A Tweak-Cube Color Image Encryption Scheme Jointly Manipulated by Chaos and Hyper-Chaos

Round 1

Reviewer 1 Report

The work "A tweak-cube colour image encryption scheme jointly manipulated by chaos and hyper-chaos", has contributions that need to be improved, following changes:

In the introduction, it is necessary to clearly describe which problems and how this work will contribute. Highlight the contributions;

A scenario and / or applicability environment with the contribution of this work is necessary, try to describe a scenario with the proposed solutions;

Improving the quality of figures 11 and 12, it is unclear the differences between them;

In the conclusions, it is necessary to make clear how the work contributes scientifically in relation to the state of the art. What are the main points and approaches that the work produces meaning in relation to what was developed;

Review recent bibliographical references, search for articles published in 2019. I suggest at least 3 or 5 publications this year.

Author Response

Please see the attachment.

Author Response File: Author Response.docx

Reviewer 2 Report

The research paper is well structured and I recommend it for publication.

Author Response

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Author Response File: Author Response.docx

Reviewer 3 Report

1.

Introduction lines 20-24 are obsolete because they state well-known facts, which are not directly relevant to the paper.

2.

Authors state:

The bifurcation graph can objectively reflect the region and state of the chaotic behavior of a map. 

However, in addition, they are calculating Lyapunov's exponents as well. So, it seems that bifurcation graph has some disadvantages as a tool for "objective reflection". Please clarify.

3.

Please clarify how did you calculated Lyapunov exponents in subsection 2.1.2. Namely, you are using lim(n->infinity), which obviously had to be adapted due to the finite nature of real calculation. There are maybe some more peculiarities of calculation which should be clarified as well, e.g. did you try to calculate with various initial conditions, etc.

4.

Did authors consider to use NIST randomness tests? Please elaborate the answer.

Author Response

Response to Reviewer 3 Comments

Point 1: Introduction lines 20-24 are obsolete because they state well-known facts, which are not directly relevant to the paper.


Response 1: Thank you for your helpful comments. We are sorry that we did not give a good start to Introduction. In order to make up for this shortcoming, we removed the contents of lines 20-24 of the section and rewrote as follows:

Digtal image is one of the most popular multimedia forms at present, and it is widely used in politics, economy, national defense and education. Since images contain a lot of information, we must worry about data being leaked, falsified and counterfeited.

Point 2: Authors state:

The bifurcation graph can objectively reflect the region and state of the chaotic behavior of a map.

However, in addition, they are calculating Lyapunov's exponents as well. So, it seems that bifurcation graph has some disadvantages as a tool for "objective reflection". Please clarify.

Response 2: Thanks for your constructive comments. We are sorry that “objective reflection” has brought you misunderstanding. The bifurcation graph does reflect the region and state of the chaotic behavior of a map. This is because after the bifurcation, a discontinuous transition occurs between the different states of the map, which is a mutation. After consecutive diversion, the final state reached by the map is chaos. Nevertheless, the bifurcation graph cannot reflect the degree of chaotic characteristics of the map. Therefore, it is necessary to calculate the Lyapunov exponent. It is not only an important indicator to judge whether the map is chaotic, but also can measure the quality of chaos. That is, as described in our manuscript, the larger the Lyapunov exponent, the better the chaotic properties of the map.

Point 3: Please clarify how did you calculated Lyapunov exponents in subsection 2.1.2. Namely, you are using lim(n->infinity), which obviously had to be adapted due to the finite nature of real calculation. There are maybe some more peculiarities of calculation which should be clarified as well, e.g. did you try to calculate with various initial conditions, etc.

Response 3: Thank you for your helpful comments. We are sorry that the description of subsection is relatively simple. In the revised manuscript, we have supplemented the exposition of the calculation for Lyapunov exponents, as is shown below.

Since the Lyapunov exponent  is independent of the initial value , we arbitrarily choose . Moreover, in order to observe the change between the adjacent iteration points as a whole, the number of iterations  should be chosen as large as possible, so we set . Then calculate the derivatives of the LFHCM, Logistic map, and Fraction map according to Equations (1-3), and substitue the results into Equation (4) to get the Lyapunov exponent plots exhibited in Figure 2.

Point 4: Did authors consider to use NIST randomness tests? Please elaborate the answer.

Response 4: Thanks for your constructive comments. We did consider using the NIST randomness test, and the original Test of Mono-bit Frequency in subsection 2.1.4 is the first sub-item of the NIST test. We apologize for the lack of comprehensive testing. To compensate for this shortcoming, in the revised manuscript we replaced the Test of Mono-bit Frequency in section 2.1.4 with the NIST test, which is marked in blue and described as:

2.1.4. NIST Test

The NIST (National Institue of Standards and Technology) SP800-22 Revla test consists of 15 sub-items that are used to detect the randomness of the sequences [28]. The method of detection is by comparing the calculated  of each sub-item to a prescribed significant level 0.01, and the generated values are all expected to greater than 0.01 to pass the test. Iterate Equation (1) with different initial values to generate 100 binary streams with 1000000 bits, then calculate their average values. The results are shown in Table 3. It is obvious that the calculated  of each sub-item falls into , that is, the outputs generated by LFHCM can pass all the tests of NIST.

Table 3. ApEn values of 1D chaotic maps.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

I suggest performing writing review for final version