A method and system for generating a non-degenerate chaotic sequence with limited precision

Abstract

The application discloses a kind of finite precision under non-degradation chaotic sequence generation method and system, belong to chaotic password technical field;The present application generates disturbance sequence using certain external entropy source, every specified iteration number, the corresponding numerical value in sequence is used to disturb system state variable and then input to discrete chaotic system for chaotic mapping.Equipped with carefully designed external entropy source and generation sequence, this disturbance scheme solves the degradation problem of chaotic system under finite precision, and can obtain theoretically strictly proven, achievable, low-cost non-degradation chaotic sequence under finite precision.Not only can it simply and effectively solve the chaotic degradation of digital chaotic system caused by finite precision effect, but also can guarantee the immunity of chaotic system to finite precision, and can flexibly construct chaotic sequences that meet multiple scenarios according to security application requirements.In addition, the present application can be applied in cryptography, secure communication, pseudo-random number design, privacy protection and other fields.

Description

Technical Field

[0001] This invention belongs to the field of chaotic cryptography, and more specifically, relates to a method and system for generating non-degenerate chaotic sequences with finite precision. Background Technology

[0002] Chaos is an inherent random phenomenon occurring in nonlinear deterministic systems. Chaotic systems possess properties such as local instability, initial condition sensitivity, pseudo-randomness, and ergodicity, which are similar to the two fundamental principles of diffusion and confusion required by cryptography. Therefore, chaos theory provides new insights for the development of cryptography and has broad application prospects in fields such as secure communication.

[0003] The most distinctive feature of chaotic sequences generated by chaotic systems is their sensitivity to initial conditions, and they operate on an infinite, continuous set of real numbers. This is a significant difference from cryptographic systems, which operate on a finite, discrete set. In practical systems, the finite precision limitations of computers and digital circuits can easily cause chaotic systems to deviate from their theoretically disordered state, thus degenerating into "short-period" cyclic phenomena.

[0004] To prevent the degradation of the dynamic characteristics of digital chaotic systems due to the effective precision effect of computers, researchers have proposed a series of methods to address this problem. One method uses higher finite precision to slow down the degradation rate as the number of iterations increases, but this method cannot fundamentally solve the degradation problem of chaotic systems and has high implementation costs. Another method designs combinations between chaotic systems, which can effectively extend the period of chaotic systems, but the distribution of the generated chaotic sequences is not ideal. Yet another method uses higher-dimensional chaotic systems or fractal structures, which can effectively alleviate the degradation problem, but the overall structure is complex and theoretical analysis is difficult. Summary of the Invention

[0005] To address the aforementioned deficiencies or improvement needs of existing technologies, this invention provides a method and system for generating non-degenerate chaotic sequences with finite precision. This addresses the technical problem that existing technologies cannot simply and effectively solve the problem of chaotic systems degenerating into short-period cycles with finite precision while ensuring the chaotic characteristics of the chaotic system.

[0006] To achieve the above objectives, in a first aspect, the present invention provides a non-degenerate chaotic sequence generation system with finite precision, comprising: a perturbation sequence generation module, a perturbation module, and a discrete chaotic system;

[0007] The perturbation sequence generation module is used to generate an aperiodic sequence as a perturbation sequence after receiving the start command, and outputs the perturbation values ​​in the perturbation sequence to the perturbation module every specified number of iterations, starting from the first iteration of the non-degenerate chaotic sequence generation system.

[0008] The perturbation module is used to initiate the first iteration of the non-degenerate chaotic sequence generation system after receiving the start command, and to receive the initial chaotic value input from the outside as the system state variable for the first iteration. In each iteration: when a perturbation value is received from the perturbation sequence generation module, the system state variable for the current iteration is perturbed using the current input perturbation value to obtain the intermediate state variable for the current iteration, and then output to the discrete chaotic system; when no perturbation value is received from the perturbation sequence generation module, the system state variable for the current iteration is directly output as the intermediate state variable for the current iteration to the discrete chaotic system.

[0009] Discrete chaotic systems are used to perform chaotic mapping on the intermediate state variables of the current iteration using its iterative mapping function in each iteration, obtain the chaotic value of the current iteration and output it, and at the same time use the chaotic value of the current iteration as the system state variable of the next iteration and feed it back to the disturbance module.

[0010] More preferably, when a disturbance exists in the nth iteration, the non-degenerate chaotic sequence generates the system state variable f in the (n+1)th iteration. DC (f P (x(n), S(m)), μ);

[0011] When there is no disturbance in the nth iteration, the system state variable x(n+1)=f of the non-degenerate chaotic sequence generation system in the (n+1)th iteration. DC (x(n), μ);

[0012] Among them, f DC (·) represents the iterative mapping function of the discrete chaotic system; μ represents the parameters of the discrete chaotic system; f P (x(n), S(m)) represents the intermediate state variables in the nth iteration, specifically the result of perturbing the system state variable x(n) in the nth iteration with the perturbation value S(m); the perturbation value S(m) is the mth perturbation value in the perturbation sequence input into the perturbation module in the nth iteration.

