A four-dimensional rotationally symmetric conservative chaotic system

Abstract

The present invention provides a four-dimensional rotationally symmetric conservative chaotic system, which is implemented based on a four-dimensional rotationally symmetric conservative chaotic circuit. The four-dimensional rotationally symmetric conservative chaotic circuit includes a first-channel circuit, a second-channel circuit, a third-channel circuit, and a fourth-channel circuit. The present invention is implemented based on a conservative chaotic circuit with a special mathematical structure and excellent performance, and has excellent trajectory traversal, ensuring the security of data encryption.

Description

Technical Field

[0001] The present invention relates to a chaotic system applied to the field of data encryption. Specifically, it particularly relates to a four-dimensional rotationally symmetric conservative chaotic system. Background Art

[0002] Since the first chaotic system was proposed in 1963, the chaos theory has attracted extensive discussions among scholars, and new systems have been continuously proposed. With the integrated application of chaotic systems in the field of data encryption, general chaotic systems can no longer meet the actual requirements of encryption systems. Chinese Patent with Application No. 202011009469.3 discloses "A Generalized Sprott-A System with Double-Cluster Conservative Chaotic Flows and Its Construction Method". Although this system can exhibit the dynamic characteristics of double-cluster conservative chaotic flows, due to the general performance of its mathematical model, the generated phase trajectory diagram has poor ergodicity in the phase plane. In the field of data encryption, data encryption based on this system is easily cracked. Summary of the Invention

[0003] In view of the technical problem that the general chaotic model proposed above is easily cracked when used for data encryption, a four-dimensional rotationally symmetric conservative chaotic system is provided. The present invention is implemented based on a conservative chaotic circuit with a special mathematical structure and excellent performance, and has excellent trajectory ergodicity to ensure the security of data encryption.

[0004] The technical means adopted by the present invention are as follows:

[0005] A four-dimensional rotationally symmetric conservative chaotic system, the system model of which is:

[0006]

[0007] where x, y, z, and w are state variables;

[0008] The system model is implemented based on a four-dimensional rotationally symmetric conservative chaotic circuit, and the four-dimensional rotationally symmetric conservative chaotic circuit includes a first channel circuit, a second channel circuit, a third channel circuit, and a fourth channel circuit, where:

[0009] The first channel circuit includes a first non-linear voltage converter, first to seventh resistors, first to third operational amplifiers, and a first capacitor;

[0010] The second channel circuit includes a second non-linear voltage converter, eighth to fourteenth resistors, fourth to sixth operational amplifiers, and a second capacitor;

[0011] The third channel circuit includes a third non-linear voltage converter, fifteenth to twenty-first resistors, seventh to ninth operational amplifiers, and a third capacitor;

[0012] The fourth-channel circuit includes a fourth non-linear voltage converter, resistors R22 to R28, operational amplifiers U10 to U12, and a fourth capacitor.

[0013] Further, in the first-channel circuit:

[0014] The first input terminal of the first non-linear voltage converter is the -z signal. The inverted voltage source at the output terminal is grounded, and the non-inverted voltage source is connected to the left end of the first resistor. The left end of the second resistor is connected to the -y signal, and the left end of the third resistor is connected to the w signal. The right ends of the first to third resistors are connected in parallel and then connected to the inverting input terminal of the first operational amplifier.

[0015] The non-inverting input terminal of the first operational amplifier is grounded, and the fourth resistor connects the inverting input terminal of the first operational amplifier to the output terminal.

[0016] The non-inverting input terminal of the second operational amplifier is grounded. The inverting input terminal is connected to the output terminal of the first operational amplifier through the fifth resistor and also connected to the output terminal of the second operational amplifier through the first capacitor. The output terminal is the -x signal.

[0017] The non-inverting input terminal of the third operational amplifier is grounded. The inverting input terminal is connected to the output terminal of the second operational amplifier through the sixth resistor and also connected to the output terminal of the third operational amplifier through the seventh resistor. The output terminal is the x signal.

[0018] Further, in the second-channel circuit:

[0019] The first input terminal of the second non-linear voltage converter is the -w signal. The inverted voltage source at the output terminal is grounded, and the non-inverted voltage source is connected to the left end of the eighth resistor. The left end of the ninth resistor is connected to the -z signal, and the left end of the tenth resistor is connected to the x signal. The right ends of the eighth to tenth resistors are connected in parallel and then connected to the inverting input terminal of the fourth operational amplifier.

[0020] The non-inverting input terminal of the fourth operational amplifier is grounded, and the eleventh resistor connects the inverting input terminal of the fourth operational amplifier to the output terminal.

