Method for constructing pseudorandom sequence generator of conservative hyperchaotic system based on FPGA (Field Programmable Gate Array)

Abstract

The invention discloses a method for constructing a pseudorandom sequence of a conservative hyperchaotic system based on an FPGA (Field Programmable Gate Array), which belongs to the field of information security, and comprises the following steps: firstly, respectively constructing a four-dimensional conservative hyperchaotic system and a five-dimensional conservative hyperchaotic system, and then fixing initial values and parameters of the two chaotic systems; the method comprises the following steps of: intercepting low 12 bits output by two hyperchaotic signals as a random source of a pseudo-random sequence generator, dividing each dimension of output signal into high 6 bits and low 6 bits, executing exclusive OR and connection operations, and finally combining results to obtain final output of a pseudo-random sequence. Two conservative hyper-chaotic systems are adopted to design the pseudo-random sequence generator, the safety and throughput are improved by means of the high sensitivity of the pseudo-random sequence generator to an initial value and the pseudo-random characteristic of the pseudo-random sequence generator, the randomness of the sequence is guaranteed by extracting low 12 bits output in each dimension, the utilization rate of chaotic signals is improved, the use of a multiplier is reduced by adopting simple bit operation, and the cost is reduced. Resource occupation is reduced, hardware implementation is facilitated, and the advantage of large throughput is achieved.

Description

Technical Field

[0001] The present invention belongs to the field of information security, and in particular relates to a method for constructing a conservative hyperchaotic system pseudo-random sequence generator based on FPGA. Background Art

[0002] Pseudorandom sequences have numerous applications in fields such as information security, cryptography, and communication systems. Chaotic systems are widely used in these fields due to their inherently high randomness and unpredictability, among other desirable properties. In practical engineering applications, due to the limited computational precision of hardware systems, the implementation of dissipative chaotic systems is often subject to truncation errors, causing the system state to collapse within a finite domain. This phenomenon makes it difficult for hardware-implemented chaotic systems to exhibit ideal chaotic properties. Compared to other chaotic systems, conservative chaotic systems are more sensitive to initial values, exhibit greater unpredictability, and exhibit richer chaotic dynamical properties, making them more suitable for the design of chaos-based pseudorandom sequence generators.

[0003] Currently, most pseudo-random sequence generators are designed based on dissipative chaotic systems, characterized by negative divergence. Because dissipative chaotic systems have fixed attractors, they are susceptible to reconfiguration. Conservative chaotic systems, on the other hand, lack specific attractors and are characterized by the sum of their Lyapunov exponents being zero. Therefore, pseudo-random sequence generators based on conservative chaos are more secure and more suitable for applications in secure communications. Currently, there are relatively few pseudo-random sequence generators based on conservative chaotic systems.

[0004] Conservative chaotic systems include two types, namely Hamiltonian conservative chaotic systems and non-Hamiltonian conservative chaotic systems. The former has Hamiltonian energy conservation and phase volume conservation (i.e. H(X) = 0 and ), while the latter refers to a chaotic system in which one or both of the energy and volume are zero (i.e. H(X)≠0 or ). Currently, most Hamiltonian conservative chaotic systems use the construction method of generalized Hamiltonian chaotic systems, the specific contents are as follows:

[0005] For an n-dimensional smooth nonlinear differential system, the Hamiltonian equivalent is used to convert it into a matrix differential system, and the generalized Hamiltonian system formula is obtained. Where x is the state variable and x∈R n , the structure matrix J(x) is a skew-symmetric matrix, representing the energy conservation part of the system, satisfying J(x)∈R n×n And J(x)=-J T (x); It is the gradient vector of the Hamiltonian function H(x). The change of H(x) is used to represent the energy exchange between the system and the outside world. H(x)∈R n×nThe gradient vector is expressed as follows: Among them, g(x l ) is about x l The variable skew-symmetric matrix is ​​expressed as follows:

[0006]

[0007] Among them, f i,j To exclude x i and x j function.

