A multi-dimensional controllable multi-scroll chaotic circuit based on a modular function feedback structure

Abstract

The application discloses a multi-dimensional controllable multi-scroll chaotic circuit based on a modulus function feedback structure, and belongs to the technical field of information security. A four-dimensional multi-scroll chaotic system is generated by adopting a generalized Hamilton system model and a modulus function feedback item, a modulus function circuit is constructed, the top layer of the modulus function feedback circuit is designed, and finally, the multi-dimensional controllable multi-scroll chaotic circuit is constructed. The multi-level design of the chaotic system circuit is performed from top to bottom by adopting a state machine control management mode. The three-dimensional control of the number, position and size of the scroll can be realized by the modulus function feedback structure, the modulus operation of a large bit-width signed fixed-point number can be realized by using less hardware resources, the circuit has higher decimal precision after 1 million iterations, and higher data throughput can be realized by using less hardware resources.

Description

Technical Field

[0001] This invention belongs to the field of information security technology, specifically relating to a multidimensional controllable multi-vortex chaotic circuit based on a modular function feedback structure. Background Technology

[0002] The nonlinear and highly sensitive properties of chaotic systems align perfectly with the principles of confusion and diffusion in cryptography. This makes chaotic systems widely applicable in secure communication, particularly in pseudo-random generator design and data encryption. Chaotic systems, by introducing multiple vortex attractors into phase space to form multi-vortex structures, possess high dimensionality and complex trajectories, making them highly valuable for secure communication. In recent years, a rich variety of multi-vortex chaotic systems have been constructed based on methods such as hyperbolic tangent functions, voltage-controlled memristors, and sign functions. However, the mathematical models of most of these systems involve complex feedback mechanisms, making direct implementation impossible on high-speed hardware platforms like FPGAs without embedded operating systems. This limits the application of multi-vortex chaotic systems in information security. Therefore, constructing highly complex chaotic systems using concise mathematical models is crucial for secure communication applications.

[0003] The controllability of multi-vortex chaotic systems is crucial to the unpredictability of their behavior. Flexible controllability allows multi-vortex chaotic systems to exhibit more complex nonlinear motions, increasing unpredictability while also providing a larger key space for cryptographic applications. Therefore, constructing multi-vortex chaotic systems with good controllability has become an important trend in current chaotic system design. Summary of the Invention

[0004] Based on the above shortcomings, this invention proposes a multidimensional controllable multi-vortex chaotic circuit based on a modal function feedback structure to solve the problems of weak multidimensional controllability of chaotic systems and the inability to implement it on an FPGA platform due to the complexity of nonlinear structures.

[0005] The technical solution adopted in this invention is as follows: a multidimensional controllable multi-vortex chaotic circuit based on a modulo function feedback structure, the construction steps of which are as follows:

[0006] Step 1: Using the generalized Hamiltonian system model and the module function feedback term, a four-dimensional multi-vortex chaotic system is generated, as shown in formula (1).

[0007]

[0008] In the formula, f(x), g(y), and h(z) are feedback functions with respect to state variables x, y, and z, respectively; a and b are constant parameters.

[0009] Among them, the modulus function feedback term The definition is shown in formula (2).

[0010]

[0011] In the formula, n, T>0, mod(ξ,T) represents taking the modulus of T with respect to ξ;

[0012] make With g(y) = y and h(z) = z, a one-dimensional controllable multi-vortex chaotic system is obtained. The control parameters (a, b, T) = (6, 6, 6) are set, and the system forms 2n vortices along the x-axis.

[0013] make h(z) = z, resulting in a two-dimensional controllable multi-vortex chaotic system. The control parameters (a,b,T) = (6,6,4) are set, and the system forms 2n×2n vortices along the x-axis and y-axis directions.

[0014] make A three-dimensional controllable multi-vortex chaotic system is obtained. The parameters (a,b,T)=(6,6,2) are set. The system forms 2n×2n×2n vortices along the x-axis, y-axis and z-axis directions.

