A new chaotic discrete dynamical system, built on trigonometric functions, is proposed. With intent to use this system within cryptographic applications, we proved with the aid of specific tools from chaos theory (e.g., Lyapunov exponent, attractor’s fractal dimension, and Kolmogorov-Smirnov test) and statistics (e.g., NIST suite of tests) that the newly proposed dynamical system has a chaotic behavior, for a large parameter’s value space, and very good statistical properties, respectively. Further, the proposed chaotic dynamical system is used, in conjunction with a binary operation, in the designing of a new pseudorandom bit generator (PRBG) model. The PRBG is subjected, by turns, to an assessment of statistical properties. Theoretical and practical arguments, rounded by good statistical results, confirm viability of the proposed chaotic dynamical system and newly designed PRBG, recommending them for usage within cryptographic applications.
Nowadays, more and more, it appears that skilful genesis of chaos turns out to be a key issue in many technological application fields such as engineering, medicine, communications, information storage, and, with particular importance, cryptography [
Designing of dynamical systems, intended to be used as base of cryptosystems, must be done so as to ensure the use of a set of associated control parameters’ values that leads to chaos [
Since 1963, when Lorenz found the first chaotic attractor in a three-dimensional autonomous system while studying atmospheric convection [
Motivated by the extent of previous work, the present paper aims to present a new chaotic discrete dynamical system which, furthermore, may be included in the wide family of PRBGs through a simple, interesting, and yet complex new PRBG model, based on binary operation.
The rest of this paper is organized as follows. Section
Newly dynamical system introduced in paper uses (
Therefore, the newly proposed one-dimensional discreet dynamic system, which is defined with respect to form (
In the following, dynamical behavior of newly proposed chaotic system is investigated, by both theoretical analysis and numerical simulation (e.g., by means of Lyapunov exponent, attractor’s geometric shape and fractal structure, and system’s ergodicity, i.e., Kolmogorov-Smirnov tests, etc.).
The behavior of the proposed discrete dynamic system (
From the above equations, it can be noticed that for any
Bifurcation diagram of
Analysis of dynamical system’s attractor’s shape can provide meaningful information about system behavior in time, for certain values of its parameters. The attractor of a dynamical system with a periodic behavior has a regular shape, while the one corresponding to a chaotic dynamical system has a complex structure, of fractal type, called strange attractor [
Figure
Attractor of
Fractal structure of an attractor is indicated by a fractional value of its fractal dimension, which is a ratio that provides a statistical index of complexity comparing how in detail a pattern changes with the scale at which it is measured or, alternatively, by a measure of the space-filling capacity of a pattern, telling how a fractal scale is different than the space in which it is embedded. There are several types of fractal dimensions, which can be theoretically and empirically estimated, such as Hausdorff dimension, Minkowski-Bouligand dimension, box-counting dimension, information dimension, and correlation dimension [
In this subsection, using Birkhoff’s theorem [
The Kolmogorov-Smirnov test is applied on two amounts of independent data
Due to the fact that the random values
Kolmogorov-Smirnov test is applied as follows: the maximum absolute difference between the two distribution functions for a significance level in case of
Kolmogorov-Smirnov test was performed over a sequence of
Test’s overall results are summarized in Table
Results of
Experiment iteration(s) | Parameter value | KS test value | Test result |
---|---|---|---|
1–49 | — |
|
PASSED |
50 | 2.805107703154233 | 0.956 | PASSED |
51–99 | — |
|
PASSED |
100 | 4.342554676193408 | 0.950 | PASSED |
101–149 | — |
|
PASSED |
150 | 5.352537126362448 | 0.950 | PASSED |
151–199 | — |
|
PASSED |
200 | 5.930260965351542 | 0.966 | PASSED |
201–249 | — |
|
PASSED |
250 | 6.547089923300568 | 0.940 | PASSED |
251–299 | — |
|
PASSED |
300 | 7.300182966818144 | 0.966 | PASSED |
301–349 | — |
|
PASSED |
350 | 7.710976298365504 | 0.960 | PASSED |
351–399 | — |
|
PASSED |
400 | 8.299998361920006 | 0.950 | PASSED |
401–449 | — |
|
PASSED |
450 | 9.127429903281719 | 0.964 | PASSED |
451–499 | — |
|
PASSED |
500 | 9.858890151572155 | 0.952 | PASSED |
Based on numerical results previously obtained, using instruments from the chaos theory, we can conclude that
The chaotic behavior is a necessary but not sufficient condition to allow usage of the proposed dynamic system within cryptographic applications. System’s security level, against some statistical cryptanalytic attacks, is assessed after a statistical analysis of the randomness of values generated. There are several options available for analyzing randomness of a newly developed pseudorandom bit generator (PRBG), as it will be revealed in the following section.
