Generating Even More Chaotic Instances in Hardware

Round 1

Reviewer 1 Report

Based on the fact that multiplication inside a computer does not follow the associative property because of roundoff effects, this work use this condition to build different chaotic instances.  Also experiments are carried out to check the performance of the work. The idea and methods are of reference value to researchers in related fields.  I have some comments as follows:

1. Please improve abstract and introduction to point better the scientific contribution and novelty of the proposal. Motivation should be added in the introduction. 

2. The related works introduction to be enhanced. Not only necessary mentioning similar research. It is necessary to mention why your approach is different, and explain from other research what is your scientific contribution and how you are going beyond the state of the art.

3. About the pseudo random number generator in Section 4, NIST test standards should be introduced to conduct random test and comparison of pseudo-random sequences generated by the system, and observe whether there are performance differences among different chaotic system instances. This is an important point. 

4. There are some inappropriate sentences in the text, which need to be carefully checked and improved, such as: "In Eq. (2) is possible to ...", "In Fig. 9 is shown ....", "Observe in (4) that ....", "For example, for the Lü oscillator in (4)". And "Eq." should be added before the formula number.

5. There are some misspellings of symbols and formulas, such as: In Line 33, "(ac)c" should be "(ac)b", In Line 59, a closing parenthesis ")" is missing. In Line 92, "a_4(x_ny_n)" should be "(a_4y_n)x_n", et al.

Author Response

Reviewer # 1

Question 1. Please improve abstract and introduction to point better the scientific contribution and novelty of the proposal. Motivation should be added in the introduction.

Response 1: The following phrase has been added to the end of the
abstract:

"To best of our knowledge, it is the first paper that
 analyze the construction of new chaotic entities by
 using the roundoff effects."

Q2. The related works introduction to be enhanced. Not only necessary mentioning similar research. It is necessary to mention why your approach is different, and explain from other research what is your scientific contribution and how you are going beyond the state of the art.

Response 2: The following paragraph has been added to the end of Section 2:

"The motivation of this work was to observe that the implementation
in hardware of a chaotic entity must follow carefully the simulation in
software, or vice versa, if one requires to obtain the same results.
But also we observe that a modified implementation follows the same
dynamical behavior. Then, we try in this work to analyze how new entities,
of chaotic oscillators or chaotic maps, new in the way that produce different
results, can be constructed by modifying the order of the multiplications
in their fixed point implementations."

 

Q3. About the pseudo random number generator in Section 4, NIST test standards should be introduced to conduct random test and comparison of pseudo-random sequences generated by the system, and observe whether there are performance differences among different chaotic system instances. This is an important point.

Response 3. The following paragraph has been adeed to Section 4:

The National Institute of Standards and Technology (NIST) of the United States
of America created a free software for testing pseudo and random number generators
[(P)RNGs].  Specifically, NIST said that the test is for (P)RNGs for cryptographic
applications, where randomness is a crucial characteristic \cite{nist}.
Thus, we test if four generated sequences with the conditions
in Tab. 1 are random by applying the
NIST suite of tests \cite{nist} to 100 sequences of $10^6$ bits.
Note that these sequences only vary in the order of the multiplications
because the four sequences are generated with the same
initial conditions. The four sequences pass all the NIST tests as
it it shown in Tab. 2 (the $p$-value must be greater than 0.01,
and the proportion value must be greater than 0.96).

This paragraph is highlighted in the document.
A new row in Table 1 with code 020000 has been added to.
Also a new Table 2 with the NIST results has been inserted
in the document.

 

4. There are some inappropriate sentences in the text, which need to be carefully checked and improved, such as: "In Eq. (2) is possible to ...", "In Fig. 9 is shown ....", "Observe in (4) that ....", "For example, for the Lü oscillator in (4)". And "Eq." should be added before the formula number.

Response 4: "Eq. has been added before each formula number.
The other changes made are:
"In Eq. (2) is possible to ..." ->  "It is possible to observe in Eq. (2)"
"In Fig. 9 is shown ...."       ->  "The development board and the
              oscillator signal on the oscilloscope are shown in Fig. 9.
"Observe in (4) that ...."      ->  "Observe in Eq. (4) that ....",
"For example, for the Lü oscillator in (4)"  ->
"For example, for the Lü oscillator in Eq. (4)", etc.