[0013] More preferably, the non-degenerate chaotic sequence generated by the above-mentioned non-degenerate chaotic sequence generation system is composed of chaotic values ​​under multiple iterations.

[0014] More preferably, the perturbation interval between two adjacent perturbations is determined by a specified number of iterations;

[0015] The number of iterations can be a preset fixed value or a non-negative integer value that changes dynamically with the number of perturbations.

[0016] When the specified iteration number is a non-negative integer value that dynamically changes with the number of perturbations, for each perturbation, the specified number of iterations between the next perturbation and the current perturbation is used as the current specified iteration number.

[0017] More preferably, the discrete chaotic system is obtained by discretizing the continuous chaotic system.

[0018] Secondly, the present invention also provides the application of the above-mentioned non-degenerate chaotic sequence generation system in chaotic cryptography, chaotic digital modulation, or chaotic pseudo-random number generators.

[0019] Thirdly, the present invention provides a method for generating non-degenerate chaotic sequences with finite precision, comprising:

[0020] After receiving the start command, the first iteration begins, and the initial chaotic value input from the outside is received as the system state variable under the first iteration. At the same time, an aperiodic sequence is generated as a perturbation sequence. Starting from the first iteration of the system generated from the non-degenerate chaotic sequence, the perturbation values ​​in the perturbation sequence are used as the perturbation values ​​under the corresponding iteration in sequence every specified number of iterations.

[0021] In each iteration:

[0022] When there is a corresponding perturbation value in the current iteration, the perturbation value in the current iteration is used to perturb the system state variable in the current iteration to obtain the intermediate state variable in the current iteration. Then, the intermediate state variable in the current iteration is chaotically mapped using a discrete chaotic system to obtain the chaotic value in the current iteration and output it. At the same time, the chaotic value in the current iteration is used as the system state variable in the next iteration.

[0023] When there is no corresponding perturbation value in the current iteration, the discrete chaotic system is directly used to perform chaotic mapping on the system state variables in the current iteration to obtain the chaotic value in the current iteration and output it. At the same time, the chaotic value in the current iteration is used as the system state variable in the next iteration.

[0024] More preferably, when a disturbance exists in the nth iteration, the non-degenerate chaotic sequence generates the system state variable f in the (n+1)th iteration. DC (f P (x(n), S(m)), μ);

[0025] When there is no disturbance in the nth iteration, the system state variable x(n+1)=f of the non-degenerate chaotic sequence generation system in the (n+1)th iteration. DC (x(n), μ);

[0026] Among them, f DC (·) represents the iterative mapping function of the discrete chaotic system; μ represents the parameters of the discrete chaotic system; f P (x(n), S(m)) represents the intermediate state variables in the nth iteration, specifically the result of perturbing the system state variable x(n) in the nth iteration with the perturbation value S(m); the perturbation value S(m) is the mth perturbation value in the perturbation sequence input into the perturbation module in the nth iteration.

[0027] More preferably, the non-degenerate chaotic sequence generated by the above-mentioned non-degenerate chaotic sequence generation method is composed of chaotic values ​​under multiple iterations.

[0028] More preferably, the perturbation interval between two adjacent perturbations is determined by a specified number of iterations;

[0029] The number of iterations can be a preset fixed value or a non-negative integer value that changes dynamically with the number of perturbations.

[0030] When the specified iteration number is a non-negative integer value that dynamically changes with the number of perturbations, for each perturbation, the specified number of iterations between the next perturbation and the current perturbation is used as the current specified iteration number.

[0031] More preferably, the discrete chaotic system is obtained by discretizing the continuous chaotic system.

[0032] Fourthly, the present invention also provides the application of the above-mentioned non-degenerate chaotic sequence generation method in chaotic cryptography, chaotic digital modulation, or chaotic pseudo-random number generators.

[0033] In summary, the above-described technical solutions conceived in this invention can achieve the following beneficial effects:

[0034] 1. This invention provides a system and method for generating non-degenerate chaotic sequences with finite precision. Every specified number of iterations, the system state variables are perturbed using corresponding values ​​from an aperiodic sequence before being input into a discrete chaotic system for chaotic mapping. By utilizing a defined external entropy source to perturb the state variables during the iteration process, a theoretically rigorous, achievable, and low-cost scheme for generating non-degenerate chaotic sequences with finite precision is obtained. This method can simply and effectively solve the problem of chaotic degeneration into short periods in digital chaotic systems due to finite precision effects and other drawbacks, while ensuring the chaotic characteristics of the chaotic system.

[0035] 2. The system and method for generating non-degenerate chaotic sequences with finite precision provided by this invention offer a wide range of perturbation schemes due to the variety of perturbation methods and the abundance of perturbation sequence choices. Furthermore, through multiple experimental selections and comparisons, perturbation sequences with good perturbation effects and easy implementation on hardware systems have been obtained. Meanwhile, by calculating the Lyapunov exponent of the perturbated system, it is shown that the method can guarantee the chaotic characteristics of the sequence.