[0021] The non-inverting input terminal of the fifth operational amplifier is grounded. The inverting input terminal is connected to the output terminal of the fourth operational amplifier through the twelfth resistor and also connected to the output terminal of the fifth operational amplifier through the second capacitor. The output terminal is the -y signal.

[0022] The non-inverting input terminal of the sixth operational amplifier is grounded. The inverting input terminal is connected to the output terminal of the fifth operational amplifier through the thirteenth resistor and also connected to the output terminal of the sixth operational amplifier through the fourteenth resistor. The output terminal is the y signal.

[0023] Further, in the third-channel circuit:

[0024] The first input terminal of the third non-linear voltage converter is the -x signal. The inverting voltage source at the output terminal is grounded, and the non-inverting voltage source is connected to the left end of the fifteenth resistor. The left end of the sixteenth resistor is connected to the -w signal, and the left end of the seventeenth resistor is connected to the y signal. The right ends of the fifteenth to seventeenth resistors are connected in parallel and then connected to the inverting input terminal of the seventh operational amplifier;

[0025] The non-inverting input terminal of the seventh operational amplifier is grounded, and the eighteenth resistor connects the inverting input terminal of the seventh operational amplifier to the output terminal;

[0026] The non-inverting input terminal of the eighth operational amplifier is grounded. Its inverting input terminal is connected to the output terminal of the seventh operational amplifier through the nineteenth resistor and also connected to the output terminal of the eighth operational amplifier through the third capacitor. The output terminal is the -z signal;

[0027] The non-inverting input terminal of the ninth operational amplifier is grounded. Its inverting input terminal is connected to the output terminal of the eighth operational amplifier through the twentieth resistor and also connected to the output terminal of the ninth operational amplifier through the twenty-first resistor. The output terminal is the z signal.

[0028] Furthermore, in the fourth channel circuit:

[0029] The first input terminal of the fourth non-linear voltage converter is the -y signal. The inverting voltage source at the output terminal is grounded, and the non-inverting voltage source is connected to the left end of the twenty-second resistor. The left end of the twenty-third resistor is connected to the -x signal, and the left end of the twenty-fourth resistor is connected to the z signal. The right ends of the twenty-second to twenty-fourth resistors are connected in parallel and then connected to the inverting input terminal of the tenth operational amplifier;

[0030] The non-inverting input terminal of the tenth operational amplifier is grounded, and the twenty-fifth resistor connects the inverting input terminal of the tenth operational amplifier to the output terminal;

[0031] The non-inverting input terminal of the eleventh operational amplifier is grounded. Its inverting input terminal is connected to the output terminal of the tenth operational amplifier through the twenty-sixth resistor and also connected to the output terminal of the eleventh operational amplifier through the fourth capacitor. The output terminal is the -w signal;

[0032] The non-inverting input terminal of the twelfth operational amplifier is grounded. Its inverting input terminal is connected to the output terminal of the eleventh operational amplifier through the twenty-seventh resistor and also connected to the output terminal of the twelfth operational amplifier through the twenty-eighth resistor. The output terminal is the w signal.

[0033] Compared with the prior art, the present invention has the following advantages:

[0034] 1. The mathematical model of the chaotic system constructed by the present invention in a cyclic symmetric form is constructed by state equations with the same structure but different positions of state variables, and has characteristics such as conservation of phase volume, no attractor, and integer-dimensional system dimension. It has better ergodicity compared to dissipative chaotic systems.

[0035] 2. The present invention introduces a cyclic symmetric structure into a conservative chaotic system and gives a realization of a four-dimensional cyclic symmetric conservative chaotic circuit. Due to the special cyclic symmetric structure, the chaotic system constructed by the conservative chaotic circuit has excellent performance, and the generated phase diagram has no attractor, further improving the ergodicity of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0037] Figure 1 It is the circuit diagram of the four-dimensional cyclic symmetric conservative chaotic circuit in the present invention.

[0038] Figure 2 It is the X-Y phase diagram of the four-dimensional cyclic symmetric conservative chaotic system in the present invention.

[0039] Figure 3 It is the X-Z phase diagram of the four-dimensional cyclic symmetric conservative chaotic system in the present invention.

[0040] Figure 4 It is the X-W phase diagram of the four-dimensional cyclic symmetric conservative chaotic system in the present invention.