[0008] Chaotic systems designed using these methods often have numerous nonlinear terms. For example, the structural matrix of a four-dimensional chaotic system might contain 12 elements, while that of a five-dimensional chaotic system might contain 20. This excessive number of elements complicates the implementation of chaotic systems in practical circuits and increases the resource consumption of pseudo-random sequence generators implemented on FPGAs. Pseudo-random sequence generators based on conservative hyperchaotic system designs that offer high throughput, high security, and low hardware resource consumption are rare. Summary of the Invention

[0009] Based on the above shortcomings, the purpose of the present invention is to propose a method for constructing a conservative hyperchaotic system pseudo-random sequence generator based on FPGA to solve the current problems of insufficient throughput, low security, and high resource consumption of pseudo-random sequence generators.

[0010] The technical solution adopted by the present invention is as follows: a method for constructing a conservative hyperchaotic system pseudo-random sequence generator based on FPGA, the steps are as follows:

[0011] Step 1: Construct a four-dimensional conservative hyperchaotic system containing 6 terms and a five-dimensional conservative hyperchaotic system containing 6 terms respectively;

[0012] Step 2: Fix the initial values ​​and parameters of the two conservative hyperchaotic systems, design the top-level architecture, use 32-bit signed fixed-point representation, replace multiplication with left shift, and replace division with right shift;

[0013] Step 3: Convert the chaotic sequence output from step 2 into binary representation. The conversion formula is shown in formula (1):

[0014]

[0015] Among them, <<< and They represent left shift and rounding functions respectively, and the function dec2bin(·) converts decimal to binary;

[0016] Step 4: Based on the valid signal outputted in step 3, the lower 12 bits of the output signals of the two hyperchaotic systems are intercepted as the source of randomness of the pseudo-random sequence generator;

[0017] Step 5: Split and connect: Split each dimension of the output signal from step 4 into the upper 6 bits and lower 6 bits, perform XOR and connect operations, and the rules are as follows:

[0018] in, and Respectively represent the high 6 bits of the i-th dimension output of step 4; and Respectively represent the lower 6 bits of the i-th dimension output of step 4; represents bitwise exclusive OR operation, and {·} represents sequential connection;

[0019] Step 6: Combine the results of step 5 to obtain the final output of the pseudo-random sequence.

[0020] Furthermore, in step 1, the equation of the four-dimensional conservative hyperchaotic system is shown in formula (2),

[0021]

[0022] Among them, a, b, c represent parameters, (x1, x2, x3, x4) are system state variables;

[0023] The equation of the five-dimensional conservative hyperchaotic system is shown in formula (3),

[0024]

[0025] Where a, b, and c represent parameters, and (x1, x2, x3, x4, x5) are system state variables.

[0026] Furthermore, in step 2, the initial values ​​of the four-dimensional conservative hyperchaotic system are (1.3, 1.9, 1.7, 2.1), and the parameters are (a, b, c) = (2, 2.3, 2); the initial values ​​of the five-dimensional conservative hyperchaotic system are (1, 1, 1, 1, 1), and the parameters are (a, b, c) = (5, 5, 5).

[0027] Another object of the present invention is achieved through an FPGA-based encryption system, which executes the above-mentioned method for constructing a conservative hyperchaotic system pseudo-random sequence generator based on FPGA, generates an encryption key after the initial key of the host computer passes through the constructed pseudo-random sequence generator, and then encrypts it with the plaintext data sent by the host computer; the decryption process uses the same initial key as the encryption, generates a decryption key through the pseudo-random sequence generator, and decrypts it with the ciphertext data sent by the host computer.

[0028] The present invention leverages the high sensitivity of conservative chaotic systems to initial values ​​and their pseudo-random nature to enhance the security of pseudo-random sequences and improve throughput. This approach ensures the randomness of the output sequence while increasing the utilization of the chaotic system's output signal. The present invention employs simple bitwise operations, reducing the number of multipliers required. This reduces resource utilization, facilitates hardware implementation, and offers high throughput. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Figure 1 This is a schematic diagram of the structural framework of a conservative hyperchaotic system pseudo-random sequence generator based on FPGA;

[0030] Figure 2 MATLAB simulation of the implementation example shows the phase diagram of the 4-dimensional conservative hyperchaotic system and the corresponding Poincare section diagram. (a) Phase diagram of the 4-dimensional conservative hyperchaotic system, (b) Poincare section diagram of the 4-dimensional conservative hyperchaotic system;

[0031] Figure 3 MATLAB simulation of the implementation example shows the phase diagram of the 5-dimensional conservative hyperchaotic system and the corresponding Poincare section diagram. (a) Phase diagram of the 5-dimensional conservative hyperchaotic system, (b) Poincare section diagram of the 5-dimensional conservative hyperchaotic system.