[0036] 3. The non-degenerate chaotic sequence generation system and method with finite precision provided by the present invention can be used to construct chaotic signals, which are particularly suitable for secure communication. Furthermore, the chaotic signal sequence can be used as a pseudo-random sequence in encryption algorithms. Based on this, the present invention can participate in the construction of pseudo-random number generators and can also improve the performance of chaotic cryptographic systems.

[0037] 4. Based on the careful design of the external entropy source, this invention can set a specific entropy source according to different needs, such as balance, nonlinearity, algebraic immunity, etc., which is easy to control and adjust, flexible in deployment, can solve the degradation problem, and is applicable to all chaotic systems. Attached Figure Description

[0038] Figure 1 A schematic diagram of the structure of the non-degenerate chaotic sequence generation system with finite precision provided by the present invention;

[0039] Figure 2 The bifurcation diagrams of the original Logistic chaotic map provided by the present invention under different precision environments are shown below; (a) is the bifurcation diagram of the original Logistic chaotic map under 8 precision environment; (b) is the bifurcation diagram of the original Logistic chaotic map under 32 precision environment.

[0040] Figure 3 The present invention provides autocorrelation plots of the original Logistic chaotic sequence and Logistic chaotic sequences perturbed by different methods under different precision environments; wherein, (a) is the autocorrelation plot of the original Logistic chaotic sequence under 8 precision environment; (b) is the autocorrelation plot of the original Logistic chaotic sequence under 16 precision environment; (c) is the autocorrelation plot of the Logistic chaotic sequence perturbed by method 1 under 8 precision environment; (d) is the autocorrelation plot of the Logistic chaotic sequence perturbed by method 2 under 8 precision environment; (e) is the autocorrelation plot of the Logistic chaotic sequence perturbed by method 3 under 8 precision environment; and (f) is the autocorrelation plot of the Logistic chaotic sequence perturbed by method 4 under 8 precision environment.

[0041] Figure 4The graphs show the Lyapunov exponent variations of the original Logistic chaotic map and the Logistic chaotic maps perturbed by different methods under different accuracy environments provided by this invention. Specifically, (a) shows the Lyapunov exponent variation of the original Logistic chaotic map under 8 accuracy; (b) shows the Lyapunov exponent variation of the original Logistic chaotic map under 16 accuracy; (c) shows the Lyapunov exponent variation of the Logistic chaotic map perturbed by method 1 under 16 accuracy; (d) shows the Lyapunov exponent variation of the Logistic chaotic map perturbed by method 2 under 16 accuracy; (e) shows the Lyapunov exponent variation of the Logistic chaotic map perturbed by method 3 under 16 accuracy; and (f) shows the Lyapunov exponent variation of the Logistic chaotic map perturbed by method 4 under 16 accuracy.

[0042] Figure 5 The present invention provides phase diagrams of the Baker chaotic mapping under different precision environments; wherein, (a) is the phase diagram of the Baker chaotic mapping under 8 precision environment; (b) is the phase diagram of the Baker chaotic mapping under 16 precision environment; and (c) is the phase diagram of the Baker chaotic mapping under 32 precision environment.

[0043] Figure 6 The present invention provides bifurcation diagrams of the Baker chaotic map under different directions and different precision environments; wherein, (a) is the bifurcation diagram of the Baker chaotic map in the x direction under 8 precision environment; (b) is the bifurcation diagram of the Baker chaotic map in the y direction under 8 precision environment; (c) is the bifurcation diagram of the Baker chaotic map in the x direction under 32 precision environment; and (d) is the bifurcation diagram of the Baker chaotic map in the y direction under 32 precision environment.

[0044] Figure 7 The present invention provides phase diagrams of Baker chaotic maps perturbed by different methods under different precision environments; wherein, (a) is the phase diagram of Baker chaotic map perturbed by method 1 under 8 precision environment; (b) is the phase diagram of Baker chaotic map perturbed by method 2 under 8 precision environment; (c) is the phase diagram of Baker chaotic map perturbed by method 3 under 8 precision environment; and (d) is the phase diagram of Baker chaotic map perturbed by method 4 under 8 precision environment.

