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The author has incorporated all comments and it is fine-tuned now.
[# PeerJ Staff Note - this decision was reviewed and approved by Vicente Alarcon-Aquino, a PeerJ Section Editor covering this Section #]
I think the paper is now ready for publication
The research quality and the presentation style is very suitable for the journal
All the results seem valid and noval
good
ok
good
good
Authors updated the paper and no further comment from my side.
Good
Good
Authors are asked to update the methodology with visualization and correct the minor grammar errors.
[# PeerJ Staff Note: The review process has identified that the English language must be improved. PeerJ can provide language editing services - please contact us at copyediting@peerj.com for pricing (be sure to provide your manuscript number and title) #]
[# PeerJ Staff Note: Please ensure that all review comments are addressed in a rebuttal letter and any edits or clarifications mentioned in the letter are also inserted into the revised manuscript where appropriate. It is a common mistake to address reviewer questions in the rebuttal letter but not in the revised manuscript. If a reviewer raised a question then your readers will probably have the same question so you should ensure that the manuscript can stand alone without the rebuttal letter. Directions on how to prepare a rebuttal letter can be found at: https://peerj.com/benefits/academic-rebuttal-letters/ #]
The content and language are in good condition. The references are cited well.
The research question is well-defined and the investigation has been performed to a high standard.
I have not checked all results but the ones I have checked are correct and I believe that the rest is also correct.
The article has interesting results. This paper proposes a hybrid method combining search and direct construction of involutory MDS matrices in a finite field. The authors also conduct experiments to find these matrices, in addition, they also evaluate the number of XOR operations of the found matrices and compare with other works.
However, the authors need to correct the following comments.
- In the Introduction, the authors should introduce some other methods to build MDS matrix such as: circulant matrix, circulant-like matrix, and recursive MDS matrix.
- Lines 175-176, the authors should change to “The following are properties of an MDS matrix.”
- Line 187, should edit “a_i s” => “a_i's”.
- Line 198, edit “b_i s” => “b_i's”.
- With the representative involutory MDS matrix size 2×2, the authors proved in Lemma 3, and showed that the matrix RIM_(2×2) is the representative involutory MDS matrix of size 2×2. With size 4×4, the authors give the matrix R_1 as an involutory MDS matrix of size 4×4 and prove it through Lemma 6, however, it is necessary to put the matrix R_1 in front of Lemma 6.
Good
Good
This paper contains a too much mathematical content without background, making it challenging for general readers to comprehend. In my opinion, the authors should consider incorporating additional explanations of the mathematics involved. There are several notations that are unclear, and the abstract itself is quite complex to grasp. Furthermore, the paper lacks background information or context for the equations presented. Here are some suggestions to improve the clarity and readability of the paper written in LaTeX:
Provide a clear and concise overview: Begin the paper by providing a brief overview of the problem statement and the main contribution of the proposed hybrid method for generating involutory Maximum Distance Separable (MDS) matrices.
Explain the significance of the research: Clearly explain why generating involutory MDS matrices over specific finite fields (such as $\mathbf{F}{2^3}$, $\mathbf{F}{2^4}$, and $\mathbf{F}_{2^8}$) is important and how it contributes to the field of cryptography or other relevant applications.
Define technical terms: Ensure that all technical terms, such as "involutory," "MDS matrices," and "search space complexity," are defined and explained in a concise manner. This will make the paper more accessible to readers who may not be familiar with the specific terminology.
Provide more details on the proposed hybrid method: Elaborate on the combination of search-based methods and direct construction methods in the proposed hybrid method. Explain how this approach reduces the search space complexity at the level of $\sqrt{n}$, where $n$ represents the number of invertible matrices to be searched for.
Present the results in a clear manner: Clearly present the results achieved by the proposed method, particularly in terms of generating involutory/non-involutory MDS matrices over $\mathbf{F}{2^3}$, $\mathbf{F}{2^4}$, and $\mathbf{F}_{2^8}$. Specify the XOR count and depth required for each matrix, highlighting their significance and efficiency.
Highlight the novelty of the findings: Emphasize the novelty of generating the lightest involutory/non-involutory MDS matrices known over the specified finite fields. Discuss how the proposed global optimization technique and the use of higher input XOR gates contribute to achieving these results.
Clarify the contribution related to Hadamard matrix: Clearly explain the new property of the Hadamard matrix and its role in generating a small subset of involutory MDS matrices. Highlight the specific value of this contribution, particularly regarding matrices with dimensions $2^k \times 2^k$, where $k>1$.
Proofread and revise for clarity: Ensure that the paper is thoroughly proofread for grammar, punctuation, and sentence structure errors. Simplify complex sentences and rephrase unclear or ambiguous statements to improve overall clarity and readability.
By implementing these suggestions, the paper will become more accessible, effectively communicate the proposed method and findings, and contribute to the existing body of knowledge in the field.
no experiment performed.
hard to find.
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