A two-directional grid multiscroll hidden attractor based on piecewise linear system and its application in pseudo-random bit generator

doi:10.1016/j.vlsi.2021.04.011

Integration

Volume 81, November 2021, Pages 34-42

https://doi.org/10.1016/j.vlsi.2021.04.011 Get rights and content

Highlights

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A methodology to design dynamical systems that display two-directional grid multiscroll hidden attractors.
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A self-excited multiscroll attractors allow the appearance of a hidden attractor.
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The methodology allows us to determine the number of scrolls in the hidden attractor.
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The hidden attractor is used as the core of a cryptographic safe pseudo-random bit generator.
•
A pseudo-random bit generator is introduced based on continuous dynamical system.

Abstract

In this work, we present a methodology to design dynamical systems that display two-directional grid multiscroll hidden attractors. Self-excited attractors based on piecewise linear systems emerging from its equilibria in order to exhibit in one-directional (1D) and two-directional (2D) grid multiscroll hidden attractors. The particularity of this methodology is that the self-excited multiscroll attractor is separated into several double scroll self-excited attractors to allow the appearance of a hidden attractor. Next, the hidden attractor is used as the core of a cryptographic safe pseudo-random bit generator and statistical evidence is provided to ensure its convenience in cryptographic applications.

Introduction

The generation of pseudo-random sequences to be used in cryptographic applications has been a subject that has been widely studied in the last decades [[1], [2], [3], [4], [5]]. The use of pseudo-random bit generators based on chaotic systems is a very popular design due to the intrinsic characteristics of them, like the high sensibility to initial conditions, dense periodic orbits, irregular motion in the phase space, among others [[6], [7], [8], [9]]. In Ref. [10] the relationships between these properties are given; for instance, confusion is related to ergodicity, the diffusion property with sensitivity to initial conditions, and the deterministic dynamic with the deterministic pseudo-randomness. Because of a large number of attacks on the information to be transmitted, it is important to adequately protect this information and avoid possible misuse.

There is a necessity for different approaches to have secure cryptographic systems. In general, cryptosystems can be divided into two classes: stream cipher and block cipher. A stream cipher takes a bit and transforms it into an output bit using a pseudo-random bit generator. Both discrete chaotic maps and continuous chaotic dynamical systems have been used for cryptography purposes. In this work, we introduce a pseudo-random bit generator based on a continuous dynamical system that can be used as the core of a cryptographic system.

Continuous chaotic systems can be classified into two categories: (1) self-excited attractors (attractors whose basin of attraction intersects with any arbitrary neighborhood of an unstable equilibrium point) and (2) hidden attractors. The Lorenz [11], Chen [12], Lü [13] and Rössler [14] systems are well-known self-excited attractors, while the second group, the hidden attractors, have been developing theoretically and practically in the last years [[15], [16], [17]]. The coexistence of hidden attractors has been studied, for example, Sharma et al. discuss the control of multistability in the hidden attractor through the scheme of linear augmentation in Ref. [18], Dudkowsky et al. reported in Ref. [19] a review of hidden attractors, discuss their theoretical properties and experimental observations. In Ref. [20] Cang et al. reported the finding of unusual hidden and self-excited coexisting dynamical behaviors in an existing Lorenz-like system. Pham et al. [21] reported a hidden hyperchaotic attractor in a memristive neural network. Hidden attractors also have been studied in fractional order systems [[22], [23], [24], [25], [26]], showing that the same classification is applicable in that type of systems. A simpler way to generate multiscroll attractors is to use piecewise linear systems instead of one-piece non-linear systems, piecewise linear systems allow to define beforehand the number and scale of the attractors in multiple directions [[27], [28], [29], [30]].

At the beginning of the study of systems with hidden attractors, specific computational procedures were developed to identify the hidden attractors coexisting with self-excited attractors. The equilibrium points do not help in the localization of hidden attractors due to an initial condition near to an equilibrium point lead to the trajectory to a self-excited attractor. On the other hand, at the same time, the generation of chaotic attractors in vectors fields without equilibrium points was developed [31]. Naturally this kind of systems satisfies the hidden attractor definition. Now, there are several classes of systems that fulfills the hidden attractor definition. Some examples of this kind of system are chaotic dynamical systems with only stable equilibria, with curves of equilibria, with surfaces of equilibria, and with non-hyperbolic equilibria. In this work, we propose an approach to generate multiscroll hidden attractor in vector fields with equilibrium points.

