An Efficient Certificate-Based Linearly Homomorphic Signature Scheme for Secure Network Coding

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper presents a new certificate-based linearly homomorphic signature scheme designed to secure network coding systems against pollution and eavesdropping attacks. The work addresses a recognised challenge in the field by attempting to combine the benefits of certificate-based cryptography with the efficiency demands of homomorphic operations in resource-constrained environments like mobile crowdsensing. The paper tackles a relevant and non-trivial problem. The proposed scheme is clearly described, and the authors have made a commendable effort to provide a formal security model, a security proof in the random oracle model, and a performance comparison.

However, the paper in its current form has significant shortcomings which need to be addressed before it can be accepted for publication.

The most significant issue is lack of contextualisation within the current state-of-the-art. There is no "Related Work" or "Literature Review" section which is usually the second section of papers like this one. After a short "Introduction" section, it directly jumps to preliminaries and definitions of the proposed scheme, without fully discussing current progresses and situations in the literature.

The research design is broadly appropriate and fundamentally sound, but it has notable weaknesses that should be addressed. The paper clearly defines its objectives, presents a formal security model, describes the construction of the certificate-based linearly homomorphic signature scheme, and provides both correctness and security analyses. However, the design could be improved by more clearly explaining parameter choices, providing broader benchmarking against additional recent schemes, and offering more diverse experimental scenarios. The methodology is sound, but a deeper and more comprehensive evaluation would enhance the credibility and robustness of the results.

The methods in the paper are generally well described, with formal definitions, algorithmic steps, and security proofs presented in a structured manner. The construction of the certificate-based linearly homomorphic signature scheme is laid out in detail, and the security model is clearly stated. However, some aspects could be explained more clearly for full reproducibility. In particular, certain parameter choices, implementation details, and assumptions are not fully justified, and the performance evaluation methodology would benefit from a more explicit description of the experimental setup.

The results are clearly presented overall, with tables and figures that summarise computational performance, storage requirements, and comparative efficiency. However, some figures and tables would benefit from more detailed captions, clearer axis labelling, and additional explanation in the text to help readers interpret the significance of the results. The comparisons focus on a limited set of existing schemes, and expanding these benchmarks could further clarify the advantages of the proposed approach. In general, the results are understandable but could be made more informative and visually polished.

The conclusions are generally supported by the results, but with some limitations. The paper claims improved computational efficiency and practical applicability of the proposed certificate-based linearly homomorphic signature scheme. These claims are consistent with the theoretical complexity analysis and the experimental timing results presented. However, the evaluation focuses on a relatively narrow set of comparisons and experimental scenarios. A broader and more diverse set of benchmarks would strengthen the evidence base. Despite this, the reported results do align with the main conclusions, and no claims appear to exceed what the data reasonably supports.

The figures and tables are mostly clear and adequately presented, but there is room for improvement. In general, they convey the intended information, and the numerical comparisons are easy to follow. However, several elements could be enhanced: some captions are brief and do not fully explain the context; axis labels and units are missing or too small in certain plots; and formatting consistency across tables could be improved.

Comments on the Quality of English Language

The paper requires thorough proofreading for English grammar and clarity. There are numerous awkward phrasings and grammatical errors that hinder comprehension (e.g., "an extra expensive secret channel is needs," "the certifier responses a certificate").

Author Response

Comments 1: The most significant issue is lack of contextualisation within the current state-of-the-art. There is no "Related Work" or "Literature Review" section which is usually the second section of papers like this one. After a short "Introduction" section, it directly jumps to preliminaries and definitions of the proposed scheme, without fully discussing current progresses and situations in the literature.

Response: We sincerely thank the reviewer for this insightful suggestion. We agree that a comprehensive contextualization is essential for a technical paper. In the revised manuscript, we have added a dedicated Section 2: Related Work. This section provides a thorough review of the current state-of-the-art in linearly homomorphic signatures and certificate-based cryptography. We have also included a discussion on the evolution of these schemes to better situate our proposed work within the existing literature.