[0032] Figure 4 The long-term Hamiltonian energy curve and system phase diagram of the 4-dimensional conservative hyperchaotic system are obtained by MATLAB simulation of the implementation example;

[0033] Figure 5 The long-term Hamiltonian energy curve diagram of the 5-dimensional conservative hyperchaotic system is obtained by MATLAB simulation of the implementation example;

[0034] Figure 6 The Lyapunov exponent diagram and phase diagram of the 4-dimensional conservative hyperchaotic system are obtained by MATLAB simulation of the implementation example;

[0035] Figure 7 The Lyapunov exponent diagram and bifurcation diagram of the 5-dimensional conservative hyperchaotic system are obtained by MATLAB simulation of the implementation example;

[0036] Figure 8 Two chaotic system oscilloscope display results are obtained for the implementation example using FPGA; (a) 4-dimensional conservative hyperchaotic system and its time series, (b) 4-dimensional conservative hyperchaotic system, (c) 5-dimensional conservative hyperchaotic system and its time series, (d) 5-dimensional conservative hyperchaotic system;

[0037] Figure 9 The RTL circuit diagram obtained for the pseudo-random sequence generator using FPGA;

[0038] Figure 10 Waveform captured by ILA for the FPGA implementation of the pseudo-random sequence generator;

[0039] Figure 11 This is the top-level architecture diagram of the encryption system;

[0040] Figure 12 Figure 1 is the encryption / decryption result diagram of the encryption system; (a) initial key, (b) plaintext, (c) encryption result, (d) decryption key, (e) ciphertext, and (f) decryption result. DETAILED DESCRIPTION

[0041] The present invention will be described in further detail below with reference to implementation examples and accompanying drawings, but the implementation of the present invention is not limited thereto.

[0042] Example 1

[0043] like Figure 1 As shown in Figure 1, a pseudo-random sequence construction method for a conservative chaotic system based on FPGA is as follows:

[0044] First, a four-dimensional conservative hyperchaotic system is constructed and its chaotic characteristics are verified. The software simulation platform uses Matlab2022b software, and the embedded ODE45 is used to solve the differential equation. The step size is set to 0.01, the number of iterations is 2000, and the relative error and absolute error are 10 -8 .

[0045] The specific formula is as follows:

[0046]

[0047] Among them, a, b and c represent parameters, (x1, x2, x3, x4) are system state variables,

[0048] It can be seen that the designed four-dimensional conservative hyperchaotic system formula contains only 6 terms. Assuming the initial values ​​of the system are (1.3, 1.9, 1.7, 2.1), the parameters are (a, b, c) = (2, 2.3, 2), the Lyapunov index (LE index) of the system is LE1 = 0.53, LE2 = 0, LE3 = 0, LE4 = -0.53, the sum of the LE index of the system is 0, the Kaplan Yorke dimension is 4, which is equal to the system dimension, verifying that the system is a conservative chaotic system. The specific phase diagram and Poincare section diagram are as follows Figure 2 When the parameters of the four-dimensional conservative hyperchaotic system are (a, b, c) = (2, b∈[-50,50], 2) and the initial value is IC = [1.1, 1.9, 1.7, 2.1], the system LE index diagram and phase diagram are as follows: Figure 6 As shown, it is proved that the four-dimensional conservative chaotic system equation has good chaotic characteristics.