[0041] In the figure: R1 to R28 are the first to twenty-eighth resistors, A1 to A4 are the first to fourth non-linear voltage converters, U1 to U12 are the first to twelfth operational amplifiers; C1 to C4 are the first to fourth capacitors. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0042] In order to enable those skilled in the art to better understand the solution of the present invention, the following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0043] As Figure 1 shown, the embodiment of the present invention provides a four-dimensional cyclic symmetric conservative chaotic system, and the system model is:

[0044]

[0045] Among them, x, y, z, and w are state variables.

[0046] The system model is implemented based on a four-dimensional rotationally symmetric conservative chaotic circuit. The four-dimensional rotationally symmetric conservative chaotic circuit includes a first-channel circuit, a second-channel circuit, a third-channel circuit, and a fourth-channel circuit. The first-channel circuit includes a first non-linear voltage converter, first to seventh resistors, first to third operational amplifiers, and a first capacitor. The second-channel circuit includes a second non-linear voltage converter, eighth to fourteenth resistors, fourth to sixth operational amplifiers, and a second capacitor. The third-channel circuit includes a third non-linear voltage converter, fifteenth to twenty-first resistors, seventh to ninth operational amplifiers, and a third capacitor. The fourth-channel circuit includes a fourth non-linear voltage converter, twenty-second to twenty-eighth resistors, tenth to twelfth operational amplifiers, and a fourth capacitor.

[0047] Specifically, the first-channel circuit includes first to third operational amplifiers U1 to U3, a first capacitor C1, a first non-linear voltage converter A1, and first to seventh resistors R1 to R7. The first input terminal V(1) of the first non-linear voltage converter A1 is the -z signal. The inverting voltage source at the output terminal is grounded, and the non-inverting voltage source is connected to the left end of the first resistor R1. The left end of the second resistor R2 is connected to the -y signal. The left end of the third resistor R3 is connected to the w signal. The right ends of the first to third resistors R1 to R3 are connected in parallel and then connected to the inverting input terminal of the first operational amplifier U1. The non-inverting input terminal of the first operational amplifier U1 is grounded. The fourth resistor R4 connects the inverting input terminal of the first operational amplifier U1 to the output terminal. The non-inverting input terminal of the second operational amplifier U2 is grounded. Its inverting input terminal is connected to the output terminal of the first operational amplifier U1 through the fifth resistor R5 and is also connected to the output terminal of the second operational amplifier U2 through the first capacitor C1. The output terminal is the -x signal. The non-inverting input terminal of the third operational amplifier U3 is grounded. Its inverting input terminal is connected to the output terminal of the second operational amplifier U2 through the sixth resistor R6 and is also connected to the output terminal of the third operational amplifier U3 through the seventh resistor R7. The output terminal is the x signal.

[0048] The second channel circuit includes fourth to sixth operational amplifiers U4 to U6, a second capacitor C2, a second non-linear voltage converter A2, and eighth to fourteenth resistors R8 to R14. The first input terminal V(1) of the second non-linear voltage converter A2 is the -w signal, the inverting voltage source at the output terminal is grounded, the non-inverting voltage source is connected to the left end of the eighth resistor R8, the left end of the ninth resistor R9 is connected to the -z signal, the left end of the tenth resistor R10 is connected to the x signal. The right ends of the eighth to tenth resistors R8 to R10 are connected in parallel and then connected to the inverting input terminal of the fourth operational amplifier U4. The non-inverting input terminal of the fourth operational amplifier U4 is grounded. The eleventh resistor R11 connects the inverting input terminal and the output terminal of the fourth operational amplifier U4. The non-inverting input terminal of the fifth operational amplifier U5 is grounded. Its inverting input terminal is connected to the output terminal of the fourth operational amplifier U4 through the twelfth resistor R12 and is also connected to the output terminal of the fifth operational amplifier U5 through the second capacitor C2. The output terminal is the -y signal. The non-inverting input terminal of the sixth operational amplifier U6 is grounded. Its inverting input terminal is connected to the output terminal of the fifth operational amplifier U5 through the thirteenth resistor R13 and is also connected to the output terminal of the sixth operational amplifier U6 through the fourteenth resistor R14. The output terminal is the y signal.

