An Encryption Application and FPGA Realization of a Fractional Memristive Chaotic System

Round 1

Reviewer 1 Report

The article under review is devoted to the investigation of fractional-order memristive chaotic system with transcendental nonlinearity. In this manuscript, the authors propose the FPGA implementation of this system subject to image incription application. The authors perform bifurcational analysis and spectral entropy estimation for determination of chaotic modes, and encription efficiency check through NIST tests, key sensitivity analysis and resistance to differential attacks analysis. There are some comments below:

1. The authors added the term "memristor system" straight to the title, however, they mention it only referring to the base articles. Sections of the introduction and the description of the investigated dynamical system need to be expanded by consideration of the electrical engineering background of the circuit with memristor, for which the equations of the system (3) were derived. What is the initial physical meaning of the state variables, what nonlinearities characterize the memristivity of the system?

2. What is the point of displaying information about local minima and maxima at the same bifurcation diagrams (figure 2)? Both of them characterize the amplitude characteristics of the time series. It would be much more useful to demonstrate peak-to-peak intervals in order to track bifurcations in phase, or to calculate the number of periods for the exhibited dynamical modes.

3. Please provide the formula for calculating the spectral entropy.

4. To improve the readability of the article, I would recommend the authors to rearrange figures and tables in the course of the text, otherwise it is quite difficult to track the presented content jumping between multiple pages.

5. Obviously, there is a problem in the formatting of table 1.

Author Response

Comments and Suggestions for Authors

The article under review is devoted to the investigation of fractional-order memristive chaotic system with transcendental nonlinearity. In this manuscript, the authors propose the FPGA implementation of this system subject to image incription application. The authors perform bifurcational analysis and spectral entropy estimation for determination of chaotic modes, and encription efficiency check through NIST tests, key sensitivity analysis and resistance to differential attacks analysis. There are some comments below:

 

1. The authors added the term "memristor system" straight to the title, however, they mention it only referring to the base articles. Sections of the introduction and the description of the investigated dynamical system need to be expanded by consideration of the electrical engineering background of the circuit with memristor, for which the equations of the system (3) were derived. What is the initial physical meaning of the state variables, what nonlinearities characterize the memristivity of the system?

 

Reply: According to the fact that many works extended integer-order chaotic systems into fractional- order domain [5,6], this work extends the memristive chaotic system in [23] into fractional-order domain to enhance its properties in encryption applications.

 

Please check [23] for more details, where (according to the equation numbering in [23])

 

• the authors added the trigonometric function (cos x) in z equation in system (2) to enhance theLyapunov exponents of the original system in (1).
• Additionally, the authors added the memristor term in a form of hyperbolic function in (3) to the system in (2) to maintain the structure of conditional symmetry and enhance the Lyapunov exponents.
• The memristive chaotic system is presented in (4).

 

In this work, system (4) in [23], is extended to fractional-order domain studying its chaotic characteristics through bifurcation and SE diagrams.

 

The x, y, z, w were mentioned as the voltage states of system (4) in [23]. Besides, the (tanh w) term in y equation represents the voltage and current constraints in the memristor, which is a typical 8-like hysteresis loop. Please refer to the circuit schematic in Fig.13 in [23].

The memristivity of the system is characterized by the hyperbolic function term (tanh w) in y equation.

 

2. What is the point of displaying information about local minima and maxima at the same bifurcation diagrams (figure 2)? Both of them characterize the amplitude characteristics of the time series. It would be much more useful to demonstrate peak-to-peak intervals in order to track bifurcations in phase, or to calculate the number of periods for the exhibited dynamical modes.

Reply: To the best of our knowledge, most bifurcation diagrams for continuous-time dynamical systems are based on the analysis of local maxima as presented in [14,16,17,26]; yet, the local minima also need to be considered, which is explained in [24]. This work considers bifurcation with both local minima and maxima.

For bifurcation tracking in phase and calculating the number of periods for a dynamical system, they can be investigated and considered in future work.

3. Please provide the formula for calculating the spectral entropy.

Reply: The formula of calculating the spectral entropy is added in Section 2 in blue color.

4. To improve the readability of the article, I would recommend the authors to rearrange figures and tables in the course of the text, otherwise it is quite difficult to track the presented content jumping between multiple pages.

Reply: All figures and tables are re-arranged in the paper.

5. Obviously, there is a problem in the formatting of table 1.

Reply: Table 1 is better organized and formatted than it was in the previous version.

