In recent years, there are numerous studies on chaotic systems with special equilibrium curves having various shapes such as circle, butterfly, heart and apple. This paper describes a new 3-D chaotic dynamical system with a capsule-shaped equilibrium curve. The proposed chaotic system has two quadratic, two cubic and two quartic nonlinear terms. It is noted that the proposed chaotic system has a hidden attractor since it has an infinite number of equilibrium points. It is also established that the proposed chaotic system exhibits multi-stability with two coexisting chaotic attractors for the same parameter values but differential initial states. A detailed bifurcation analysis with respect to variations in the system parameters is portrayed for the new chaotic system with capsule equilibrium curve. We have shown MATLAB plots to illustrate the capsule equilibrium curve, phase orbits of the new chaotic system, bifurcation diagrams and multi-stability. As an engineering application, we have proposed a speech cryptosystem with a numerical algorithm, which is based on our novel 3-D chaotic system with a capsule-shaped equilibrium curve. The proposed speech cryptosystem follows its security evolution and implementation on Field Programmable Gate Array (FPGA) platform. Experimental results show that the proposed encryption system utilizes 33% of the FPGA, while the maximum clock frequency is 178.28 MHz.

Chaos theory has been applied in various areas of science and engineering such that found in nonlinear oscillatory systems [

By definition, dynamical systems with a positive Lyapunov index number are called chaotic [

Speech signal encryption is one of the most widely used techniques for ensuring the security of the verbal transmission of data. To address information security concerns, various cryptographic standards and protocols have been proposed, including speech encryption algorithms such as Data Encryption Standard (DES) [

Each standard encryption technique has pluses and minuses; however, these cryptographic techniques do not address the distribution of encryption keys. The signal being aperiodic, broadband, and spacious spectrum for housing the secret message, the use of chaotic systems in cryptography brought new life into encryption systems [

This study presents a novel 3-D chaotic system consisting of six nonlinear terms. A novelty in our research work is that the proposed chaotic system exhibits a capsule-shaped equilibrium curve. In addition, an illustration of the system’s dynamic properties including signal plots and bifurcation diagram are given. For nonlinear dynamical systems, multi-stability refers to the coexistence of chaotic attractors for the same values of parameters but different values of initial states [

Section 2 presents the modelling of our proposed chaotic system with capsule equilibrium curve. In Section 3, we analyze the proposed system from the perspective of bifurcation and multi-stability properties. We show that the new chaotic system has two coexisting chaotic attractors for the same parameter set but different sets of initial sets. A speech encryption scheme based on the proposed system is introduced in Section 4. Finally, conclusions are dressed in Section 5.

This study proposed a 3-D chaotic dynamical system having six nonlinear terms.

In the system

To determine the equilibrium points of system

Solving the equations

MATLAB’s Lyapunov Index (LI) values spectrum analysis was used to establish the chaotic existence in the three-dimensional dynamical system

To perform time-series analysis for the system’s state

Based on Wolf’s procedure [

The chaotic property of system

The negative sum of all LEI values in

Moreover, the Kaplan-Yorke dimension for system

The higher the value of

By fixing the parameter set _{0} = (0.02, 0.01, 0.02), we determine the values of LI and signal plots for the system

In this section, we use bifurcation diagrams to analyze the dynamics of the proposed chaotic system

For the first case, let

For both cases, we performed bifurcation analysis for the state variables ξ and

Multi-stability is an intrinsic property of nonlinear dynamical systems, which reveals the abundant dynamic characteristics of the system

By fixing

This section illustrates the speech-based cryptosystem using the proposed 3-D chaotic system

The encryption process consists of two phases. Firstly, the speech signal

Then this signal is masked again using the second chaotic system B output, to produce the encrypted signal _{2}(

We note that the outputs of systems

In the decryption process (see _{2}(

Then the un-masking of the produced signal _{1}(_{2}(

We assume that the key generators on the decryption side are identical and synchronized with the systems on the encryption side. Based on this assumption, the recovered signal _{2}(

The simulation results were obtained using MATLAB/SIMULINK. Various tests were performed to examine the security and efficiency of the proposed system, such as waveforms, spectrogram analysis, Signal to Noise Ratio (SNR), and correlation tests. Speech signals have an 8000 Hz using eight quantization bits.

The proposed speech cryptosystem can be visually analyzed using the spectrogram’s distribution of energy in the time-frequency plane. As shown in

Noteworthy, the proposed chaotic system consists of two parameters and three initial conditions. The combination of these five parameters makes up the keyspace. They are responsible for the robustness of the proposed algorithm against many attacks. Based on key sensitivity test, it was shown that a slight change in one parameter alters the decrypted signal entirely. For example,

The Correlation Coefficient (CC) and Signal to Noise Ratio (SNR) are used to determine the strength of the proposed algorithm against statistical attacks.