Comments 2: The research design is broadly appropriate and fundamentally sound, but it has notable weaknesses that should be addressed. The paper clearly defines its objectives, presents a formal security model, describes the construction of the certificate-based linearly homomorphic signature scheme, and provides both correctness and security analyses. However, the design could be improved by more clearly explaining parameter choices, providing broader benchmarking against additional recent schemes, and offering more diverse experimental scenarios. The methodology is sound, but a deeper and more comprehensive evaluation would enhance the credibility and robustness of the results.

Response: We appreciate the reviewer’s positive assessment of our research design and the constructive feedback on its weaknesses. We have updated our performance evaluation with more detailed benchmarking. Regarding the comparison with other schemes, we would like to clarify that most existing linearly homomorphic signature (LHS) schemes are constructed under the Public Key Infrastructure (PKI) or Identity-Based (IB) settings. Since our scheme is Certificate-Based (CB), a direct quantitative comparison with PKI-based schemes might be misleading due to different trust models and overhead structures. However, we have added a qualitative comparison in Section 2 to highlight the specific advantages of our certificate-based approach.

Comments 3: The methods in the paper are generally well described, with formal definitions, algorithmic steps, and security proofs presented in a structured manner. The construction of the certificate-based linearly homomorphic signature scheme is laid out in detail, and the security model is clearly stated. However, some aspects could be explained more clearly for full reproducibility. In particular, certain parameter choices, implementation details, and assumptions are not fully justified, and the performance evaluation methodology would benefit from a more explicit description of the experimental setup.

Response: Thank you for pointing out the need for better reproducibility. We have taken the following steps to ensure that our results can be independently verified: We have added a detailed description of our experimental setup in Section 6.1. This includes the use of the Java Pairing-Based Cryptography (jPBC) library and the specific configuration of the Type F elliptic curve, which provides a standard security level for modern cryptographic protocols.

Comments 4: The results are clearly presented overall, with tables and figures that summarise computational performance, storage requirements, and comparative efficiency. However, some figures and tables would benefit from more detailed captions, clearer axis labelling, and additional explanation in the text to help readers interpret the significance of the results. The comparisons focus on a limited set of existing schemes, and expanding these benchmarks could further clarify the advantages of the proposed approach. In general, the results are understandable but could be made more informative and visually polished.

Response: We appreciate the reviewer's feedback on the visual presentation of our data. We have overhauled all figures and tables in the revised manuscript. Specifically, we have: Expanded all captions to provide sufficient context for independent interpretation. Standardized axis labels and units for better readability. Added detailed textual explanations for each figure to guide the reader through the significance of the observed performance trends, particularly emphasizing the efficiency gains of our scheme.

Comments 5: The conclusions are generally supported by the results, but with some limitations. The paper claims improved computational efficiency and practical applicability of the proposed certificate-based linearly homomorphic signature scheme. These claims are consistent with the theoretical complexity analysis and the experimental timing results presented. However, the evaluation focuses on a relatively narrow set of comparisons and experimental scenarios. A broader and more diverse set of benchmarks would strengthen the evidence base. Despite this, the reported results do align with the main conclusions, and no claims appear to exceed what the data reasonably supports.

Response: We thank the reviewer for this observation. We have refined the conclusion to more accurately reflect the scope of our findings. In the revised Section 6, we have elaborated on our evaluation metrics. In cryptographic scheme analysis, the primary focus is typically on the computational overhead of critical phases, such as signature generation and homomorphic verification (data packet detection). While we have focused on these bottlenecks, we have added a discussion acknowledging that broader network-scale scenarios could be an area for future work, thereby ensuring our claims remain strictly within the bounds of our experimental data.

Comments 6: The figures and tables are mostly clear and adequately presented, but there is room for improvement. In general, they convey the intended information, and the numerical comparisons are easy to follow. However, several elements could be enhanced: some captions are brief and do not fully explain the context; axis labels and units are missing or too small in certain plots; and formatting consistency across tables could be improved.

Response: We thank the reviewer for the meticulous attention to detail. We have conducted a comprehensive review of all visual elements. All tables have been reformatted for consistency (e.g., aligning decimal points and using uniform font sizes). We have ensured that every plot now includes clear axis labels with correct units and that all captions are self-explanatory.

