Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In my opinion, the paper is strong in terms of idea and mathematical basis, but needs clarification in notation and more generalizability. With small modifications, it could have a greater impact on the research community. In my opinion, the strengths of the paper are: The idea of ​​“symmetric folding” to increase the density of witnesses and improve random search is innovative and inspired by the physical world. The mathematical modeling and presentation of precise theorems and conclusions provide a solid theoretical basis for the proposed method. Numerical experiments show that the proposed method reduces the computation time in practice.

However, in my opinion, the following weaknesses and suggestions for revision should be considered and implemented:
Some symbols (such as ⊂⊂, ∩∩, ⇌⇌) are ambiguous and non-standard. It is suggested to use more conventional symbols or to define each symbol fully.

Although the method is demonstrated on the problem of number decomposition, it claims to be general. It would be better to provide more diverse examples from other domains.

Direct comparisons with classical stochastic algorithms (such as Monte Carlo) are lacking. Adding such a comparison would increase the validity of the method.

The computational cost of the folding process itself has not been investigated for large data. It would be useful to point out this issue and ways to optimize it.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript presents an interesting study inspired by the physical process of paper folding. It develops mathematical models for symmetric folding operations and applies them to matrix-based datasets, demonstrating their potential to improve the efficiency of random search. While the topic is conceptually appealing, the current version of the paper requires significant revision before it can be considered for publication.

• In the abstract, revise it to align with the style and structure typically used in Symmetry It currently reads as a sequence of technical statements rather than a concise summary of motivation, methodology, key results, and implications. Please ensure logical flow and coherence, highlighting the novelty, main findings, and potential applications of the proposed approach.
• In the Intro, References such as “Reference [19]” or “Reference [20]” are not used correctly in the current form. Please indicate the authors’ names.
• The aim of the study and the research questions (RQs) should be clearly and explicitly stated at the end of the introduction.
• A literature review section is missing. This section should synthesize prior research, discuss the current state of the field, and clearly define how the present work contributes to or advances existing knowledge.
• A flowchart of the proposed algorithm may be provided to better illustrate the use.
• In Section 2.2, the definitions currently provided lack theoretical grounding or proofs. Each definition should either be supported by a brief theoretical justification or illustrated with examples to clarify meaning and context. Mathematical rigor should be demonstrated through either formal proofs or explanatory examples.
• In line 161, " The statement "This method makes it simple to determine if a randomly chosen planar lattice is a multiple of a prime number" would benefit from a concrete example. Please illustrate this process step by step using a simple numerical or geometric example.
• The manuscript currently lacks a discussion section where results are interpreted, compared with previous studies, and critically analyzed.
• Moreover, managerial or practical implications of the proposed symmetric folding method should be discussed. For instance, in which domains (e.g., data science, computational geometry, cryptography, optimization) could this approach be realistically applied? This addition would significantly strengthen the paper’s relevance and impact.
• Please check language clarity and ensure consistent terminology.

Comments on the Quality of English Language

• Please check language clarity and ensure consistent terminology.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This paper presents an innovative symmetric folding method inspired by paper folding to accelerate witness-based random search, with rigorous mathematical modeling, and theoretical proofs in odd composite integer divisor identification. The core idea of enhancing witness density through folding is promising, and the numerical experiments demonstrate improved computational efficiency. However, there are specific issues regarding theoretical coherence, experimental detail, and clarity of key definitions that need to be addressed to strengthen the paper’s rigor and readability.

 

1. The introduction briefly mentions the inefficiency of existing methods (e.g., [22], [23]) but lacks in-depth analysis of their specific limitations. It only states "some results are still inefficient" without quantifying or elaborating on whether the inefficiency stems from witness distribution gaps, computational complexity, or other factors. This makes it difficult for readers to fully grasp the targeted innovation of the symmetric folding method and its advantages over prior work.

 

2. Definition 3 (Symmetric folding of strips) is incomplete. The text writes "the symmetric folding of P to (onto) Q, denoted by P Γ Q, meaning Q is the base, is given by" but fails to provide the subsequent mathematical formulation. As a core definition for understanding the folding operation, this omission creates ambiguity about how strip folding is computationally implemented, hindering the reproducibility of the proposed method.

 

3. The numerical experiments (Tables 1–5) lack critical details on experimental design. There is no explanation of whether the reported "Time" values are averages of multiple repeated trials, nor is there a discussion of the parameter selection for g (e.g., g=2 is used in examples but no justification is given for this choice or how varying g affects search efficiency). Additionally, the experiments only focus on integer factorization, and it is unclear if the method’s performance generalizes to other witness-based search scenarios mentioned in the introduction (e.g., cryptography, data mining).

 

4. The future work section is overly vague. The proposed direction of "developing random search algorithms on two-dimensional hybrid folded regions" does not build on the current experimental results or theoretical findings. For example, the paper does not analyze potential challenges such as computational overhead of multi-dimensional folding, optimal folding sequences, or how to handle irregular datasets, making the future research plan lack feasibility and continuity.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have replied to the comments, therefore it can be accepted in its current form.