Author Response File: Author Response.pdf

Reviewer 2 Report

Regardless of the general interest evoked by the article, the work dramatically lacks details. It is especially applicable to the introduction, experiment description, and analysis of the results presented. Moreover, the number of pages is lower than in a usually submitted paper, so authors could use these volumes to append previously omitted aspects. In the present form, from the reviewer's point of view, it is too difficult to assess the quality of the research.

 

Some comments:

1. The maximum frequency of the design is surprisingly low for the Artix 7 chips, which indicates suboptimal HDL implementation of the algorithm.
2. Arnold's map can not be considered a tool for cryptographic strength improvement since its operation does not involve a key and is known to an adversary.
3. An approach to how the authors lower differential attack vulnerability is not apparent.
4. The paper requires an analysis of the vulnerabilities of the proposed algorithms given by the authors.
5. The results of NIST tests are suspiciously good. Much wider attention should be paid to their description. 
6. The paper requires an analysis of the reduced domain of keys and the divergence of the results for similar keys.

 

Author Response

Regardless of the general interest evoked by the article, the work dramatically lacks details. It is especially applicable to the introduction, experiment description, and analysis of the results presented. Moreover, the number of pages is lower than in a usually submitted paper, so authors could use these volumes to append previously omitted aspects. In the present form, from the reviewer's point of view, it is too difficult to assess the quality of the research.

 

1. The maximum frequency of the design is surprisingly low for the Artix 7 chips, which indicates suboptimal HDL implementation of the algorithm.

Reply: To the best of our knowledge, the maximum frequency of the Artix 7-implementations for the chaotic systems, which don’t include transcendental nonlinearities, in the previous works ranges between 38 to 52 MHz such as in [7], the chaotic systems achieve max frequencies of 38 and 48.387 MHz and in [22], the chaotic systems achieve max frequencies of 40.237 and 51.214 MHz.

In addition, the maximum frequency of the Artix 7-implementations for the chaotic systems, which include only trigonometric functions, in the previous works ranges between 13 to 23 MHz as in [20], the chaotic systems achieve max frequencies of 13.85, 16.349, 16.348 and 22.907 MHz.

In this work and [21], the chaotic system include transcendental nonlinearities in the three modes of CORDIC, which increase the complexity of Artix-7 implementation, increase the consumed resourses and decrease the achieved frequencies to be in the ranges of 12.368 and 14 MHz

 

2. Arnold's map can not be considered a tool for cryptographic strength improvement since its operation does not involve a key and is known to an adversary.

Reply: The role of Arnold’s map appears in the permutation of the output image after the substitution phase not in the key, while the cryptographic strength improvement in this scheme appears in the fractional memristive system, which include four fractional orders, four initial conditions and four parameters acting as twelve sub-keys. Each computed value is the addition of fixed and key parts, for example, , where  is the same  value given near (4) and  the corresponding subkey scaled by .

3. The paper requires an analysis of the vulnerabilities of the proposed algorithms given by the authors

Reply: The differential attack is enhanced through plain image dependence (sum of channels), where  and the multiplexer, where the channel order is selected by their own LSBs based on the table in Fig.4 [26,27].

 

4. The results of NIST tests are suspiciously good. Much wider attention should be paid to their description.

Reply: Similar chaos based schemes succeeded in NIST test as in [26,27]. Additionally, the description of NIST test is added in Section 3 in blue color.

5. The paper requires an analysis of the reduced domain of keys and the divergence of the results for similar keys.

Reply:  Eight of the sub-keys are limited to 21-bit, while the other four are limited to 22-bit resulting in a key space of . Computing the chaotic system parameters as the sum of fixed and key parts, for example, , (as explained in the reply to comment 2) guarantees that all 256 bits and all key values can be used because the scaling in  prevents drifting from chaotic behavior. Table 3 demonstrates key sensitivity through MSE and entropy values, which indicate a wrong decrypted image when the LSB in the first sub-key is changed. Similar results are reported for the rest of the subkeys.

We hope the revised version meets your expectations, and thanks so much for your time and valuable comments.

Author Response File: Author Response.pdf

Reviewer 3 Report

1.   The authors extend a memristive chaotic system with transcendental nonlinearities to the fractional-order domain. The proposed system is implemented on the Artix-7 FPGA board, achieving a throughput of 0.396 Gbit/s.