Since the work reported by Suykens & Vandewalle [32] about n-Double scroll self-excited attractor from the Chua's system, there is some interest in generating chaotic or hyperchaotic self-excited attractors with multiple scrolls. There are different approaches to yield multi-scroll self-excited attractors which ranged from modifying the Chua's system by replacing the nonlinear part with different nonlinear functions to the use of nonsmooth nonlinear functions such as, hysteresis, saturation, threshold and step functions. These approaches allow the visualization of 2D, and 3D grid scroll self-excited attractors. For example, Yalcin et al. [33], reported that a 1D, 2D and 3D-grid of scrolls may be introduced locating them around the equilibrium points in space using a step function. Lü et al. [34] presented an approach using hysteresis that enables the creation of 1D, 2D, and 3D grid scrolls self-excited chaotic attractors. In this work, we present a generalized theory that is capable of generating 1D and 2D grid multiscroll hidden attractor.

In [35] introduced a dynamical system by means of piecewise linear systems. The system can display four-scroll self-excited attractor due to four unstable hyperbolic focus-saddle equilibria with stability index of type I, i.e., a negative real eigenvalue and a pair of complex conjugated eigenvalues with positive real part. The four-scroll attractor is located around the four focus saddle equilibria ${x_{1}^{*}, x_{2}^{*}, x_{3}^{*}, x_{4}^{*}}$ which are located at the same distance $d (x_{1}^{*}, x_{2}^{*}) = d (x_{2}^{*}, x_{3}^{*}) = d (x_{3}^{*}, x_{4}^{*})$ and collinear. It is possible to split the four-scroll attractor in two double-scroll attractors if the distance $d (x_{2}^{*}, x_{3}^{*})$ is properly increased and the distances $d (x_{1}^{*}, x_{2}^{*}) = d (x_{3}^{*}, x_{4}^{*})$ are kept. Thus, the system can present monostability and bistability and the resulting basin of attraction presents a significantly widening during this process. Then, the equilibria are surrounded by the basin of attraction of the double-scroll self-excited attractors. Using the aforementioned methodology to generate systems with multiscroll attractors based on the separation of double-scroll self-excited attractors, we propose a generalization to generate 1D and 2D grid multiscroll hidden attractor. Furthermore, we build a secure cryptographic pseudo-random bit generator with statistical evidence of its suitability in cryptographic applications. The rest of this work is organized as follows: In section 2 the basic definitions are described. Section 3 defines the problem. Section 4 presents the description of the methodology to obtain systems with multiscroll hidden attractors. In Section 5 the pseudo-random bit generator is described and statistical evidence is provided to show the convenience in cryptographic applications. Finally, the conclusions are presented in section 6.

Section snippets

Preliminaries

Consider a class of affine linear system given by $\dot{x} = A x + B,$ where $x = {(x_{1}, x_{2}, x_{3})}^{T} \in R^{3}$ is the state vector, $B = {(b_{1}, b_{2}, b_{3})}^{T} \in R^{3}$ stands for a real vector, $A = (α_{i j}) \in R^{3 \times 3}$ denotes a linear operator, and the equilibrium point is located at $x^{*} = - A^{- 1} B$ . The dynamic of the system is given by matrix $A$ which has a stable manifold $E^{s}$ and another unstable $E^{u}$ . According to the above it is possible to define an unstable dissipative system (UDS) as follows:

Definition 1

A system given by (1) with eigenvalues $λ_{i}$ , $i = 1,2,3,$ satisfying $\sum i = 1$

System description and problem statement

Consider the affine linear dissipative system (2) with a switching law $B$ as in (3), where its companion matrix $A$ is given in the Jerk form $A = (\begin{array}{l} 0 & 1 & 0 \\ 0 & 0 & 1 \\ - α_{1} & - α_{2} & - α_{3} \end{array}) .$