[0015] make Furthermore, a multi-vortex structure with different amplitudes in each dimension is constructed in three-dimensional space. When the parameters n and T in the three dimensions take different values, the multi-vortex arrangement of the system presents a cuboid structure, and the shape and position will change. Similarly, the two-dimensional controllable multi-vortex chaotic system can obtain multi-vortex chaotic systems with different shapes and positions by setting different parameters.

[0016] Step 2: Constructing the Modular Function Circuit: Design the top layer of the modular function feedback circuit, where x_in is the modulus, n and T are the parameters of the chaotic system, both defined as signed fixed-point numbers in 42Q32 format, and fx_out is the output value of the feedback function, which is a signed fixed-point number in 43Q32 format. The Verilog HDL hardware description pseudocode of the modular function feedback circuit is shown in Algorithm 1. The always block uses the sensitive signal x_in as the trigger condition. When the feedback circuit receives x_in, it calculates the response output fx_out.

[0017]

[0018]

[0019] Step 3: Construct a multi-dimensional controllable multi-vortex chaotic circuit. A top-down, multi-layered design of the chaotic system circuit is adopted using a state machine control management model. The top layer of the system includes an initial value selection module XS, a modular function feedback module F_FUN, a state control module ASF, a counter module CNT, and a chaotic iteration module CI. The chaotic iteration module CI includes four lower-level modules X_ITER, Y_ITERA, Z_ITERA, and U_ITERA that perform iterative operations. The states of each module need to be transitioned under the precise timing control of the state control module ASF. Based on the timing design, the transmission process of timing signals for the initial value selection module XS, the modular function feedback module F_FUN, the counter module CNT, and the chaotic iteration module CI under the control of the state control module ASF is given. From the time the chaotic system receives the valid start signal start_ms at time T1 to the time T3 when the chaotic sequence is generated, two clock latency periods have elapsed. Based on the flow of timing signals, the state transition process of the state control module ASF is designed, containing six states. In each state, the control signal performs precise timing control, including the following:

[0020] 1) After the system is powered on and reset, the state control module ASF enters the S0 state. When a valid start_ms signal is received, the state control module ASF jumps to the S1 state; otherwise, it retains the state.

[0021] 2) The state control module ASF enters the S1 state, sets the iterative operation enable signal ena_ms, the initial value valid signal init_valid, the data selection signal selx, and the counter enable signal en_cnt to one, and then jumps to the S2 state.

[0022] 3) In state S2, the state control module ASF clears init_valid and selx to zero, and gives a valid chaotic flow flag str_flag, and then jumps to state S3.

[0023] 4) When the state control module ASF detects that the length of the generated chaotic sequence has reached n in state S3, it jumps to state S4; otherwise, it retains the state.

[0024] 5) After the state control module ASF enters the S4 state, it clears ena_ms and en_cnt and then enters the next state;

[0025] 6) After the state control module ASF clears str_flag to zero in state S5, it jumps back to state S0, which means that the chaotic flow generation is over.

[0026] The beneficial effects and advantages of this invention are as follows: The system constructed using the multi-vortex chaotic model has a relatively simple mathematical model, and the number, position, and size of vortices can be controlled in three dimensions through a modular function feedback structure. Modulo operations on large-bit-width signed fixed-point numbers can be performed using relatively few hardware resources. Experimental results show that the circuit of this invention not only maintains high decimal precision after 1 million iterations, but also achieves high data throughput with fewer hardware resources. Attached Figure Description

[0027] Figure 1 The phase diagram of a multidimensional controllable multi-vortex chaotic system in the xyz plane;

[0028] Figure 2 The Lyapunov exponent spectra of system parameters a and b;

[0029] Figure 3 The bifurcation graph of the system with parameters a∈[-10,10] and T∈[0,10] is shown.