[0049] Then, a five-dimensional conservative hyperchaotic system is constructed and its chaotic characteristics are verified. The software simulation platform uses Matlab2022b software, and the embedded ODE45 is used to solve the differential equation. The step size is set to 0.01, the number of iterations is 2000, and the relative error and absolute error are 10 -8 The specific formula is as follows:

[0050]

[0051] Among them, a, b and c represent parameters, (x1, x2, x3, x4, x5) are system state variables, and it can be seen that the designed five-dimensional conservative hyperchaotic system formula contains only 6 terms. Assuming the initial value of the system is (1, 1, 1, 1, 1), the parameters are (a, b, c) = (5, 5, 5), and the Lyapunov index (LE index) of the system is LE1 = 1.28, LE2 = 0.01, LE3 = 0, LE4 = -0.01, LE5 ​​= -1.28. The sum of the LE index of the system is 0, and the Kaplan Yorke dimension is 5, which is equal to the system dimension, verifying that the system is a conservative chaotic system. The specific phase diagram and Poincare section diagram are as follows Figure 3 When the parameters (a, b, c) = (a∈[-5,5],5,5), IC = (1,1,1,1,1), the system LE index diagram and bifurcation diagram are as follows. Figure 7 As shown, it is proved that the five-dimensional conservative chaotic system equation has good chaotic characteristics.

[0052] Verify the conservatism of the system's Hamiltonian energy:

[0053] For the rate of change of the phase volume of a four-dimensional conservative chaotic system at time t and divergence can be expressed as follows:

[0054]

[0055] Combining the above two formulas, we can get This means that the phase volume of the four-dimensional conservative chaotic system is conserved. At the same time, the Hamiltonian energy of the system at time t is expressed as:

[0056]

[0057] The rate of change of the Hamiltonian energy of the system is:

[0058]

[0059] available This means that the Hamiltonian energy of the four-dimensional conservative hyperchaotic system is conserved, and its Hamiltonian energy is always determined by the initial value of the system, H1(X(t))=H1(X(0)). The long-term Hamiltonian energy curve and phase diagram of the system are as follows: Figure 4 As shown, it is proved that the Hamiltonian energy of the system is conserved for a long time. The specific calculation formula is as follows:

[0060]

[0061] The same method can be used to verify the five-dimensional conservative chaotic system, and the Hamiltonian energy of the system is conserved and always determined by the initial value of the system, H2(X(t)) = H2(X(0)). To avoid repetition, we will not elaborate on it. The long-term Hamiltonian energy curve and phase diagram of the system are as follows: Figure 5 As shown, it is proved that the Hamiltonian energy of the system is conserved for a long time. The specific calculation formula is as follows:

[0062]

[0063] Next, we implemented the proposed four- and five-dimensional conservative hyperchaotic systems on an FPGA and used an oscilloscope for visualization. The primary platform used was an AX7103 development board with the XC7A100T-2FGG484I chip, an AN9767 dual-channel 14-bit DA output module, and an oscilloscope. We first discretized the two chaotic states using the improved Euler algorithm, using the following formula:

[0064]

[0065] It can be seen that using the improved Euler algorithm to solve the chaotic system equation only involves two steps, which can reduce resource consumption. Then connect the AX7103 development board to the dual-channel 14-bit DA output module AN9767, and then connect the oscilloscope to observe the chaotic system implementation results on the oscilloscope. Since the value range of the four-dimensional conservative hyperchaotic system is [-5,5], it is necessary to add a positive value of 5, and at the same time truncate the integer digits by 3 and the decimal digits by 11 to display on the oscilloscope; and the value range of the five-dimensional conservative hyperchaotic system is [-3,3], so add a positive value of 3, truncate the integer digits by 2 and the decimal digits by 12 to display on the oscilloscope, and the display result is as follows Figure 8 As shown in Figure 2, it can be seen that the oscilloscope display results are roughly consistent with the Matlab software simulation results.

[0066] Example 2

[0067] Implement the pseudo-random sequence generator in hardware and analyze the results:

[0068] After the system is implemented on hardware, the RTL level (register transfer level) view synthesized by Xilinx Vivado 2019.2 is as follows Figure 9 As shown, the proposed pseudo-random sequence generator algorithm includes a chaotic signal generator module based on four-dimensional and five-dimensional conservative chaotic systems, a truncation module, a segmentation and union module, and a PRNG union module. The proposed pseudo-random sequence generator algorithm is implemented in Verilog HDL (Hardware Description Language) using an AX7103 development board. Furthermore, the proposed pseudo-random sequence generator algorithm uses bit operations, making it easy to implement on a digital platform. The overall circuit is simple and efficient.