[0045] Figure 8The present invention provides autocorrelation maps of unperturbed Baker chaotic maps and Baker chaotic maps perturbed by different methods in different directions and with different precision environments; wherein, (a) is the autocorrelation map of the sequence generated in the x-direction of the unperturbed Baker chaotic map in an 8-precision environment; (b) is the autocorrelation map of the sequence generated in the x-direction of the unperturbed Baker chaotic map in a 32-precision environment; (c) is the autocorrelation map of the sequence generated in the x-direction of the perturbed Baker chaotic map by method 1 in an 8-precision environment; (d) is the autocorrelation map of the sequence generated in the x-direction of the perturbed Baker chaotic map by method 2 in an 8-precision environment; (e) is the autocorrelation map of the sequence generated in the x-direction of the perturbed Baker chaotic map by method 3 in an 8-precision environment; and (f) is the autocorrelation map of the sequence generated in the x-direction of the perturbed Baker chaotic map by method 4 in an 8-precision environment. Detailed Implementation

[0046] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0047] To address the problem of chaotic systems degenerating into short periods due to the finite precision effect, this invention provides a provable scheme for improving the degeneration problem by perturbing a chaotic system with an aperiodic sequence under finite precision. First, the aperiodic sequence perturbation solves the problem of chaotic systems degenerating into periods under finite precision. Second, it ensures that the perturbed system remains chaotic, as demonstrated in the experimental results.

[0048] like Figure 1 As shown, the present invention provides a non-degenerate chaotic sequence generation system with finite precision, comprising: a perturbation sequence generation module, a perturbation module, and a chaotic sequence generation module.

[0049] The perturbation sequence generation module is used to generate an aperiodic sequence as a perturbation sequence after receiving the start command, and outputs the perturbation values ​​in the perturbation sequence to the perturbation module every specified number of iterations, starting from the first iteration of the non-degenerate chaotic sequence generation system.

[0050] The perturbation module is used to initiate the first iteration of the non-degenerate chaotic sequence generation system after receiving the start command, and to receive the initial chaotic value input from the outside as the system state variable for the first iteration. In each iteration: when a perturbation value is received from the perturbation sequence generation module, the system state variable for the current iteration is perturbed using the current input perturbation value to obtain the intermediate state variable for the current iteration, and then output to the discrete chaotic system; when no perturbation value is received from the perturbation sequence generation module, the system state variable for the current iteration is directly output as the intermediate state variable for the current iteration to the discrete chaotic system.

[0051] Discrete chaotic systems are used to perform chaotic mapping on the intermediate state variables of the current iteration using its iterative mapping function in each iteration, obtain the chaotic value of the current iteration and output it, and at the same time use the chaotic value of the current iteration as the system state variable of the next iteration and feed it back to the disturbance module.

[0052] It should be noted that the perturbation interval between two adjacent perturbations is determined by the specified number of iterations;

[0053] The number of iterations can be a preset fixed value (such as a fixed non-negative integer value t), or a non-negative integer value that changes dynamically with the number of perturbations;

[0054] When the specified iteration number is a non-negative integer value that dynamically changes with the number of perturbations, for each perturbation, the specified iteration number between the next perturbation and the current perturbation is used as the current specified iteration number. A certain non-negative integer sequence T can be preset. m ={t1, t2, t3, ..., t m}, for each perturbation, from the non-negative integer sequence T m The iteration count between the next perturbation and the current perturbation is extracted sequentially; where m = 1, 2, 3, ...

[0055] Specifically, in this invention, when t≥0 or t m After ≥0 system iterations, the perturbation module performs one perturbation operation.

[0056] Specifically, the perturbation sequence generation module generates a perturbation sequence of sufficient length based on the general term formula of an aperiodic sequence; the mathematical model of the perturbation sequence generation module is:

[0057] S(m)=f PGen (m)

[0058] In the formula, f PGen (·) represents the general term formula for the perturbation sequence, and the generated number sequence {S} m} is an aperiodic sequence, such as cos(2m)+1, round(·) represents the rounding function; S(m) represents the m-th number in the perturbation sequence. To facilitate the implementation of the generated perturbation sequence on a hardware system, this invention preferably employs... This is used to generate an aperiodic sequence. It's important to note that in practice, the perturbation sequence generated using the general formula must remain aperiodic even under finite precision conditions.

[0059] Specifically, in each iteration, the perturbation module mixes the system state variables of the current iteration with the corresponding sequence values ​​in the perturbation sequence to obtain the perturbed state variables, which are denoted as the intermediate state variables of the current iteration. The mathematical model of the perturbation module is as follows:

[0060]

[0061] In the formula, f P (·) represents the perturbation method; x(n) represents the system state variable in the nth iteration (it should be noted that the system state variable x(1) = x0 in the first iteration; x0 is the initial chaotic value set); X(n) represents the intermediate state variable in the nth iteration, and X(n) becomes the state variable x(n+1) in the (n+1)th iteration after entering the discrete chaotic system; the perturbation value S(m) is the mth perturbation value in the perturbation sequence, corresponding to the nth iteration of the system; it should be noted that there can be multiple perturbation methods, for example, X(n) = (x(n) + S(m)) mod 1. This means that before the non-degenerate chaotic sequence generation system enters the next iteration, the current state variable xn is added to the nth number of the perturbation sequence using ordinary addition, and then modulo 1 is taken to obtain its fractional part. For example, X(n) = bitxor(x(n), S(m)), where bitxor(·) refers to converting the d-bit fractions of x(n) and S(m) to binary at a specific precision d, then performing an XOR operation on the common bits of the two binary representations, and finally converting the result to decimal. It is important to note that when perturbing the state variable, the original numerical range of the state variable cannot be changed. For example, in the Logistic chaotic mapping, all state variables belong to the range [0, 1], so the perturbed state variable still needs to satisfy X(n) ∈ [0, 1].