[0049] The third channel circuit includes seventh to ninth operational amplifiers U7 to U9, a third capacitor C3, a third non-linear voltage converter A3, and fifteenth to twenty-first resistors R15 to R21. The first input terminal of the third non-linear voltage converter A3 is the -x signal, the inverting voltage source at the output terminal is grounded, the non-inverting voltage source is connected to the left end of the fifteenth resistor R15, the left end of the sixteenth resistor R16 is connected to the -w signal, the left end of the seventeenth resistor R17 is connected to the y signal. The right ends of the fifteenth to seventeenth resistors R15 to R17 are connected in parallel and then connected to the inverting input terminal of the seventh operational amplifier U7. The non-inverting input terminal of the seventh operational amplifier U7 is grounded. The eighteenth resistor R18 connects the inverting input terminal and the output terminal of the seventh operational amplifier U7. The non-inverting input terminal of the eighth operational amplifier U8 is grounded. Its inverting input terminal is connected to the output terminal of the seventh operational amplifier U7 through the nineteenth resistor R19 and is also connected to the output terminal of the eighth operational amplifier U8 through the third capacitor C3. The output terminal is the -z signal. The non-inverting input terminal of the ninth operational amplifier U9 is grounded. Its inverting input terminal is connected to the output terminal of the eighth operational amplifier U8 through the twentieth resistor R20 and is also connected to the output terminal of the ninth operational amplifier U9 through the twenty-first resistor R21. The output terminal is the z signal.

[0050] The fourth-channel circuit includes operational amplifiers U10 to U12, a fourth capacitor C4, a fourth non-linear voltage converter A4, and resistors R22 to R28. The first input terminal of the fourth non-linear voltage converter A4 is the -y signal, the inverted voltage source at the output terminal is grounded, and the non-inverted voltage source is connected to the left end of the twenty-second resistor R22. The left end of the twenty-third resistor R23 is connected to the -x signal, and the left end of the twenty-fourth resistor R24 is connected to the z signal. The right ends of the twenty-second to twenty-fourth resistors R22 to R24 are connected in parallel and then connected to the inverted input terminal of the tenth operational amplifier U10. The non-inverted input terminal of the tenth operational amplifier U10 is grounded. The twenty-fifth resistor R25 connects the inverted input terminal of the tenth operational amplifier to its output terminal. The non-inverted input terminal of the eleventh operational amplifier U11 is grounded. Its inverted input terminal is connected to the output terminal of the tenth operational amplifier U10 through the twenty-sixth resistor R26 and is also connected to the output terminal of the eleventh operational amplifier U11 through the fourth capacitor C4. The output terminal is the -w signal. The non-inverted input terminal of the twelfth operational amplifier U12 is grounded. Its inverted input terminal is connected to the output terminal of the eleventh operational amplifier U11 through the twenty-seventh resistor R27 and is also connected to the output terminal of the twelfth operational amplifier U12 through the twenty-eighth resistor R28. The output terminal is the w signal.

[0051] The non-linear voltage converters A1 to A4 all use NONLINEAR_DEPENDENT in Multisim software.

[0052] The models of the operational amplifiers U1 to U12 are all TL082IP. The inverted power supply voltage VDD = -12V, and the non-inverted power supply voltage VCC = 12V.

[0053] The selected values of the resistors and capacitors with relatively good experimental effects in the circuit have been indicated in Figure 1 the figure.

[0054] The working principle of the present invention is as follows:

[0055] Taking the first-channel circuit as an example, the first to third resistors R1 to R3 are connected in parallel and then connected to the inverted input terminal of the first operational amplifier U1. The inverted input terminal of the first operational amplifier U1 is connected to its output terminal through the third resistor R3, forming an adder circuit. It should be noted that the left end of the first resistor R1 is connected to the non-linear voltage converter A1. The inverted input terminal of the second operational amplifier U2 is connected to the output terminal of the first operational amplifier U1 through the fifth resistor R5 and is also connected to its output terminal through the first capacitor C1, forming an integrator circuit to generate the -x signal. The sixth resistor R6 is connected to the inverted input terminal of the third operational amplifier U3, and the seventh resistor R7 connects the inverted input terminal of the third operational amplifier U3 to its output terminal, forming an inverter to convert the -x signal into the x signal.

[0056] The working principle of other channel circuits is the same as that of the first channel circuit, which will not be elaborated here.

[0057] Figures 2 to 4 It proves the rationality of the circuit of the present invention. The phase diagram generated by the circuit of the present invention has no attractor and has good ergodicity, which conforms to the characteristics of a conservative chaotic system.