Reviewer 2 Report

Comments and Suggestions for Authors

An Efficient Certificate-based Linearly Homomorphic Signature Scheme for Secure Network Coding

Decision: Reject

The problem in question is relevant in the domain of cybersecurity. Presenting such a study under the concept of a comparative analysis is important for the continued development of the procedures that govern the security mechanisms related to existing mobile crowdsensing systems (MSC). However, the methodology developed and presented as the main contribution is not described in a thorough and scientifically accepted manner to be adequate for publication, although the paper  as follows:

1. The paper’s title needs to be specific regarding the paper’s contributions. The authors should avoid sticking to the notion of a comparative study and compare the executed evalutation experiments over the results of other already published papers on the same field of study.
2. The related work section is absent and the authors fail to introduce the reader to the main topic upon which the paper under evaluation outperforms and contributed compared to the already conducted state of the art published papers.
3. The authors neglect to present as open source the JPBC scripts through which the proposed contributions have been implemented.
4. The structure of the paper could be considered more as a Master or Postgraduate thesis rather than a paper adequate for publication due to the numerous bullet styled mathematical expressions and definitions which although enclose valued confuse the reader not only to understand the main points of the conducted solution but also to follow the great number of abbreviations which by the way are poorly explained.

5. Finally, no significant discussion upon the extracted results is conducted constraining the conducted research paper into comparing models without any novelty or significant contribution. The authors should also try to compare their conducted experiments to the results of other prominent works in the same field of study to add value to their work.

My final proposition is Reject.

Comments on the Quality of English Language

As in the review text that precedes.

Author Response

Comment 1: The paper’s title needs to be specific regarding the paper’s contributions. The authors should avoid sticking to the notion of a comparative study and compare the executed evalutation experiments over the results of other already published papers on the same field of study.

Response: We thank the reviewer for this constructive suggestion regarding the title and the comparative analysis.   We follow the reviewer’s advice to move beyond a simple internal evaluation. In the revised Section 6, we have included a comparative study that aligns our experimental results with those reported in recent literature. This comparison demonstrates our scheme’s efficiency in terms of computation overhead relative to the current state-of-the-art.

 

Comment 2: The related work section is absent and the authors fail to introduce the reader to the main topic upon which the paper under evaluation outperforms and contributed compared to the already conducted state of the art published papers.

Response: We sincerely thank the reviewer for this insightful suggestion. We agree that a comprehensive contextualization is essential for a technical paper. In the revised manuscript, we have added a dedicated Section 2: Related Work. This section provides a thorough review of the current state-of-the-art in linearly homomorphic signatures and certificate-based cryptography. We have also included a discussion on the evolution of these schemes to better situate our proposed work within the existing literature.

Comment 3: The authors neglect to present as open source the JPBC scripts through which the proposed contributions have been implemented.

Response: We appreciate the reviewer's emphasis on open science and reproducibility. We would like to clarify that our implementation utilizes the standard Java Pairing-Based Cryptography (jPBC) library. To address the reviewer's concern, we have significantly expanded the "Implementation Details" section in the manuscript, providing the specific parameters, curve types (Type F), and algorithmic flow used in our experiments. We are currently cleaning up the source code and scripts to meet documentation standards. We intend to make these scripts available in a public repository (e.g., GitHub) upon the formal publication of the paper.

Comment 4: The structure of the paper could be considered more as a Master or Postgraduate thesis rather than a paper adequate for publication due to the numerous bullet styled mathematical expressions and definitions which although enclose valued confuse the reader not only to understand the main points of the conducted solution but also to follow the great number of abbreviations which by the way are poorly explained.

Response: We sincerely apologize for any clarity issues caused by the presentation style and the use of abbreviations. We have restructured the manuscript to improve the narrative flow. Specifically, we have converted some of the bulleted mathematical definitions into a more cohesive textual format, ensuring that the formal rigor required for a cryptographic paper does not hinder readability. Furthermore, we have ensured that every abbreviation is defined at its first mention in the text and provided more intuitive explanations for the key components of our proposed solution to help readers follow the core logic more easily.

Comment 5: Finally, no significant discussion upon the extracted results is conducted constraining the conducted research paper into comparing models without any novelty or significant contribution. The authors should also try to compare their conducted experiments to the results of other prominent works in the same field of study to add value to their work.