[0062] The mathematical model of a discrete chaotic system is:

[0063] x(n+1)=f DC (X(n), μ)

[0064] Where μ represents the parameters of the discrete chaotic system. DC This represents the iterative mapping function for discrete chaotic systems.

[0065] It should be noted that discrete chaotic systems can be discrete chaotic systems of various dimensions, such as Logistic chaotic systems, Baker chaotic systems, Sine chaotic systems, Tent chaotic systems, and Henon chaotic systems. They can also be obtained by discretizing continuous chaotic systems. Specifically, for continuous chaotic systems, Euler's algorithm, Runge-Kutta's algorithm, etc., can be used to discretize them to obtain the corresponding discrete chaotic systems. Continuous chaotic systems can include Lorenz chaotic systems, Chua's circuit systems, Chen chaotic systems, Arnold chaotic systems, etc.

[0066] Specifically, taking the Lorenz chaotic system as an example, the Lorenz chaotic system is a continuous chaotic system, and its equations are as follows:

[0067]

[0068] Discretize it using Euler's algorithm to obtain the discretized iterative equation:

[0069]

[0070] In general, the mathematical model for the generation of aperiodic chaos with finite precision is as follows:

[0071]

[0072] It is worth noting that the above processes were all carried out under conditions of limited precision.

[0073] Furthermore, the non-degenerate chaotic sequence generation system generates a digital chaotic sequence by iterating through a sufficient number of chaotic values ​​(set by the user as needed), thereby forming a chaotic signal.

[0074] This invention also provides a method for generating non-degenerate chaotic sequences with finite precision, comprising:

[0075] After receiving the start command, the first iteration begins, and the initial chaotic value input from the outside is received as the system state variable under the first iteration. At the same time, an aperiodic sequence is generated as a perturbation sequence. Starting from the first iteration of the system generated from the non-degenerate chaotic sequence, the perturbation values ​​in the perturbation sequence are used as the perturbation values ​​under the corresponding iteration in sequence every specified number of iterations.

[0076] In each iteration:

[0077] When there is a corresponding perturbation value in the current iteration, the perturbation value in the current iteration is used to perturb the system state variable in the current iteration to obtain the intermediate state variable in the current iteration. Then, the intermediate state variable in the current iteration is chaotically mapped using a discrete chaotic system to obtain the chaotic value in the current iteration and output it. At the same time, the chaotic value in the current iteration is used as the system state variable in the next iteration.

[0078] When there is no corresponding perturbation value in the current iteration, the discrete chaotic system is directly used to perform chaotic mapping on the system state variables in the current iteration to obtain the chaotic value in the current iteration and output it. At the same time, the chaotic value in the current iteration is used as the system state variable in the next iteration.

[0079] Specifically, this invention determines the perturbation interval based on the required perturbation interval. At system iteration times where perturbation is required, the perturbation value under the current iteration is used to perturb the system state variables under the current iteration to obtain intermediate state variables under the current iteration. A discrete chaotic system is then used to perform chaotic mapping on the intermediate state variables under the current iteration to obtain chaotic values ​​under the current iteration and output them. At the same time, the chaotic values ​​under the current iteration are used as system state variables under the next iteration. At other system iteration times where perturbation is not required, a discrete chaotic system is directly used to perform chaotic mapping on the current system state variables to obtain chaotic values ​​under the current iteration and output them. At the same time, the chaotic values ​​under the current iteration are used as system state variables under the next iteration.

[0080] The related technical solutions are the same as the non-degenerate chaotic sequence generation system with finite precision provided by this invention, and will not be described in detail here.

[0081] Furthermore, this invention also provides applications of the aforementioned nondegenerate chaotic sequence generation system or method in chaotic cryptography, chaotic digital modulation, or chaotic pseudo-random number generators. The specific application is similar to that of conventional chaotic systems; the nondegenerate chaotic sequence output by the nondegenerate chaotic sequence generation system can be transformed into binary random numbers through some nonlinear transformations, and then these binary random numbers can be used in stream ciphers (e.g., encryption and decryption can be achieved through XOR operations).

[0082] The following five examples will provide a more detailed explanation of the specific implementation methods.