[0030] Figure 4 Phase diagrams for systems with different initial values ​​x0;

[0031] Figure 5 Phase diagrams of the system in the xy plane with different initial values;

[0032] Figure 6 This is a graph showing the difference between two sequence pairs (x1, y1, z1, u1) and (x2, y2, z2, u2) after n iterations, given small differences in initial values ​​and parameters.

[0033] Figure 7 A comparison of the spectral entropy values ​​of the system as a function of parameters a and b;

[0034] Figure 8 A comparison chart of the C0 complexity of the system as parameters n and T change;

[0035] Figure 9 The modular function algorithm model for signed fixed-point numbers;

[0036] Figure 10 This is the top-level design diagram of the modulo function feedback circuit;

[0037] Figure 11 The Modelsim simulation diagram of the modulo function feedback circuit is shown.

[0038] Figure 12 This is a diagram of the system's top-level architecture.

[0039] Figure 13 This is a timing simulation diagram of the system;

[0040] Figure 14The timing signal flow diagram of each module in the system under ASF control is shown.

[0041] Figure 15 This is the system state transition diagram;

[0042] Figure 16 The system's Modelsim waveform simulation diagram;

[0043] Figure 17 Block diagram of the oscilloscope verification platform;

[0044] Figure 18 A system multi-vortex chaotic flow graph captured by an oscilloscope. Detailed Implementation

[0045] The invention will now be described in further detail with reference to the accompanying drawings.

[0046] Example 1

[0047] A multidimensional controllable multi-vortex chaotic circuit based on a modulo function feedback structure is constructed as follows:

[0048] Step 1: Using the generalized Hamiltonian system model and the module function feedback term, a four-dimensional multi-vortex chaotic system is constructed, as shown in formula (1). This system can realize multi-dimensional control of the number, position and amplitude of vortices.

[0049]

[0050] In the formula, f(x), g(y), and h(z) are feedback functions with respect to the state variables x, y, and z, respectively, and a and b are constant parameters.

[0051] Among them, the modulus function feedback term The definition is shown in formula (2).

[0052]

[0053] Where n, T>0, mod(ξ,T) represents finding the modulus of T with respect to ξ.

[0054] make Given g(y) = y and h(z) = z, a one-dimensional controllable multi-vortex chaotic system is obtained. The control parameters are set as (a, b, T) = (6, 6, 6). Figure 1 (ad) presents the phase diagram of a one-dimensional controllable multi-vortex chaotic system in the xyz plane for different values ​​of control parameter n. It can be seen from the figure that as n changes, the system forms 2n vortices along the x-axis.

[0055] make h(z) = z, thus obtaining a two-dimensional controllable multi-vortex chaotic system. The control parameters are set as (a,b,T) = (6,6,4). Figure 1 (eh) presents the phase diagram of a two-dimensional controllable multi-vortex chaotic system in the xyz plane for different values ​​of control parameter n. It can be seen from the figure that as n changes, the system forms 2n×2n vortices along the x-axis and y-axis.

[0056] make The equilibrium point of the three-dimensional controllable multi-vortex chaotic system is obtained as follows: The Jacobian matrix is ​​as shown in formula (3).

[0057]

[0058] X can be determined through eigenvalues. e3 It belongs to the center point. Similarly, it can be deduced that the equilibrium points of both one-dimensional and two-dimensional controllable multi-vortex chaotic systems are center points. Let the parameters (a,b,T) = (6,6,2). Figure 1 (il) presents the phase diagram of MCMCCS-3D in the xyz plane for different values ​​of the control parameter n. It can be seen from the figure that as n changes, the system's vortices form 2n×2n×2n vortices along the x-axis, y-axis and z-axis directions.

[0059] If let It is possible to construct multi-vortex structures with different amplitudes in each dimension in three-dimensional space. Figure 1 (mn) presents the xyz phase diagrams of (a,b)=(6,6) and the feedback functions f(x), g(y), h(z) for different values ​​of parameters n and T. As shown in the diagrams, when the parameters n and T in the three dimensions take different values, the new system's multi-vortex structure exhibits a cuboid structure, with changes in both shape and position. Similarly, a two-dimensional controllable multi-vortex chaotic system can also be obtained with different shapes and positions by setting different parameters. This helps to further enhance the controllability of the chaotic model and expand the key space, thereby improving the unpredictability and randomness of the system.