[0058] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions recorded in the foregoing embodiments, or perform equivalent replacements on some or all of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A four-dimensional rotationally symmetric conservative chaotic system for data encryption, characterized in that, the system model is: where x, y, z, w are state variables; the system model is implemented based on a four-dimensional rotationally symmetric conservative chaotic circuit, and the four-dimensional rotationally symmetric conservative chaotic circuit includes a first-channel circuit, a second-channel circuit, a third-channel circuit, and a fourth-channel circuit; In the first-channel circuit: The first input terminal of the first non-linear voltage converter is the -z signal, the inverting voltage source at the output terminal is grounded, the non-inverting voltage source is connected to the left end of the first resistor, the left end of the second resistor is connected to the -y signal, the left end of the third resistor is connected to the w signal, and the right ends of the first to third resistors are connected in parallel and then connected to the inverting input terminal of the first operational amplifier; The non-inverting input terminal of the first operational amplifier is grounded, and the fourth resistor connects the inverting input terminal and the output terminal of the first operational amplifier; The non-inverting input terminal of the second operational amplifier is grounded, the inverting input terminal is connected to the output terminal of the first operational amplifier through the fifth resistor, and is also connected to the output terminal of the second operational amplifier through the first capacitor, and the output terminal is the -x signal; The non-inverting input terminal of the third operational amplifier is grounded, the inverting input terminal is connected to the output terminal of the second operational amplifier through the sixth resistor, and is also connected to the output terminal of the third operational amplifier through the seventh resistor, and the output terminal is the x signal; In the second-channel circuit: The first input terminal of the second non-linear voltage converter is the -w signal, the inverting voltage source at the output terminal is grounded, the non-inverting voltage source is connected to the left end of the eighth resistor, the left end of the ninth resistor is connected to the -z signal, the left end of the tenth resistor is connected to the x signal, and the right ends of the eighth to tenth resistors are connected in parallel and then connected to the inverting input terminal of the fourth operational amplifier; The non-inverting input terminal of the fourth operational amplifier is grounded, and the eleventh resistor connects the inverting input terminal and the output terminal of the fourth operational amplifier; The non-inverting input terminal of the fifth operational amplifier is grounded, its inverting input terminal is connected to the output terminal of the fourth operational amplifier through the twelfth resistor, and is also connected to the output terminal of the fifth operational amplifier through the second capacitor, and the output terminal is the -y signal; The non-inverting input terminal of the sixth operational amplifier is grounded, its inverting input terminal is connected to the output terminal of the fifth operational amplifier through the thirteenth resistor, and is also connected to the output terminal of the sixth operational amplifier through the fourteenth resistor, and the output terminal is the y signal; In the third-channel circuit: The first input terminal of the third non-linear voltage converter is the -x signal, the inverting voltage source at the output terminal is grounded, the non-inverting voltage source is connected to the left end of the fifteenth resistor, the left end of the sixteenth resistor is connected to the -w signal, the left end of the seventeenth resistor is connected to the y signal, and the right ends of the fifteenth to seventeenth resistors are connected in parallel and then connected to the inverting input terminal of the seventh operational amplifier; The non-inverting input terminal of the seventh operational amplifier is grounded, and the eighteenth resistor connects the inverting input terminal and the output terminal of the seventh operational amplifier; The non-inverting input terminal of the eighth operational amplifier is grounded, its inverting input terminal is connected to the output terminal of the seventh operational amplifier through the nineteenth resistor, and is also connected to the output terminal of the eighth operational amplifier through the third capacitor, and the output terminal is the -z signal; The non-inverting input terminal of the ninth operational amplifier is grounded. Its inverting input terminal is connected to the output terminal of the eighth operational amplifier through the twentieth resistor and is also connected to the output terminal of the ninth operational amplifier through the twenty-first resistor. The output terminal is the z signal; In the fourth channel circuit: The first input terminal of the fourth non-linear voltage converter is the -y signal. The inverting voltage source at the output terminal is grounded, and the non-inverting voltage source is connected to the left end of the twenty-second resistor. The left end of the twenty-third resistor is connected to the -x signal, and the left end of the twenty-fourth resistor is connected to the z signal. The right ends of the twenty-second to twenty-fourth resistors are connected in parallel and then connected to the inverting input terminal of the tenth operational amplifier; The non-inverting input terminal of the tenth operational amplifier is grounded, and the twenty-fifth resistor connects the inverting input terminal and the output terminal of the tenth operational amplifier; The non-inverting input terminal of the eleventh operational amplifier is grounded. Its inverting input terminal is connected to the output terminal of the tenth operational amplifier through the twenty-sixth resistor and is also connected to the output terminal of the eleventh operational amplifier through the fourth capacitor. The output terminal is the -w signal; The non-inverting input terminal of the twelfth operational amplifier is grounded. Its inverting input terminal is connected to the output terminal of the eleventh operational amplifier through the twenty-seventh resistor and is also connected to the output terminal of the twelfth operational amplifier through the twenty-eighth resistor. The output terminal is the w signal.

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