[0069] The pseudo-random sequence generator designed in this example contains 9 32-bit initial values ​​and 6 parameters. If the parameters are not considered, the key space is 2 32×9 =2 288 However, if the parameters are considered, the key space of the pseudo-random sequence is greater than 2 288 , much larger than 2 100 , which can effectively resist brute-force decryption attacks. Therefore, the pseudo-random sequence generator designed by the present invention can generate a pseudo-random sequence with a larger key space and higher security.

[0070] In order to test the randomness of the generated pseudo-random sequence, the NIST SP800-22 randomness test suite of the National Institute of Standards and Technology (NIST) was used as a statistical analysis tool. The test suite consists of 15 statistical tests designed to comprehensively evaluate the randomness of the binary sequence generated by the random number generator. The distribution of P-value and Pass rate is used to test the randomness of the tested sequence. If P-value>0.01 and Pass rate>0.98, the sequence passes the sub-test. The present invention tested 108 sequences of length 10 6 The test results of binary sequences are shown in Table 1, which proves that the binary sequences generated by the proposed pseudo-random sequence generator have good randomness, because they pass all test items and achieve good results.

[0071] Table 1 NIST SP800-22 test results

[0072]

[0073] (*):average value

[0074] Data throughput and resource consumption are important indicators for measuring a good pseudo-random sequence generator. The amount of data generated per unit time is the data throughput. The greater the data throughput of a random sequence, the more pseudo-random sequences are generated in the same amount of time. A good pseudo-random sequence generator should be able to generate sequences with stronger randomness, while having greater throughput and less resource consumption. After implementing the proposed pseudo-random sequence generator on the FPGA, the digital signal waveform inside the FPGA was captured in real time using the Vivado ILA tool, as shown in Figure 2. Figure 10 As shown, the data signals x1_out, y1_out, …, z2_out, and w2_out are the output signals of the chaotic signal generator, prng_out and prng_out_valid are the output data and output valid signal of the pseudo-random sequence generator, respectively. clk_sys is the FPGA system clock (50 MHz). Since the ILA captures signals at a fixed frequency, it cannot accurately capture certain high-level signals. However, the intervals between the cyan cursors show that an average of 108 bits of pseudo-random sequence is output every three system clocks, resulting in a data throughput of 1800 Mb / s. This demonstrates the excellent performance of the pseudo-random sequence generator designed in this invention.

[0075] Example 3

[0076] Apply this to a pseudorandom sequence generator and display the results:

[0077] An encryption system was designed as an application example, centered around a conservative hyperchaotic pseudo-random sequence generator based on an FPGA. After completing simulation and RTL synthesis of each submodule in the Vivado 2019.1 design suite, the entire system netlist was synthesized. Finally, the system was downloaded to an FPGA development board, where the encryption and decryption process could be observed on the board's built-in LCD screen. A state machine control and management model was adopted, with each module in the system controlled by a master state machine module.

[0078] The encryption system of this invention generates an encryption key by passing the host computer's initial key through a conservative hyperchaotic pseudorandom sequence generator. This key is then used to encrypt the plaintext data sent by the host computer. The decryption process uses the same initial key used for encryption, generates a decryption key through the chaotic system, and decrypts the ciphertext data sent by the host computer. The decryption process is the inverse of encryption. This system uses XOR encryption, using the same system for both encryption and decryption. The principle behind this is that after two XOR cycles, the original data is restored. Figure 11 This is the top-level architecture diagram of the encryption system. This design follows the top-down design principle and includes seven modules: transceiver module enc_uart, main state machine module enc_fsm, conservative hyperchaotic system pseudo-random sequence generator module key_core, data cache module data_function, key cache module key_ram, encryption module enc_function, and LCD driver module dync_lcd.