[0083] The example selected two different discrete chaotic systems: a one-dimensional Logistic chaotic map and a two-dimensional Baker chaotic map. The mathematical model of the Logistic chaotic map is shown below:

[0084] x(n+1)=μx(n)(1-x(n))

[0085] When the parameter μ = 4 and the initial value x0 = 0.4 is selected, the bifurcation diagrams of the Logistic chaotic mapping in the 8-precision environment, the bifurcation diagram in the 32-precision environment, the Lyapunov exponent variation diagram in the 8-precision environment, and the Lyapunov exponent variation diagram in the 16-precision environment are respectively as follows: Figure 2 Figure (a) in the middle Figure 2 Figure (b) in the middle Figure 4 Figure (a) and Figure 4 As shown in Figure (b), it is easy to see that the chaotic effect of the system is significantly reduced under low precision conditions. Meanwhile, the autocorrelation plots of the Logistic chaotic sequence under 8-precision and 16-precision conditions are shown in Figure (b). Figure 3 Figure (a) and Figure 3 As shown in Figure (b), at low precision, the Logistic chaotic system degenerates, and the generated sequence has obvious periodicity.

[0086] The mathematical model for the Baker chaotic map is shown below:

[0087]

[0088] When the parameters μ = 0.4, and the initial values ​​x0 = 0.2 and y0 = 0.3 are selected, the phase diagrams of the Baker chaotic map in 8-precision and 32-precision environments are respectively as follows: Figure 5 Figures (a) and (b) in the diagram are shown. The bifurcation diagrams of the Baker chaotic map in the x-direction under 8-precision, y-direction under 8-precision, x-direction under 32-precision, and y-direction under 32-precision environments are respectively shown in Figures (a) and (b). Figure 6 As shown in (a)-(d) above, the effects of finite precision on chaotic systems are all evident. On the other hand, Figure 8 Figures (a) and (b) show the autocorrelation graphs of the generated sequences of the Baker chaotic map in the x-direction under 8-precision and 16-precision conditions, respectively. These images demonstrate the significant impact of the finite precision effect on Baker chaotic systems.

[0089] Example 1: Taking the Logistic chaotic mapping as an example, the mathematical model of the perturbation sequence generation module is as follows:

[0090]

[0091] Here, `round(·)` represents the rounding function, which can produce a number sequence of the form 1, 1 / 2, 2 / 2, 1 / 3, 2 / 3, 3 / 3... Furthermore, the time t=0 when the interval between perturbation iterations is a fixed value, that is, a perturbation is performed once for each system iteration, and the mathematical model of the perturbation module is set as follows:

[0092] X(n) = (x(n) + S(m)) mod 1

[0093] This means that before the chaotic system enters the next iteration, the current system state variable x is... n The fractional part is obtained by adding the nth number of the perturbation sequence to it using ordinary addition and then taking the modulo 1. This perturbation mode is called Method 1. Figure 3 Figure (c) shows the autocorrelation plot of the Logistic chaotic sequence after perturbation by Method 1 under 8-precision conditions. Figure 3 Figures (a) and (b) show the sequences generated by the unperturbed Logistic chaotic system in 8-precision and 16-precision environments, respectively. The periodicity of the sequences generated by the chaotic system is significantly reduced. On the other hand, Figure 4 Figure (c) shows the change of the Lyapunov exponent in the Logistic chaotic system after perturbation by Method 1 with varying μ under a precision of 16. The figure demonstrates that the present invention can still guarantee the chaotic properties of the system even with limited precision.

[0094] Example 2: Taking the Logistic chaotic mapping as an example, the mathematical model of the perturbation sequence generation module is as follows:

[0095] S(m)=cos(2m)+1

[0096] The experimental setup, including the perturbation module and perturbation interval, is the same as in Example 1. The perturbation mode under this setup is referred to as Method 2. Figure 3 Figure (d) shows the autocorrelation plot of the Logistic chaotic sequence after perturbation by Method 2 under 8-precision conditions. It can be seen that Method 2 alleviates the periodicity of the original Logistic chaotic sequence under 8-precision conditions. On the other hand, Figure 4 Figure (d) shows the change of the Lyapunov exponent in the Logistic chaotic system after perturbation by Method 2 under 16-precision conditions, as μ changes. From the range of positive Lyapunov exponents in the figure, it can be seen that the perturbation by Method 2 results in a wider range of parameters from which chaos can occur. Therefore, this invention allows for the selection of different perturbation sequences according to different needs.

[0097] Example 3: Taking the Logistic chaotic mapping as an example, the mathematical model of the perturbation sequence generation module is as follows:

[0098] S(m)=log2(m+2)

[0099] The experimental setup, including the perturbation module and perturbation interval, is the same as in Example 1. The perturbation mode under this setup is referred to as Method 3. Figure 3Figure (e) shows the autocorrelation plot of the Logistic chaotic sequence after perturbation by Method 3 under 8-precision conditions. It can be seen that Method 3 alleviates the periodicity of the original Logistic chaotic sequence under 8-precision conditions. On the other hand, Figure 4 Figure (e) shows the change of the Lyapunov exponent in the Logistic chaotic system perturbed by Method 3 under a precision of 16, as μ changes. This indicates that the numbers in the perturbed sequence can be unbounded, and the present invention imposes few restrictions on the perturbed sequence.