[0060] Step 2: Analyze the dynamic characteristics of the multidimensional controllable multi-vortex chaotic system.

[0061] S21: The divergence of the divergence system is shown in equation (4).

[0062]

[0063] As the system volume remains constant over time, this means that the non-dissipative nature of the chaotic system can maintain the complex dynamic structure of the phase space during long-term iterations, thereby increasing the complexity and randomness of the chaotic key stream.

[0064] S22: Simulation of Lyapunov exponent spectrum and bifurcation diagram

[0065] Set the initial value (x0, y0, z0, u0) = (4, 3, 2, 1). Figure 2 (ab) presents the Lyapunov exponent spectra of the system for parameters a∈[-25,25] and T∈[0,60]. The Lyapunov exponents of parameters a and T are greater than 0 over a relatively wide parameter range, indicating that the system can maintain a chaotic state within a large parameter space. The bifurcation diagrams of the system for parameters a∈[-10,10] and T∈[0,10] are shown below. Figure 3 As shown. According to Figure 3 (a) Except near parameter a = 0, the bifurcation diagram of the system state variable x shows a densely distributed cluster of points. This indicates that the system exhibits chaotic characteristics, with highly sensitive and unpredictable state values. From Figure 3 (b) It can be seen that as the parameter T increases, the range of values ​​for the chaotic orbit of the state variable x gradually increases. This indicates that when the parameter n remains constant, the parameter T can affect the magnitude of the vortex.

[0066] S23: Multistable

[0067] Set the system control parameters (a,b,T,n) = (-0.01,6,4,2). Figure 4 Phase diagrams of the system in the xyz coordinate system with different initial values ​​of x0 are presented. When the value of a is near 0, the system trajectory exhibits a quasi-periodic state. Changing the initial value of x0 will cause the system phase diagram to exhibit different quasi-periodic flow modes.

[0068] Set the control parameters (a,b,T,n) = (1,1,4,2). The xy-plane phase diagrams of the system's evolution over time under different initial conditions are shown below. Figure 5 As shown. According to Figure 5 (bd), changing the initial values ​​y0 or z0 can also yield some coexisting quasi-periodic orbits. Furthermore, comparing... Figure 4 (a) and Figure 5 (a) It can be seen that as parameter a increases and b decreases, the system has entered a chaotic state under certain specific parameters.

[0069] Step 3: Test the sequences generated by the system to verify the nonlinear characteristics of the system.

[0070] S31: Sensitivity and Relevance

[0071] By introducing initial values ​​and parameters with slight differences, the sensitivity of the proposed system to the initial values ​​and parameters is tested using the difference method. Figure 6Figures (a) and (b) show the changes in the difference between two sets of sequences (x1, y1, z1, u1) and (x2, y2, z2, u2) after n iterations, given small differences in initial values ​​and parameters. According to the figures, the system undergoes 10 iterations with varying initial values ​​and parameters. -14 After a tiny change of orders of magnitude, the system exhibits completely different outputs after approximately 550 iterations. Table 1 lists the system outputs after 10 changes in initial values ​​and parameters. -14 The table shows the correlation calculation results of the two sets of output sequence pairs before and after the order-of-magnitude change. As can be seen from the table, regardless of small changes in the system's initial values ​​or parameters, the correlation of all four sequence pairs approaches 0. This analysis indicates that the system is highly sensitive to both initial values ​​and parameters.

[0072] Table 1. Correlation coefficients of output sequence pairs

[0073]

[0074] The difference between the initial value and the initial parameter is 1e-14, and the length of the test sequence is 50000.