Name | CC | SNR in dB |
---|---|---|

Speech 1 | 7.2801e^{−4} |
−62.7157 |

Speech 2 | 7.8085e^{−4} |
−61.5435 |

Speech 3 | 8.9026e^{−4} |
−61.1175 |

Speech 4 | 7.5163e^{−4} |
−62.0466 |

Measuring the encryption speed of a speech signal is affected by the operating system, programming language, CPU frequency, and system complexity. The encryption speed results for a speech sample using the proposed speech cryptosystem are (0.12 s). Where the length of speech sample is (7.9 s) and the specifications of the used PC are; Intel(R) Core (TM) i7-8565U CPU @ 1.80 GHz hp LAPTOP-UFTCGPRC, 8 GB RAM, Windows11.

The algorithm complexity for the encryption of a speech signal consists of n samples is mainly determined by the following sub operations: masking phases 1 and 2. The complexity of each masking phase is O(n). Therefore, the overall complexity of the proposed speech cryptosystem is O(2n).

In this test, we analyze the randomness of the encrypted speech signal using NIST 800-22 test package provided by the National Institute of standards and technology (NIST) of the United States. Which is a statistical randomness test consisting of fifteen tests. This paper mainly tests the randomness of the original and the encrypted speech signals. The fifteen tests were applied to examine the randomness of each signal as shown in

Name | Original speech signal | Encrypted speech signal |
---|---|---|

Frequency | Fail | Pass |

Block frequency | Fail | Pass |

Cumulative sum | Fail | Pass |

Approximate entropy | Fail | Pass |

Runs | Fail | Pass |

Longest run | Fail | Pass |

Rank | Fail | Pass |

Fast fourier transform | Fail | Pass |

Non-overlapping template | Pass | Pass |

Overlapping template | Fail | Pass |

Universal | Fail | Pass |

Serial | Fail | Pass |

Linear complexity | Pass | Pass |

Random excursions | Fail | Pass |

Random excursions variant | Fail | Pass |

In this part of the study, the speech-based cryptosystem has been realized on the FPGA platform. The synthesizable Very High-Speed Integrated Circuit Hardware Description Language (VHDL) code was generated using a Hardware Description Language (HDL) coder. MATLAB toolbox is responsible for converting MATLAB/SIMULINK design into VHDL or Verilog code. The generated VHDL code has been tested using the QUARTUS program using CYCLON V FPGA chip. The Quartus includes full support for all popular methods of entering a description of the desired circuit into a Computer Aided Design (CAD) system, such as the Verilog, VHDL, and based on a schematic diagram.

In this section, we illustrate in detail how to build the speech-based cryptosystem using the proposed 3D chaotic system with a capsule equilibrium curve

The discretization step size parameter h is chosen to be 0.001 with

Using Simulink, we can draw the top-level block diagram of the discretized 3D chaotic system

The top-level block diagram of speech encryption process is shown in

Similarly, the top-level block diagram for the decryption is shown in

Then, the hardware implementation of the proposed cryptosystem is done using the VHDL code. Each system is realized on the FPGA Cyclon V platform as shown in

With regards to FPGA implementation, all variables and constants makes use of the 32-bit fixed-point arithmetic representation. This representation can be divided into three parts, 1-bit for the sign, 7-bits for the integer part, and 24-bits for the fractional part.

Encryption system | Decryption system | |||
---|---|---|---|---|

Units | Utilization | Units | Utilization | |

Logic utilization | 18451 | 33% | 18508 | 33% |

Total registers | 193 | – | 193 | – |

Total pins | 68 | 25% | 68 | – |

Total DSP blocks | 36 | 23% | 36 | 23% |

Maximum frequency (Hz) | 178.28 | – | 178.28 | – |

Power consumption (mW) | 365.60 | – | 356.61 | – |

This research describes a new chaotic dynamical system with a capsule equilibrium curve. The proposed chaotic system is made up of six nonlinear terms. The proposed chaotic system exhibits a hidden attractor and multi-stability with coexisting attractors. A detailed bifurcation analysis with respect to parameters was described for the proposed chaotic system with capsule equilibrium curve. Finally, we have proposed speech cryptosystem and algorithm, which is based on our novel chaotic system with a capsule equilibrium curve and follows its security evolution and implementation on the FPGA platform. The measures for encrypted and decrypted signal quality showed that the proposed algorithm has high quality of recovered speech signal. Experimental results showed that our speech encryption system utilized 33% of the Field Programmable Gate Array (FPGA), while the maximum clock frequency was 178.28 MHz.

Chaotic systems have several applications in science and engineering. In this work, we focused only on a speech encryption cryptosystem. As future research work, we plan to develop more engineering applications of our chaotic system such as watermarking, data hiding, steganography, etc.

The authors would like to thank the Universiti Sultan Zainal Abidin for supporting this project.

This project is partially funded by the Center for Research Excellence, Incubation Management Center, Universiti Sultan Zainal Abidin via an internal grant UniSZA/2021/SRGS-IC/07.

The authors declare that they have no conflicts of interest to report regarding the present study.