Response: We appreciate the reviewer’s positive assessment of our research design and the constructive feedback on its weaknesses. We have updated our performance evaluation with more detailed benchmarking. Regarding the comparison with other schemes, we would like to clarify that most existing linearly homomorphic signature (LHS) schemes are constructed under the Public Key Infrastructure (PKI) or Identity-Based (IB) settings. Since our scheme is Certificate-Based (CB), a direct quantitative comparison with PKI-based schemes might be misleading due to different trust models and overhead structures. However, we have added a qualitative comparison in Section 2.

Reviewer 3 Report

Comments and Suggestions for Authors

The paper proposes a certificate-based linearly homomorphic signature scheme to against pollution attacks and preserve confidentiality in the network coding system.

The following recommendations for improvement should be considered to improve the quality of the paper:

• A literature review section should be included in the paper (after the introduction section). In the literature review, you need to discuss and critically review strongly related research work. Moreover, at the end of the literature review, you need to include paragraphs that highlight current research gap that will be addressed in your research study.

 

• To provide better understanding of the performance, provide detailed analysis of the computational cost/ time delay for the various digital signature operations, including signature generation and verification operations.

 

• In the section "5.2. Experiment Analysis" includes a comparison between the proposed method and other digital signature schemes in terms of resources consumption. This will be helpful to identify the most efficient method.

 

• The similarity percentage is high and not acceptable. Therefore, the paper should be revised to avoid the similarity. If any text is taken from other sources, it should be included in double quotation. You should include citations and references for all information taken from other sources. You should declare the use of generative AI tools (if any).

 

• The paper should be revised to fix grammatical issues. Please avoid the use of "We propose" or "We say" to indicate the work you have done, because it is not appropriate for scientific writing. Instead, you can use "this research proposes" or "this study introduces".

 

• Section "2. Preliminaries and Definitions"  should have an introductory paragraph before subsection 2.1

 

• Provide an appropriate in-text citation and references for the definitions presented in section 2.1.

 

• In section 3.5, provide more explanation for the "Data Processing Operations"

 

• The following sections should be re-written according to the journal's policies and they need to be specific to reflect the case of the underlying paper and the authors.

 

1. Author Contributions
2. Funding
3. Institutional Review Board Statement
4. Data Availability Statement
5. Informed Consent Statement
6. Acknowledgments
7. Conflicts of Interest

 

 

Comments on the Quality of English Language

• The paper should be revised to fix grammatical issues. Please avoid the use of "We propose" or "We say" to indicate the work you have done, because it is not appropriate for scientific writing. Instead, you can use "this research proposes" or "this study introduces".

Author Response

Comment 1: A literature review section should be included in the paper (after the introduction section). In the literature review, you need to discuss and critically review strongly related research work. Moreover, at the end of the literature review, you need to include paragraphs that highlight current research gap that will be addressed in your research study.

Response: We sincerely thank the reviewer for this insightful suggestion. We agree that a comprehensive contextualization is essential for a technical paper. In the revised manuscript, we have added a dedicated Section 2: Related Work. This section provides a thorough review of the current state-of-the-art in linearly homomorphic signatures and certificate-based cryptography. We have also included a discussion on the evolution of these schemes to better situate our proposed work within the existing literature.

 

Comment 2: To provide better understanding of the performance, provide detailed analysis of the computational cost/ time delay for the various digital signature operations, including signature generation and verification operations.

Response: We appreciate this suggestion to enhance the technical depth of our performance evaluation. In the revised Section 6, we have included a comprehensive breakdown of the computational costs. This includes both a theoretical asymptotic analysis and experimental timing results for each critical phase. We have specifically highlighted the latency of pairing operations and modular exponentiations to provide a clearer picture of the scheme's real-world efficiency.

 

Comment 3: In the section "5.2. Experiment Analysis" includes a comparison between the proposed method and other digital signature schemes in terms of resources consumption. This will be helpful to identify the most efficient method.