[0075] S32: Complexity

[0076] The normalized spectral entropy value of the system is obtained by utilizing the energy distribution in the Fourier transform domain and based on the Shannon entropy algorithm. Figure 7 Figures (a) and (b) show the comparison of the spectral entropy values ​​of the system with some chaotic systems under varying parameters a and b, respectively. It can be seen from the figures that the system exhibits a more ideal SE over a wider parameter range, regardless of whether the parameters are a or b. The sequence is divided into regular and irregular parts to obtain the proportion of the irregular part in the chaotic sequence, i.e., the C0 complexity. This is then compared with the C0 complexity of different chaotic systems. Figure 8 As shown, compared to other systems, the parameters n and T have a more ideal C0 complexity in most parameter ranges.

[0077] Step 4: Construct a lightweight circuit for the modulo function feedback circuit:

[0078] The modular function has two input variables (modulo τ and modulo T) and one output variable (remainder r), which can be expressed as r = τ mod T. Given that the results of modular functions and chaotic systems can be decimals, and that fixed-point numbers have simpler arithmetic logic, require less storage space, and offer greater determinism compared to floating-point numbers, this section represents all data using signed fixed-point numbers in mQn format, where m is the total data bit width, n is the total decimal bit width, and the highest bit is the sign bit. For example... Figure 9As shown, a modular function algorithm model based on bit operations is presented. First, the sign bits of the modulus T and the constant parameter P are truncated. Although the modulus is generally not negative, in FPGAs, T must be defined as a signed number to ensure that arithmetic operations yield correct signed numbers. Similarly, P also needs to be assigned to a signed variable. Next, a binary subtraction is performed on the truncated T and P. Finally, a bitwise AND operation is performed between the modulus τ and the obtained difference to obtain the remainder r. This model supports modulo 2^T τ. N , Modular function operations on all signed fixed-point numbers (including integers and decimals).

[0079] like Figure 10 The diagram shows the top-level design of the modular function feedback circuit, where x_in is the modulus, n, and T are the parameters of the chaotic system, all defined as signed fixed-point numbers in 42Q32 format. fx_out is the output value of the feedback function, which is a signed fixed-point number in 43Q32 format. The Verilog HDL hardware description pseudocode of the modular function feedback circuit is shown in Algorithm 1. The always block uses the sensitive signal x_in as the trigger condition, ensuring that the feedback circuit can quickly calculate the response output fx_out when it receives x_in. This helps improve the iterative efficiency of the chaotic system.

[0080]

[0081] The Modelsim simulation of the feedback circuit's fx_out output waveform under different x_in inputs was performed. The waveform display was set to signed, Friction bits = 30, Precision = 10. The Modelsim simulation of the modulo function feedback circuit is shown below. Figure 11 As shown. Since the actual decimal width is set to 32 bits, the actual value of the simulation result is 1 / 2 of the displayed value. 2According to the diagram, parameter T = 16 / 4 = 4, NT = n × T = 32 / 4 = 8. When the modulus x_in = -43.99993895 / 4 = -10.9999847375, the remainder fx_mod = 4.00006105 / 4 = 1.0000152625 = mod(x_in, T), and the feedback output fx_out = -11.99993895 / 4 = -2.9999847375 = x_in + NT. When the modulus x_in = -23.75588203 / 4 = -5.9389705075, the remainder fx_mod = 8.244117973 / 4 = 2.06102949325 ≈ mod(x_in, T) = 2.0610294925, and the feedback output fx_out = 0.1220589867 / 4 = 0.030514746675 ≈ -1 + (2 / T) mod(x_in, T) = 0.03051474625. Simulation analysis shows that when the decimal width is 32 bits, the calculation accuracy can reach approximately 1 × 10⁻⁶. -9 This indicates that the analog-to-analytical feedback circuit can achieve computational functions with considerable accuracy.