Next, we select the equilibria $x^{*} = {(β_{1}, β_{2}, β_{3})}^{T}$ for system (2), where $β_{j}$ are piecewise functions, $j = 1,2,3$ . Thence, we determine the vector $B$ , as follows $B = - A x^{*} = (\begin{array}{l} - β_{2} \\ - β_{3} \\ α_{1} β_{1} + α_{2} β_{2} + α_{3} β_{3} \end{array}),$ where the switching law is in charge of $β_{j}^{'} s$ , in terms of the beforehand located equilibria. Thus, if $β_{j}$ is a $k_{j}$ -pieces function, then the system dynamics will

One directional grid multiscroll hidden attractors

To generate a system with a multiscroll hidden attractor in one direction is necessary to consider the equilibrium points of the form $x^{*} = {(β_{1}, 0,0)}^{T}$ . The vector $B$ is defined as follows $B = (\begin{array}{l} 0 \\ 0 \\ α_{1} β_{1} \end{array}) .$

The objective is to generate an $p$ -scroll hidden attractor based on a $k$ -scroll self-excited attractor, with $k = 2 \cdot p$ , where each scroll of the hidden attractor encloses a pair of scrolls of the self-excited attractor, i.e., a scroll of the hidden attractor oscillates around a double-scroll self-excited

Pseudo-random bit generator

In this section, we present the pseudo-random bit generator (PRBG) in the same spirit that in Ref. [41], it is based on the time series obtained from system states given by (7), (9), (10). This dynamical system displays a two-directional grid multiscroll hidden attractor. The main idea is to iterate the system n times to obtain a sequence $X$ after a certain number of iterations of the transient state (in this case $2000$ ). Taking advantage of the sensibility to initial conditions of the hidden

Conclusions

A methodology to generate two-directional grid multiscroll hidden attractors has been presented. It is possible to generate multiscroll hidden attractors via double-scroll self-excited attractors based on piecewise linear systems.

Furthermore, a pseudo-random bit generator based on the dynamics of a multiscroll hidden attractor was introduced. A statistical test suite was used on the sequences obtained to validate the viability of use in cryptographic applications proving that the sequences

Credit author statement

Jesus R. Pulido-Luna: Software, Investigation, Writing, Original Draft Preparation. Jorge A. López-Rentería: Methodology, Formal analysis, Funding acquisition. Nohe R. Cazarez-Castro: Validation, Visualization, Project administration. Eric Campos: Conceptualization, Writing – Review & Editing, Supervision, Funding acquisition.

Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: CONACYT for the financial support through Project No. A1-S-32341 and Project No. A1-S-30433 and Tecnológico Nacional de México for the financial support through Projects No. 5564.19-P, 8085.20-P and 11122.21-P.

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¹: This author wants to thank CONACYT for the DSc scholarship.

²: J. A. López-Rentería wants to thank CONACYT for the financial support Project No. A1-S-32341.

³: N. R. Cazarez-Castro acknowledges Tecnológico Nacional de México for the financial support trough Projects No. 5564.19-P, 8085.20-P and 11122.21-P.

⁴: E. Campos acknowledges CONACYT for the financial support through Project No. A1-S-30433.

View full text

A two-directional grid multiscroll hidden attractor based on piecewise linear system and its application in pseudo-random bit generator

Highlights

Abstract

Introduction

Section snippets

Preliminaries

System description and problem statement

One directional grid multiscroll hidden attractors

Pseudo-random bit generator

Conclusions

Credit author statement

Declaration of competing interest

Comput. Phys. Commun.

IFAC-PapersOnLine

Int. J. Non Lin. Mech.

Phys. Lett.

Chaos, Solitons & Fractals

IFAC Proceedings Volumes

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Commun. Nonlinear Sci. Numer. Simulat.

Chaos, Solit. Fractals

Automatica

Nonlinear Analysis: Hybrid Systems

Phys. Nonlinear Phenom.

Commun. Nonlinear Sci. Numer. Simulat.

A pseudo-random number generator for the system/360

IBM Syst. J.

A simple unpredictable pseudo-random number generator

SIAM J. Comput.

A non-linear congruential pseudo random number generator

Algorithms

Implementing a chaotic cryptosystem by performing parallel computing on embedded systems with multiprocessors

Entropy

Chaos based cryptographic pseudo-random number generator template with dynamic state change

Appl. Sci.

Some basic cryptographic requirements for chaos-based cryptosystems

International Journal of Bifurcation and Chaos