[0100] Example 4: Taking the Logistic chaotic mapping as an example, the mathematical model and perturbation interval of the perturbation sequence generation module are the same as in Example 1. The mathematical model of the perturbation module is set up as follows:

[0101] X(n) = bitxor(x(n), S(m))

[0102] Here, bitxor(·) refers to converting the d-bit fraction of x(n) to binary and the d-bit fraction of S(n) to binary at a specific precision d, then performing an XOR operation on the common bits of the two binary representations, and finally converting the final result to decimal. The perturbation mode under this setting is called Method 4. Figure 3 Figure (f) shows the autocorrelation plot of the Logistic chaotic sequence after perturbation by method 4 under 8-precision conditions. It can be seen that method 4 alleviates the periodicity of the original Logistic chaotic sequence under 8-precision conditions. On the other hand, Figure 4 Figure (f) shows the change of the Lyapunov exponent in the Logistic chaotic system perturbed by method 4 under a precision of 16. The system enters chaos at μ = 1.01, much earlier than the original chaotic system and the perturbed results of the other three methods. These results indicate that the numbers in the perturbed sequence can be unbounded, and this invention imposes relatively few restrictions on the perturbed sequence.

[0103] Example 5: The four perturbation methods described above are applied to the Baker chaotic system. Among them, Figure 7 In the diagram, (a)-(d) represent the phase diagrams obtained by perturbing the Baker chaotic system using methods 1-4 in an 8-precision environment. Figure 5 Comparing (a)-(c) in the figures, it is evident that by using the present invention to perturb the chaotic system in a low-precision environment, the problem of the chaotic system entering a short period can be effectively solved, and the chaotic effect is even better than that of the system in a higher-precision environment. Figure 8 In the figure, (c)-(f) represent the autocorrelation plots of the chaotic sequences generated after perturbation of the Baker chaotic system by methods 1-4 in an 8-precision environment, respectively, and Figure 8Comparing Figures (a) and (b) in the figure shows that, after the perturbation of the present invention, the chaotic sequence generated by the chaotic system under low-precision environment has a pseudo-randomness comparable to that of the chaotic sequence generated under high-precision environment.

[0104] Using the eight systems described in the five embodiments above, along with the original Logistic chaotic mapping and the original Baker chaotic mapping, 15 chaotic sequences of length 100,000 were generated in an 8-precision environment (the Baker mapping generated sequences in both the x and y directions). The approximate entropy and permutation entropy of these 10 chaotic sequences were then calculated. Specifically, when calculating the approximate entropy, the embedding dimension was 2, and the similarity tolerance was 0.2 times the standard deviation of each sequence group; when calculating the permutation entropy, the embedding dimension was 3, and the delay time was 1. The results are shown in Table 1 below.

[0105] Table 1

[0106]

[0107]

[0108] As shown in Table 1, under finite precision conditions, each perturbation method effectively enhances the randomness of the chaotic sequences generated by the chaotic system, i.e., effectively increases the system's complexity. Therefore, overall, the method provided by this invention can effectively generate non-degenerate chaotic sequences under finite precision conditions. Essentially, this method is a rigorously provable approach that, under finite precision conditions, perturbs a chaotic system using a specific aperiodic sequence as an external entropy source to obtain non-degenerate chaotic sequences. Furthermore, it can effectively improve the chaotic performance and security performance of the original chaotic system under resource-constrained conditions, and is easier to implement, lower in cost, and offers better security.

[0109] It should be noted that the above embodiments 1-5 can all be used as chaotic systems in fields such as chaotic cryptography, secure communication (such as secure communication, spread spectrum communication, etc.), and chaotic pseudo-random number generators.

[0110] In summary, to address the problem of chaotic systems degenerating into short periods due to the finite precision effect, this invention provides a provable method for improving the degeneration problem of chaotic systems under finite precision by perturbing them with aperiodic sequences. Firstly, the aperiodic sequence perturbation solves the problem of chaotic systems degenerating into periods under finite precision conditions. Secondly, it ensures that the perturbed system remains chaotic. Specifically, this invention uses a defined external entropy source to generate a perturbation sequence. Every specified number of iterations, the corresponding values ​​in the sequence are used to perturb the system state variables before inputting them into a discrete chaotic system for chaotic mapping. Based on a carefully designed external entropy source and the generated sequence, this perturbation scheme solves the degeneration problem of chaotic systems under finite precision, yielding theoretically rigorously provable, implementable, and low-cost non-degenerate chaotic sequences under finite precision. This not only simply and effectively solves the chaotic degradation of digital chaotic systems caused by the finite precision effect, ensuring the chaotic system is immune to finite precision, but also allows for the flexible construction of chaotic sequences to meet various scenarios according to security application requirements. In addition, this invention can also be applied to fields such as cryptography, secure communication, pseudo-random number design, and privacy protection.

