Review Reports

Round 1

Reviewer 1 Report

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Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

The article deals with randomness analysis of the sequences obtained by interleaving PN-sequences coming from different characteristic polynomials but with the same degree.

The article is interesting. Current topic. The weaknesses of the article are:
1) too short abstract and summary
2) a discussion of similar works, which is also very sparse
3) literature can be refreshed

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Summary of the Work

The aim of this work is to investigate period, linear complexity and some cryptographic properties of the "interleaving" Pseudo Noise (PN) sequences with different characteristic polynomials with the same degree. The authors compared these sequences with those ones obtained from other sequence generators with similar parameters.

Main Results Obtained

- The authors performed a randomness analysis of their sequences

- They showed that their sequences are better than the sequences obtained by "interleaving" PN-sequences with the same polynomial.

General Considerations

- The manuscript is written in a clear and pedagogical way.

- The authors have already published other works on interleaving (shifted) PN sequences.

- The first two sections of the work are propaedeutic and some parts of them have already appeared in the MDPI-Mathematics journal. Please check these Sections and change the sentences that have already been published.

- The innovative part of the work starts from Section 3. “Interleaving PN-sequences with different characteristic polynomials”. The first two Sections are, however, appreciable as the authors have explained the concepts by analysing concrete examples.

- Please check your English, some typos were found.

- Please complete the references by taking into account the suggestions below.

- There are some points concerning the cryptographic properties of the authors’ sequences that, in my opinion, need to be clarified. This is the purpose of the suggestions below.

Suggestions

1) The use of interleaving PN sequences involves some important synchronisation difficulties. In fact, since it is not periodic with a short period (ref. to Theorem 3, page 7 of the manuscript), the sequence proposed by the authors is difficult to use to synchronise communication and maintain synchronisation. For this reason, in general, it is preferable to use another public communication to solve the synchronisation problem where the hidden message is a slight distortion of the transmission. The authors are asked to comment this point.

2) It is known that the linear complexity of a sequence is the length of the shortest LFSR that generates such a sequence (for instance, in the specific example 1, the shrunken sequence shows linear complexity equals 6). For the sake of clarity, the authors are asked to explain in the manuscript - in few words - the importance of the linear complexity in terms of metric of the security of binary sequence, by making reference in particular the linear complexity of the interleaving PN-sequences with different polynomials in comparison with the ones obtained from other sequence generators with similar parameters.

3) Figure 2., illustrates the autocorrelation for 8-interleaved sequences with polynomials of degree 16. In particular, according to the authors, the Subfigure 2a shows that the values are almost zero (except for the first value which is 1), indicating random sequence. The authors' conclusion is based on the statement that if {s_i} is a random periodic sequence of period T, then |T · C(τ)| can be expected to be quite small for all values of τ with 0 < τ < T. However, from statistics we know that if we have random sequence the correlation coefficient is 0, but the converse is not true. Hence, we may object that Fig. 2a. only provides an indication about the randomness of the sequence but not the certainty. The authors are asked to dispel this possible objection.

4) Figure 3 illustrates the map of 8-interleaved sequence with different polynomials. Since this figure shows a disordered cloud, without patterns, the authors deduced that this corresponds to a chaos map that does not provide any useful information for the cryptanalysis of the sequence. However, as is well known, the usual test for determining whether a system is chaotic or non-chaotic is the calculation of the largest Lyapunov exponent. Indeed, there are several examples of maps showing messy clouds that aren't chaotic at all. In these cases, disordered clouds may hide underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals and self-organisation. A practical method of determining whether a system is chaotic or not is the following: if the system has sensitivity to initial conditions, then we can say that the system is chaotic. Did the authors verify whether the examined 8-interleaved sequence is sensitive to the initial conditions? If not, it can be objected that, even in this case, it can only be said that there is an indication of a chaos map but not certainty.

5) The concept of entropy was introduced too fleeting and introduced in the manner reported in the manuscript, the reader is completely lost. For clarity, first of all, please provide the definition of the entropy of the sequence. Successively, please show that the return map is useful to visually measure the entropy of the sequence above defined.

6) We come now to the key questions. How does your scheme protect message integrity? How is your scheme preferable to standard SSL/TLS channels providing both message integrity and confidentiality?

Conclusions

The manuscript is interesting and I enjoyed reading it. However, there are some points that need to be clarified. Authors are encouraged to make a supplementary effort by taking into account the above suggestions. In my opinion, this will contribute to increase the soundness of their work.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

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Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

The authors answered, point-to-point, the questions raised in my previous report. In my opinion, this version of the manuscript deserves to be published.

Author Response

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Author Response File: Author Response.pdf