[0082] Next, the resource consumption of the new modular function algorithm model and the traditional division module implementation of the signed fixed-point modular function feedback circuit were verified. Hardware implementations of both methods were performed on a Xilinx ARTIX-7 series XC7A100 FPGA. The traditional division module implementation of the modular function feedback circuit used 2264 lookup table units, while the newly proposed bit-operation-based modular function algorithm model can achieve the same function using only 219 lookup table units.

[0083] Step 5: Construct a multi-dimensional, controllable, multi-vortex chaotic circuit

[0084] Based on the modular function feedback circuit constructed in step four, a top-down, multi-layered design of the chaotic system circuit is performed using a state machine control and management model. The system's top-level architecture is as follows: Figure 12 As shown in the figure, the top layer of the system includes an initial value selection module XS, a modular function feedback module F_FUN, a state control module ASF, a counter module CNT, and a chaotic iteration module CI. The chaotic iteration module CI includes four lower-level modules X_ITER, Y_ITERA, Z_ITERA, and U_ITERA that perform iterative operations.

[0085] The states of each module in the system need to be transitioned under the precise timing control of the State Control Module (ASF). Figure 13The timing design of the system is presented. Based on the timing design, the transmission process of timing signals for the initial value selection module XS, the modulo function feedback module F_FUN, the counter module CNT, and the chaotic iteration module CI under the control of the state control module ASF is described as follows: Figure 14 As shown in the figure, it can be seen that from the time the chaotic system receives the valid start signal start_ms at time T1 to the time the chaotic sequence is generated at time T3, two clock latency periods have elapsed. Based on the flow of the timing signal, the state transition process of the state control module ASF is as follows: Figure 15 As shown. State transition Figure 1 It comprises six states, in which precise timing control is applied to the control signals. The workflow of a chaotic system can be summarized as follows:

[0086] 1) After the system is powered on and reset, the state control module ASF enters the S0 state. When a valid start_ms signal is received, the state control module ASF jumps to the S1 state; otherwise, it retains the state.

[0087] 2) The state control module ASF enters the S1 state, sets the iterative operation enable signal ena_ms, the initial value valid signal init_valid, the data selection signal selx, and the counter enable signal en_cnt to one, and then jumps to the S2 state.

[0088] 3) In state S2, the state control module ASF clears init_valid and selx to zero, and gives a valid chaotic flow flag str_flag, and then jumps to state S3.

[0089] 4) When the state control module ASF detects that the length of the generated chaotic sequence has reached n in state S3, it jumps to state S4; otherwise, it retains the state.

[0090] 7) After the state control module ASF enters the S4 state, it clears ena_ms and en_cnt and then enters the next state;

[0091] 8) After the state control module ASF clears str_flag to zero in state S5, it jumps back to state S0, which means that the chaotic flow generation is over.

[0092] Step Six: Simulation and Board-Level Verification of the System's Lightweight Digital Circuits

[0093] Use the Modelsim tool to simulate and test the system. Figure 16The global simulation waveform and local simulation results of the chaotic system are presented. The chaotic system generated a total of 2 million chaotic sequences through iteration without collapsing to a fixed point or a steady state. This indicates, to some extent, the stability of the system's chaotic characteristics and proves the high accuracy of the fixed-point number scheme. The hardware implementation results are compared and analyzed with the results of 64-bit floating-point operations in Matlab. The known result of 1 million iterations using Matlab is -5.812329020070686. This result was amplified by 2... 2 The result is -23.249316080282744. This result has an error of only 0.000004 compared to the FPGA hardware implementation result of -23.249320, indicating that the error between the Matlab simulation result and the FPGA implementation result is less than 10. -5 This result further demonstrates the stability and practical application potential of the novel multidimensional controllable multivolume conservative chaotic system during long-term operation. To visually verify the output waveform, an AN9767 DAC module was used to build a [system described in the original text]. Figure 17 The oscilloscope verification platform shown. Figure 18 As shown, the xy coordinate plane waveforms of MCMCCS-1D and MCMCCS-2D when n=2 are given.