[0079] like Figure 12 The following is the encryption / decryption result diagram of the encryption system. Figure 12 Figures (a)-(c) illustrate the encryption process. The host computer sends the 128-bit hexadecimal initial key 12345678_9ABCDEF1_23456789_ABCDEF12, followed by the 128-bit hexadecimal plaintext 111111111_22222222_333333333_44444444. When the encryption (key1) button is pressed on the development board, the system returns the encryption result E8DB3BBE_1CDD46A0_215D20BA_3527A9F9, which is the 128-bit hexadecimal ciphertext. Figure 12 (d)-(f) represent the decryption process. The same initial key is sent for encryption, followed by the ciphertext E8DB3BBE_1CDD46A0_215D20BA_3527A9F9 to the FPGA. The key (key2) is then decrypted to obtain the 128-bit hexadecimal decryption result 11111111_22222222_33333333_44444444. Figure 12It can be seen that the system can completely recover the plaintext 11111111_22222222_33333333_44444444, and the system can correctly perform encryption / decryption. This shows that the pseudo-random sequence generator designed by the present invention can be applied to encryption systems and can obtain better encryption results.

[0080] The above description is merely a specific example of the present invention and does not limit the manner in which the present invention is to be practiced. However, those skilled in the art will recognize that any modifications, equivalent substitutions, or improvements that violate the spirit and principles of the present invention are prohibited. Any modifications, equivalent substitutions, or improvements that are made within the spirit and principles of the present invention are intended to be within the scope of protection of the claims.

Claims

1. A method for constructing a conservative hyperchaotic system pseudo-random sequence generator based on FPGA, characterized in that: Here are the steps: Step 1: Construct a four-dimensional conservative hyperchaotic system containing 6 terms and a five-dimensional conservative hyperchaotic system containing 6 terms respectively; Step 2: Fix the initial values ​​and parameters of the two conservative hyperchaotic systems, design the top-level architecture, use 32-bit signed fixed-point representation, replace multiplication with left shift, and replace division with right shift; Step 3: Convert the chaotic sequence output from step 2 into binary representation. The conversion formula is shown in formula (1): Among them, <<< and They represent left shift and rounding functions respectively, and the function dec2bin(·) converts decimal to binary; Step 4: Based on the valid signal outputted in step 3, the lower 12 bits of the output signals of the two hyperchaotic systems are intercepted as the source of randomness of the pseudo-random sequence generator; Step 5: Split and connect: Split each dimension of the output signal from step 4 into the upper 6 bits and lower 6 bits, perform XOR and connect operations, and the rules are as follows: in, and Respectively represent the high 6 bits of the i-th dimension output of step 4; and Respectively represent the lower 6 bits of the i-th dimension output of step 4; represents bitwise exclusive OR operation, and {·} represents sequential connection; Step 6: Combine the results of step 5 to obtain the final output of the pseudo-random sequence.

2. The method for constructing a conservative hyperchaotic system pseudo-random sequence generator based on FPGA according to claim 1, characterized in that: In step one, The equation of the four-dimensional conservative hyperchaotic system is shown in formula (2), Among them, a, b, c represent parameters, (x1, x2, x3, x4) are system state variables; The equation of the five-dimensional conservative hyperchaotic system is shown in formula (3), Where a, b, and c represent parameters, and (x1, x2, x3, x4, x5) are system state variables.

3. The method for constructing a conservative hyperchaotic system pseudo-random sequence generator based on FPGA according to claim 2, characterized in that: In step 2, the initial values ​​of the four-dimensional conservative hyperchaotic system are (1.3, 1.9, 1.7, 2.1), and the parameters are (a, b, c) = (2, 2.3, 2); the initial values ​​of the five-dimensional conservative hyperchaotic system are (1, 1, 1, 1, 1), and the parameters are (a, b, c) = (5, 5, 5).

4. An encryption system based on FPGA, characterized in that: The encryption system executes the method for constructing a pseudo-random sequence generator for a conservative hyperchaotic system based on FPGA according to any one of claims 1 to 3, generates an encryption key after passing the initial key of the host computer through the constructed pseudo-random sequence generator, and then encrypts the encryption key with the plaintext data sent by the host computer; The decryption process uses the same initial key as the encryption key, generates a decryption key through a pseudo-random sequence generator, and decrypts the ciphertext data sent by the host computer.

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