2.      The manuscript can be accepted as short paper.

3.    In the figure 4, encryption scheme and multiplexing table should be demonstrated in detail.

4.    In the figure 6, histograms of R,G,B of encrypted Baboon image should be demonstrated in detail.

5.      Revise the English thoroughly before submission.

Author Response

1. The authors extend a memristive chaotic system with transcendental nonlinearities to the fractional-order domain. The proposed system is implemented on the Artix-7 FPGA board, achieving a throughput of 0.396 Gbit/s.

Reply: Yes.

2. The manuscript can be accepted as short paper.

Reply: Yes.

3. In the figure 4, encryption scheme and multiplexing table should be demonstrated in detail.

Reply: It is addressed in the reviewed manuscript version. The multiplexing table presented in Fig 4 is used to select the channel order of each pixel by their own LSBs as explained in [26,27].

 

4. In the figure 6, histograms of R,G,B of encrypted Baboon image should be demonstrated in detail.

Reply: It is addressed in the reviewed manuscript version. Figure 6 indicates a uniform histogram of R,G,B of encrypted Baboon image and so do the R,G,B of encrypted Peppers, Barb and Lak images.

 

5. Revise the English thoroughly before submission.

Reply: The English is revised in the manuscript version.

We hope the revised version meets your expectations, and thanks so much for your time and valuable comments.

Author Response File: Author Response.pdf

Reviewer 4 Report

In this paper, an encryption application and FPGA realization of a fractional memristive chaotic system is proposed.

The paper is interesting and contains useful scientific insights for researchers. Only a few suggestions for the authors:

1. Please explain all the parameters shown in (1).

2. Figure 1 is too far from the text part where it is referenced.

3. Table 1 is missing, or it has to be better organized.

Author Response

The paper is interesting and contains useful scientific insights for researchers. Only a few suggestions for the authors:

1. Please explain all the parameters shown in (1).

            Reply: The parameters are explained in blue color.

2. Figure 1 is too far from the text part where it is referenced.

Reply: Figure 1 is moved to be near the text where it is referenced.

3. Table 1 is missing, or it has to be better organized.

Reply: Table 1 is better organized than it was in the previous version.

We hope the revised version meets your expectations, and thanks so much for your time and valuable comments.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

1. Readers of Electronics may be interested in the details regarding the equivalent electrical circuit for system (3). Please add it and your comments about the physical sense of the memristor directly to the manuscript.

2. Unfortunately, I cannot agree with the authors' arguments. Reference [24] is not a peer-reviewed scientific publication. The addition of minima points in Figure 2 does not carry any information, only confuses the potential reader who may want to count the number of periods, but will not be able to because of overlapping diagrams of different colors.

3. The format of Table 1 is still not reader friendly.

The remaining comments were sufficiently addressed by the authors of the article.

Author Response

 

1. Readers of Electronics may be interested in the details regarding the equivalent electrical circuit for system (3). Please add it and your comments about the physical sense of the memristor directly to the manuscript.

 

Reply: The details regarding system (3) and the physical sense of the memristor are added in Section 2 in blue color.

Please note that the circuit schematic presented in the original paper [30] is the analog implementation of the integer order system from the mathematical representation not the electrical circuit model. In this work, the digital implementation of the fractional order extended system is obtained from the mathematical representation.

 

2. Unfortunately, I cannot agree with the authors' arguments. Reference [24] is not a peer-reviewed scientific publication. The addition of minima points in Figure 2 does not carry any information, only confuses the potential reader who may want to count the number of periods, but will not be able to because of overlapping diagrams of different colors.

Reply: According to the fact that Ref [24], in the previous version,  is not a scientific publication and both bifurcations with local maxima and minima characterize the amplitude characteristics of the time series, which means the addition of minima points in Figure 2 hasn’t powerful impact, and that most bifurcation diagrams for continuous-time dynamical systems are based on the analysis of local maxima as presented in [21,23,24,32], the bifurcation figures with local minima and Ref [24], in the previous version,  are removed in the reviewed manuscript vesion.

3. The format of Table 1 is still not reader friendly.

Reply: Table 1 is better organized and formatted than it was in the previous version.

 

The remaining comments were sufficiently addressed by the authors of the article.

We hope the revised version meets your expectations, and thanks so much for your time and valuable comments.

 

Author Response File: Author Response.docx

Round 3

Reviewer 1 Report

All the previous comments have been taken into account. The only suggestion I can make is to increase the ranges on the state variable axes in Figure 2. In its current form, the bifurcation diagrams appear to be cut.

Author Response

Reply: The figures are modified according to your valuable comments

Reply: The ranges of the state variable axes in Fig.2 are increased in the reviewed manuscript.