[0094] Step 7: Statistical testing and implementation efficiency analysis of the system's lightweight digital circuit output sequence

[0095] For the 16 tests in the NIST suite, the significance level was set to 1%. A binary sequence was considered random with a 99% confidence level when the p-value > 0.01; otherwise, it was considered non-random. To perform this test, we generated and captured sequences of length 10 from the lightweight chaotic circuit designed in step five. 6 The binary sequence was subjected to NIST testing, and the results are shown in Table 1. All 16 tests were successful, indicating that the system's output sequence has strong randomness.

[0096] Table 2. NIST Test Results

[0097] 1 Frequency 0.696109 Success 2 Block-frequency 0.803901 Success 3 Runs 0.852953 Success 4 Long runs of ones 0.679644 Success 5 Rank 0.537895 Success 6 FFT 0.101192 Success 7 Non-overlapping 0.245787 Success 8 Overlapping template 0.900908 Success 9 Universal 0.640319 Success 10 10 Linear complexity 0.541902 Success 11 Serial 0.581179 Success 12 Approximate entropy 0.687989 Success 13 Cumulative sums 0.616808 Success 14 Random excursions 0.227335 Success 15 Random excursions 0.040937 Success 16 Lempel ziv 0.696109 Success

[0098] To balance hardware resource costs and implementation efficiency, the CI module of the chaotic system does not employ a pipelined design. Even so, the lightweight chaotic circuit can operate at a clock frequency of 60MHz, approximately five times that of a traditional microcontroller. Furthermore, since the new system can generate a 168-bit chaotic sequence per clock cycle, the theoretical system throughput can reach 10.08Gbps. Tables 3 and 4 present a comparison of the FPGA implementation efficiency and resource consumption of some advanced multi-volume chaotic systems in recent years. According to the tables, although the clock frequency of the new system is lower than some other studies, it achieves a higher data throughput. Moreover, its LUT resource consumption is significantly lower than other studies. These results indicate that, compared to other studies, the lightweight chaotic circuit designed based on a modulus-function-based multi-dimensional controllable multi-volume chaotic system has certain advantages in terms of resources and throughput.

[0099] Table 3 Comparison of FPGA Implementation Efficiency for Multi-Vortex Chaotic Systems

[0100] Artix-7 Continuous 60 10080 Fixed Zynq-7000 Discrete 126.039 44.321 Floating Kintex-7 Discrete 120.51 10283.52 Fixed Virtex-II Continuous 12.649 202 Fixed Zynq-7000 Continuous 100 2400 Fixed

[0101] Table 4 Comparison of FPGA Implementation Resources for Multi-Vortex Chaotic Systems

[0102] 902 364 14 35 4373 5339 NA 10 4694 2227 224 512 16970 4677 49 33 2959 257 192 36 1514 1458 16 6

[0103] It is worth noting that the contents not described in detail in the embodiments of the present invention belong to the prior art known to those skilled in the art.

[0104] In summary, the multidimensional controllable multi-vortex chaotic system and lightweight digital circuit architecture design method using modular function feedback described in this invention have the following characteristics:

[0105] (1) A model is proposed based on the generalized Hamiltonian system and the modular function feedback function, which has a relatively simple mathematical model;

[0106] (2) The model can achieve three-dimensional control of the number, position, and amplitude of multiple vortices through a modular function feedback structure. Flexible control of chaotic systems can enhance the unpredictability of the system;

[0107] (3) It has rich multistable states, can maintain a chaotic state over a wide range of parameters, has high system complexity, and is highly sensitive to initial values ​​and parameters.

[0108] (4) The chaotic circuit designed based on the lightweight digital circuit architecture of the feedback function does not collapse after multiple iterations, has high fractional precision, and consumes less hardware resources and has higher throughput compared with the existing multi-vortex chaotic system.

