, we found and fixed a citation [141] in the beginning of Section 7.1.

Comment 3: While most figures in the paper are clear and well-presented, Figure 11 requires revision. Specifically, the numbers on top of the blue bars should be minimized—retain only a few key values (e.g., 4042, 1053, 828) to avoid clutter and improve readability.

Reply 3: We updated figures 10 and 11.