 

Q5. There are some misspellings of symbols and formulas, such as: In Line 33, "(ac)c" should be "(ac)b", In Line 59, a closing parenthesis ")" is missing. In Line 92, "a_4(x_ny_n)" should be "(a_4y_n)x_n", et al.

Response 5: Thank you for your observations. These changes have been made in the document and they
are highlighted.

 

Reviewer 2 Report

This manuscript presents an approach to generating chaotic instances in hardware. The authors assumed that it is well known that multiplication inside a computer does not follow the associative property because of roundoff effects. They propose to use this fact to generate other different chaotic instances of chaotic maps or oscillators when the multiplication of three terms appears. They discuss this topic and offer a chaotic system in software and hardware within an FPGA where another 144 different 2D map instances and 81 different Lü oscillators were created.

I believe the authors did a good job, but I have some concerns regarding the presentation of their results. Their experiment is not well described, and I feel vital information is missing to reproduce it. Moreover, It would be nice to compare their work with similar ones. I have some suggestions that may improve the final version of this manuscript.

·         The introduction should contain more details on the motivation of the described research. When citing other papers, it would be nice to have some comments pointing out the good results and missing approaches of each article.

·         An in-depth evaluation of similar proposals would enrich the present manuscript. I missed a solid state-of-the-art section.

·         Sections 2, 3, and 4 are well-written and add valuable information to understand the topic. It is not clear what are the authors' contributions. Please add more details to make it even more evident.

·         Adding some code in section 5 would make it easier for the reader to reproduce the experiment.

·         It would be nice to have insights about the development in the discussion section.

·         I also missed a comparison with other similar work.

·         The authors need to make it clear what their contributions are.

 

I believe the manuscript needs to be enhanced, but I am sure it is exciting and will contribute to many other groups researching the same topic. 

Author Response

Reviewer 2.
  
Observation 1:
·         The introduction should contain more details on the motivation of the described research. When citing other papers, it would be nice to have some comments pointing out the good results and missing approaches of each article.

Answer 1:
We describe the motivation in one paragraph at the end of Sec. 1

 "The motivation of this work was to observe that the implementation in hardware of a chaotic entity must follow carefully the simulation in software, or vice versa, if one requires to obtain the same results. But also we observe that a modified implementation follows the same dynamical behavior. Then, we try in this work to analyze how new entities, of chaotic oscillators or chaotic maps, new in the way that produce different results, can be constructed by modifying the order of the multiplications in their fixed point implementations."

 

Observation 2:
·         An in-depth evaluation of similar proposals would enrich the present manuscript. I missed a solid state-of-the-art section.

Answer 2:
The phrase:

"To best of our knowledge, it is the first paper that analyze the construction of new chaotic entities by
 using the roundoff effects."

is added at the end of the abstract. We believe that this is the first paper
about building another instances of chaotic entities by changing the
association in the multiplications.

 

Observation 3:
·         Sections 2, 3, and 4 are well-written and add valuable information to understand the topic. It is not clear what are the authors' contributions. Please add more details to make it even more evident.

Answer 3:

We believe that with the new text added to the document this observation is fulfilled.

Observation 4:
·         Adding some code in section 5 would make it easier for the reader to reproduce the experiment.

The VHDL code will be publicly available in the same URL as the C code.
This lines have been added at the end of Section 5:

"The VHDL code of the hardware designs is publicly available in
https://cs.cinvestav.mx/~fraga/Instances.tar.gz."

Observation 4:
·         It would be nice to have insights about the development in the discussion section.

The following paragraph has been added to end of the disussion section:

"We believe that in applications where several PRNG are needed, our design could
be useful. At least it is not necessary to built several PRNGs based in different
equations; if the chaotic entity has triplets terms, then several instances of
it can be built. Still we are searching for specific applications of the proposed
designs."