Response: We appreciate the reviewer’s positive assessment of our research design and the constructive feedback on its weaknesses. We have updated our performance evaluation with more detailed benchmarking. Regarding the comparison with other schemes, we would like to clarify that most existing linearly homomorphic signature (LHS) schemes are constructed under the Public Key Infrastructure (PKI) or Identity-Based (IB) settings. Since our scheme is Certificate-Based (CB), a direct quantitative comparison with PKI-based schemes might be misleading due to different trust models and overhead structures. However, we have added a qualitative comparison in Section 2.

 

Comment 4: The similarity percentage is high and not acceptable. Therefore, the paper should be revised to avoid the similarity. If any text is taken from other sources, it should be included in double quotation. You should include citations and references for all information taken from other sources. You should declare the use of generative AI tools (if any).

Response: We take the issue of manuscript originality very seriously and thank the reviewer for bringing this to our attention. We have conducted a thorough revision of the entire manuscript to significantly reduce the similarity index. We have rephrased the descriptive text and restructured several sections to ensure originality. We would like to clarify that some remaining similarities may occur in the Preliminaries and Security Proofs. In cryptography, definitions of standard hard problems (e.g., the Bilinear k-CCA problem) and formal security models follow established, standardized nomenclature to ensure mathematical precision and avoid ambiguity. However, we have ensured that all such instances are properly cited and attributed to their original sources.

 

 

Comment 5: Section "2. Preliminaries and Definitions"  should have an introductory paragraph before subsection 2.1

Response: Thank you for this suggestion regarding the structural clarity of the paper. We have added an introductory paragraph at the beginning of Section 2. This paragraph outlines the mathematical foundations and cryptographic primitives (such as bilinear pairings and the specific certificate-based framework) that will be discussed in the subsequent subsections, providing a roadmap for the reader.

 

Comment 6: Provide an appropriate in-text citation and references for the definitions presented in section 2.1.

Response: We agree that proper attribution is essential. We have added the necessary citations in Section 3.1 to credit the foundational works from which these standard definitions and mathematical properties are derived.

 

Comment 7: In section 3.5, provide more explanation for the "Data Processing Operations"

Response: Thank you for the feedback. We realized that the initial explanation was too concise. We have significantly expanded the discussion on "Data Processing Operations." Due to the restructuring of the paper, this detailed explanation can now be found in Sections 4.5 and 4.6. These sections now provide a step-by-step description of how data is pre-processed and integrated into the linearly homomorphic signature generation process.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have addressed the issues pointed out in the previous report. I would recommend that this new version of the paper is accepted.

Author Response

Response: We thank the reviewer for the positive feedback and for recommending the acceptance of our paper. We appreciate the constructive comments provided during the revision process.

Reviewer 2 Report

Comments and Suggestions for Authors

As in the review text that precedes.

The authors have amended adequately the originally submitted manuscript. However one final thorough check for any for any grammatical, spelling errors or general improvement of the syntax of certain paragraphs will be on favor of the published manuscript.  Finally, the authors should expand further the discussion upon their conducted experimental evaluation procudure. That is the core of their paper's contribution and should be as descriptive and analytical as possible.

Comments on the Quality of English Language

As in the review text that precedes.

Author Response

Comment: The authors should expand further the discussion upon their conducted experimental evaluation procedure. That is the core of their paper's contribution and should be as descriptive and analytical as possible.

Response: We appreciate the reviewer's emphasis on the experimental evaluation. While the primary contribution of this work lies in the theoretical construction and formal security proof of the proposed Certificate-based Linearly Homomorphic Signature scheme, we fully agree that a rigorous empirical evaluation is essential to demonstrate its practical feasibility. In response to this suggestion, we have significantly expanded the Experimental Evaluation section (Section 6.2) in the revised manuscript. 

Reviewer 3 Report

Comments and Suggestions for Authors

It seems that the comments are addressed and the manuscript has been improved accordingly. I have no further comments.

Comments on the Quality of English Language

• The paper should be revised to fix grammatical issues. Please avoid the use of "We propose" or "We say" to indicate the work you have done, because it is not appropriate for scientific writing. Instead, you can use "this research proposes" or "this study introduces".

Author Response

Response: We thank the reviewer for the positive feedback and for recommending the acceptance of our paper. We appreciate the constructive comments provided during the revision process.