In order to assess PRBG’s statistical properties (i.e., its true randomness and implicit suitability within cryptographic applications; see e.g., [
Chaotic cryptography deals with real numbers, so, in order to proceed and apply the battery of the statistical tests aforementioned, we have to apply a computational method to transform a chaotic sequence of real numbers into a bitstream. The discretization method that we used consisted in the extraction of the fractional parts of the generated subunitary real numbers.
CrypTool was used to compute the occurrence frequencies of any binary substring, composed of
PRBG’s
|
Substring | Frequency (%) |
|
---|---|---|---|
Histogram |
0 | 50.0734 | 0.0734 |
1 | 49.9266 | 0.0734 | |
| |||
Digram |
00 | 24.9605 | 0.0395 |
01 | 24.9660 | 0.0340 | |
10 | 24.9660 | 0.0340 | |
11 | 25.1074 | 0.1074 | |
| |||
Trigram |
000 | 12.4899 | 0.0101 |
001 | 12.4706 | 0.0294 | |
010 | 12.4293 | 0.0707 | |
011 | 12.5367 | 0.0367 | |
100 | 12.4705 | 0.0295 | |
101 | 12.4954 | 0.0046 | |
110 | 12.5367 | 0.0367 | |
111 | 12.5707 | 0.0707 | |
| |||
4-gram |
0000 | 6.2240 | 0.0260 |
0001 | 6.2659 | 0.0159 | |
0010 | 6.2015 | 0.0485 | |
0011 | 6.2691 | 0.0191 | |
0100 | 6.2085 | 0.0415 | |
0101 | 6.2208 | 0.0292 | |
0110 | 6.2568 | 0.0068 | |
0111 | 6.2799 | 0.0299 | |
1000 | 6.2658 | 0.0158 | |
1001 | 6.2047 | 0.0453 | |
1010 | 6.2278 | 0.0222 | |
1011 | 6.2676 | 0.0176 | |
1100 | 6.2620 | 0.0120 | |
1101 | 6.2746 | 0.0246 | |
1110 | 6.2799 | 0.0299 | |
1111 | 6.2908 | 0.0408 | |
| |||
12-gram |
000000000000 | 0.0265 | 0.0020 |
— |
|
| |
111111111111 | 0.0284 | 0.0039 |
It can be observed that the deviation from ideal value, of each
A RP (i.e., Recurrence Plot) holds important insights into the time evolution of
PRBG’s visual recurrence analysis: AMI graph (a), FNN graph (b), and recurrence plot (c).
Lack in clear patterns, within the RPI, indicates that consecutive samples in bitstream’s structure are much far apart and uncorrelated. More than that, RPI’s homogeneity along the major diagonal and its irregular distribution emphasizes a stationary, mostly stochastic behavior (i.e., intrinsically nondeterministic, nonintermittent, and sporadic), of the system that has generated the bitstream and, namely, a true random process (i.e., random binary strings).
VRA, through its embedded RQA (i.e., Recurrence Quantification Analysis) tool, also provides other additional measures (e.g., entropy, mean, percentage of recurrence and of determinism, etc.); some of them, the most important ones, are quantified in Table
PRBG’s general statistics.
Mean | Variance | Standard deviation | Skewness | Kurtosis | Entropy |
---|---|---|---|---|---|
0.5007 | 0.2500 | 0.5000 | −0.0029 | −1.9999 | 0.9999 |
Despite the fact that skewness has a negative value (i.e., indicating that the tail on the left side of the probability density function is longer than the right side and the bulk of the values lie to the right of the mean), being close to zero indicates that the values are relatively evenly distributed on both sides of the mean, typically (but not necessarily) implying a symmetric distribution [
Good general statistical properties revealed with the aid of VRA (either visually—evaluation of RPI’s structural properties or through different specific measures evaluation—RQA), highlights randomness of bitstreams generated using the proposed PRBG function, thus allowing advancement to other statistical test suites.