[0111] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A system for generating non-degenerate chaotic sequences with finite precision, characterized in that, It is applied in chaotic cryptography, chaotic digital modulation or chaotic pseudo-random number generator, including: perturbation sequence generation module, perturbation module and discrete chaotic system; The perturbation sequence generation module is used to generate an aperiodic sequence as a perturbation sequence after receiving a start command, and to output the perturbation values ​​in the perturbation sequence to the perturbation module every specified number of iterations, starting from the first iteration of the non-degenerate chaotic sequence generation system. The perturbation module is used to initiate the first iteration of the non-degenerate chaotic sequence generation system after receiving the start command, and to receive the initial chaotic value input from the outside as the first iteration. one The system state variables under each iteration; and in each iteration: when the perturbation value input by the perturbation sequence generation module is received, the system state variables under the current iteration are perturbed using the currently input perturbation value to obtain the intermediate state variables under the current iteration, and output to the discrete chaotic system; when the perturbation value input by the perturbation sequence generation module is not received, the system state variables under the current iteration are directly output to the discrete chaotic system as the intermediate state variables under the current iteration. The discrete chaotic system is used to perform chaotic mapping on the intermediate state variables of the current iteration using its iterative mapping function in each iteration, and output the chaotic value of the current iteration. At the same time, the chaotic value of the current iteration is used as the system state variable of the next iteration and fed back to the disturbance module. When in the n When a perturbation exists in the iteration, the non-degenerate chaotic sequence generation system in the... n System state variables under +1 iterations ; When in the n When there is no perturbation in the iteration, the non-degenerate chaotic sequence generation system in the ... n System state variables under +1 iterations ; in, The iterative mapping function for the discrete chaotic system; The parameters of the discrete chaotic system; For the first n The intermediate state variables in the next iteration are specifically determined by the perturbation value. For the first n System state variables under the next iteration The result after the perturbation operation; perturbation value For the first n In the next iteration, the perturbation sequence input into the perturbation module is the first... m One perturbation value; The non-degenerate chaotic sequence generated by the non-degenerate chaotic sequence generation system consists of chaotic values ​​under multiple iterations.

2. The non-degenerate chaotic sequence generation system according to claim 1, characterized in that, The perturbation interval between two adjacent perturbations is determined by the specified number of iterations; The specified number of iterations is either a preset fixed value or a non-negative integer value that changes dynamically with the number of disturbances; When the specified iteration number is a non-negative integer value that dynamically changes with the number of disturbances, each time a disturbance occurs, the number of iterations between the next disturbance and the current disturbance is specified as the current specified iteration number.

3. The non-degenerate chaotic sequence generation system according to claim 1, characterized in that, The discrete chaotic system is obtained by discretizing the continuous chaotic system.

4. A method for generating non-degenerate chaotic sequences with finite precision, characterized in that, The system applied to the non-degenerate chaotic sequence generation system of claim 1 includes: After receiving the start command, the first iteration begins, and the initial chaotic value from the external input is used as the basis for the second iteration. one The system state variables under the next iteration; at the same time, an aperiodic sequence is generated as a perturbation sequence, and starting from the first iteration of the system generated by the non-degenerate chaotic sequence, the perturbation values ​​in the perturbation sequence are sequentially used as the perturbation values ​​under the corresponding iteration every specified number of iterations; In each iteration: When there is a corresponding perturbation value in the current iteration, the perturbation value in the current iteration is used to perturb the system state variable in the current iteration to obtain the intermediate state variable in the current iteration. Then, the intermediate state variable in the current iteration is chaotically mapped using a discrete chaotic system to obtain the chaotic value in the current iteration and output it. At the same time, the chaotic value in the current iteration is used as the system state variable in the next iteration. When there is no corresponding perturbation value in the current iteration, the discrete chaotic system is directly used to perform chaotic mapping on the system state variables in the current iteration, and the chaotic value in the current iteration is output. At the same time, the chaotic value in the current iteration is used as the system state variable in the next iteration. When in the When a perturbation exists in the nth iteration, the nth iteration System state variables under the next iteration ; When in the n When there is no perturbation in the nth iteration, the th n System state variables under +1 iterations ; in, The iterative mapping function for the discrete chaotic system; The parameters of the discrete chaotic system; For the first n The intermediate state variables in the next iteration are specifically determined by the perturbation value. For the first n System state variables under the next iteration The result after the perturbation operation; perturbation value For the first n In the next iteration, the perturbation sequence input into the perturbation module is the first... m One perturbation value; The non-degenerate chaotic sequence generated by the non-degenerate chaotic sequence generation method consists of chaotic values ​​under multiple iterations.

5. The method for generating non-degenerate chaotic sequences according to claim 4, characterized in that, The perturbation interval between two adjacent perturbations is determined by the specified number of iterations; The specified number of iterations is either a preset fixed value or a non-negative integer value that changes dynamically with the number of disturbances; When the specified iteration number is a non-negative integer value that dynamically changes with the number of disturbances, each time a disturbance occurs, the number of iterations between the next disturbance and the current disturbance is specified as the current specified iteration number.

6. The method for generating non-degenerate chaotic sequences according to claim 4, characterized in that, The discrete chaotic system is obtained by discretizing the continuous chaotic system.

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