Observations 5 and 6:
·         I also missed a comparison with other similar work.
·         The authors need to make it clear what their contributions are.

We believe that with the new texts added to the document these two observations are fulfilled.

 

Observation 7:
I believe the manuscript needs to be enhanced, but I am sure it is exciting and will contribute to many other groups researching the same topic.

Thank you for your observations that help to improve the quality of the document.

Reviewer 3 Report

Please see the attached PDF file.

Comments for author File: Comments.pdf

Author Response

Reviewer 3
  
Question 1) In line 64, the statement “integer number that use a bits in the integer part, and b in the
fractional part,” may result confused due to integer numbers do not have fractional part. Hence, further
comments are required on it.

Answer 1: Thank you for your observation. Paragraph has been changed as:

   "Now we are going to analyze the fixed point arithmetic. The notation A( a.b) will be
used here to represent a set of numbers that uses a bits in the integer part, and b bits in the
fractional part. Each number is of size a + b + 1 bits (plus the sign bit) and indeed this
number is stored as a signed integer of size a + b + 1 bits."

Q2) Line after line 65, the given notation:
                                                x ∈ A (a . b) .
It means that there exist a set not well defined in the paper. Hence, A (a . b) should be a set well
defined correctly.

Answer 2: We believe that with our answer to your question 1, this question is also answered.

 

Q3) Please verify equation (1), or give a reference for it. This, because it can be stated as follows:
                 a         a    −b
             −2 ≤ x ≤ 2 + 2      ,
or some comments on it.

Answer 3: Thank you for your observation. The equation (1) in the paper is correct.
But your right, it is necessary to specify the representation for the negative numbers
(complement 2). The paragraph has been changed to:

"For a number x ∈ A( a.b), if the negative numbers of size a + b + 1 are represented in
complement 2 notation [9], the range of numbers that can be represented is
               a       a   −b
             −2 ≤ x ≤ 2 - 2  ."

Q4) Line 69. The statement “A microprocessor offers the sum and multiplication of two integer numbers,” Which microprocessor? The Z-80 one?

Yes. All the microprocessors, and even microcontrollers, offer the operations of sum and multiplication of integer numbers. With these simple operations the fixed point arithmetic can be used.

 

Q5) Equation (4). The Euler parameter h is a common factor on several terms of the given equation.

Answer 5: Yes. Expanding the parameter h in equation (4), five terms with three multiplication can be obtained. This observation has been added to the document.

"where h is the integration step value. Expanding the parameter h in equation (4), four terms
with two multiplication each one (and different associations on them) can be obtained. An
initial state ( x0 , y0 , z0 ) must be given to apply Eq. (4)."

Q6) Figures 4 and 5 have a notation given as sel [1,0] on the outside of the main block. Gives some words on it from the programming point of view.

Answer 6: The figures 4 and 5 and been changed a the text
"Signal Sel selects the multiplication association" has been added to the figures 4 and 5 captions.

Q7) Figure citations should be given in consecutive number. See line 183.

Thank you for your observation. Text in the document han been changed correctly.

 

8) Further discussions on initial conditions sensibility of Chaos for the real experimentation is needed.

This pharase has been added to the end of Sec. 5.

"During the experiment, the board switches changed the association of the multiplications and the change of the phase diagram on the oscilloscope can be visualized."

Q9) Is it possible to realize your experimentation in other technologies than the FPGA one?

Yes. FPGA has been seen as a intermediate design between the software and a dedicate design on an specific ASIC chip. In a microcontroler or in a Raspberry PI the experiment with the 2D map can be realized using the version in software which is publicly provided. In an architecture of 32 bits, the multiplication of two numbers of 64 bits must be programmed [11].

This observation is added at the end of the discussion section.

Round 2

Reviewer 1 Report

The authors have made a great work addressing all my concerns. I recommend the paper to be accepted.

Reviewer 2 Report

The authors succeed in solving all issues pointed out. I believe the manuscript may be published in the current state. 

Reviewer 3 Report

Thank you very much for your response to the review comments. In my opinion, the paper has been well answered.