For the numerical experimentations of the proposed pseudorandom bit generator, we have generated
For
NIST tests’ results.
Test name | Passing ratio of the test | Uniformity |
Test result |
---|---|---|---|
Frequency | 0.992 | 0.602803 | PASSED |
Block frequency | 0.990 | 0.748891 | PASSED |
Cumulative sums | 0.991 | 0.090388 | PASSED |
Runs | 0.990 | 0.939005 | PASSED |
Longest run | 0.989 | 0.592443 | PASSED |
Rank | 0.991 | 0.840367 | PASSED |
FFT | 0.989 | 0.242363 | PASSED |
Nonoverlapping template | 0.983 | 0.761719 | PASSED |
Overlapping template | 0.983 | 0.230755 | PASSED |
Universal | 0.987 | 0.050629 | PASSED |
Approximate entropy | 0.988 | 0.959347 | PASSED |
Random excursions | 0.987 | 0.614382 | PASSED |
Random excursions variant | 0.984 | 0.830939 | PASSED |
Serial | 0.986 | 0.209392 | PASSED |
Linear complexity | 0.989 | 0.764655 | PASSED |
The method to calculate the passing ratio of total test and the uniformity
Most chaos-based PRNGs (and, implicitly, their subsequent PRBGs) are based on a single chaotic system (e.g., [
We consider two one-dimensional chaotic maps (e.g., as the previously designed model, i.e., (
Also, we consider the binary operation given by the formula
The output
The real numbers obtained using the proposed PRNG were discretized extracting their fractional parts in order to apply the NIST statistical tests. For the numerical experimentations on the proposed pseudorandom numbers generator, we have generated 2.000 different binary sequences (sample size
NIST tests’ results.
Test name | Passing ratio of the test | Uniformity |
Test result |
---|---|---|---|
Frequency | 0.991 | 0.045971 | PASSED |
Block frequency | 0.991 | 0.653773 | PASSED |
Cumulative sums | 0.990 | 0.035876 | PASSED |
Runs | 0.991 | 0.235589 | PASSED |
Longest run | 0.987 | 0.937919 | PASSED |
Rank | 0.994 | 0.221898 | PASSED |
FFT | 0.988 | 0.160357 | PASSED |
Nonoverlapping template | 0.986 | 0.036592 | PASSED |
Overlapping template | 0.985 | 0.316808 | PASSED |
Universal | 0.989 | 0.703417 | PASSED |
Approximate entropy | 0.991 | 0.129620 | PASSED |
Random excursions | 0.992 | 0.806491 | PASSED |
Random excursions variant | 0.985 | 0.885727 | PASSED |
Serial | 0.987 | 0.737475 | PASSED |
Linear complexity | 0.990 | 0.325206 | PASSED |
It can be seen that the computed proportion for each test lies inside the confidence interval; hence, the tested binary sequences generated by the proposed PRBG are random with respect to all the 16 tests of NIST suite.
Development of new chaotic dynamic systems, which meet the current demands of security, is a present research direction in the field of cryptography. The main objective is to obtain a large key space, induced by the control parameter and (or) initial conditions, for which the dynamic system is in chaotic regime, is ergodic, and has a uniform distribution of the values
With respect to the aforementioned ideas, in this paper we have designed a new one-dimensional chaotic dynamic system that meets these requirements.
Moreover, a larger key space than the one of the known chaotic maps (e.g., logistic, tent, Hénon, etc.) was achieved, and, despite the fact that the implementation of trigonometric maps is little slower than the ones of other kinds of maps (e.g., polynomial, exponential, etc.), we consider that the advantage of a larger key space induced by their usage is a good compromise (i.e., a win-win situation).
Using specific mathematical and numerical tools from chaos theory and statistics, we proved that the proposed chaotic dynamic system has very good cryptographic properties. The proposed map was used in a new innovative way to design a new PRNG/PRBG model, based on a well-known binary operation.
We have performed an exhaustive testing process of the randomness of the generated binary sequences using the NIST suite to prove the viability of the proposed PRNG/PRBG.
The authors declare that there is no conflict of interests regarding the publication of this paper.