Claims

1. A multidimensional controllable multi-vortex chaotic circuit based on a modulo function feedback structure, characterized in that, The construction steps are as follows: Step 1: Using the generalized Hamiltonian system model and the module function feedback term, a four-dimensional multi-vortex chaotic system is generated, as shown in formula (1). In the formula, f(x), g(y), and h(z) are feedback functions with respect to state variables x, y, and z, respectively; a and b are constant parameters. Among them, the modulus function feedback term The definition is shown in formula (2). In the formula, n, T>0, mod(ξ,T) represents taking the modulus of T with respect to ξ; make With g(y) = y and h(z) = z, a one-dimensional controllable multi-vortex chaotic system is obtained. The control parameters (a, b, T) = (6, 6, 6) are set, and the system forms 2n vortices along the x-axis. make h(z) = z, resulting in a two-dimensional controllable multi-vortex chaotic system. The control parameters (a,b,T) = (6,6,4) are set, and the system forms 2n×2n vortices along the x-axis and y-axis directions. make A three-dimensional controllable multi-vortex chaotic system is obtained. The parameters (a,b,T)=(6,6,2) are set. The system forms 2n×2n×2n vortices along the x-axis, y-axis and z-axis directions. make Furthermore, a multi-vortex structure with different amplitudes in each dimension is constructed in three-dimensional space. When the parameters n and T in the three dimensions take different values, the multi-vortex arrangement of the system presents a cuboid structure, and the shape and position will change. Similarly, the two-dimensional controllable multi-vortex chaotic system can obtain multi-vortex chaotic systems with different shapes and positions by setting different parameters. Step 2: Constructing the Modular Function Circuit: Design the top layer of the modular function feedback circuit, where x_in is the modulus, n and T are the parameters of the chaotic system, both defined as signed fixed-point numbers in 42Q32 format, and fx_out is the output value of the feedback function, which is a signed fixed-point number in 43Q32 format. The Verilog HDL hardware description pseudocode of the modular function feedback circuit is shown in Algorithm 1. The always block uses the sensitive signal x_in as the trigger condition. When the feedback circuit receives x_in, it calculates the response output fx_out. Step 3: Construct a multi-dimensional controllable multi-vortex chaotic circuit. A top-down, multi-layered design of the chaotic system circuit is adopted using a state machine control management model. The top layer of the system includes an initial value selection module XS, a modular function feedback module F_FUN, a state control module ASF, a counter module CNT, and a chaotic iteration module CI. The chaotic iteration module CI includes four lower-level modules X_ITER, Y_ITERA, Z_ITERA, and U_ITERA that perform iterative operations. The states of each module need to be transitioned under the precise timing control of the state control module ASF. Based on the timing design, the transmission process of timing signals for the initial value selection module XS, the modular function feedback module F_FUN, the counter module CNT, and the chaotic iteration module CI under the control of the state control module ASF is given. From the time the chaotic system receives the valid start signal start_ms at time T1 to the time T3 when the chaotic sequence is generated, two clock latency periods have elapsed. Based on the flow of timing signals, the state transition process of the state control module ASF is designed, containing six states. In each state, the control signal performs precise timing control, including the following: 1) After the system is powered on and reset, the state control module ASF enters the S0 state. When a valid start_ms signal is received, the state control module ASF jumps to the S1 state; otherwise, it retains the state. 2) The state control module ASF enters the S1 state, sets the iterative operation enable signal ena_ms, the initial value valid signal init_valid, the data selection signal selx, and the counter enable signal en_cnt to one, and then jumps to the S2 state. 3) In state S2, the state control module ASF clears init_valid and selx to zero, and gives a valid chaotic flow flag str_flag, and then jumps to state S3. 4) When the state control module ASF detects that the length of the generated chaotic sequence has reached n in state S3, it jumps to state S4; otherwise, it retains the state. 5) After the state control module ASF enters the S4 state, it clears ena_ms and en_cnt and then enters the next state; 6) After the state control module ASF clears str_flag to zero in state S5, it jumps back to state S0, which means that the chaotic flow generation is over.

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