A three-dimensional OFDM constellation encryption method based on Rodriguez rotation

Abstract

The present application is about an OFDM three-dimensional constellation encryption method based on Rodriguez rotation. It includes: using a chaotic Brownian motion sequence to control the rotation axis and rotation angle of the Rodriguez rotation formula to construct a rotation matrix; the transmitting end performs serial-to-parallel conversion processing and three-dimensional modulation processing on the plaintext information in sequence to obtain a transmitted plaintext signal, and uses the rotation matrix to randomly rotate the plaintext signal to generate an encrypted signal; the encrypted signal is sequentially subjected to I / Q conversion processing, CP addition processing, IFFT processing and parallel-to-serial conversion processing to generate an encrypted transmission signal; the receiving end performs serial-to-parallel conversion processing, FFT processing, CP removal processing and inverse I / Q conversion processing on the encrypted transmission signal in sequence to obtain a signal to be decrypted; the rotation matrix is ​​used to perform random inverse rotation on the signal to be decrypted to generate a decrypted signal, and the decrypted signal is sequentially subjected to three-dimensional demodulation and parallel-to-serial conversion to obtain decrypted information. The encryption method of the present application is more comprehensive and more secure, and avoids matrix inversion operations during the decryption process, resulting in higher computational efficiency.

Description

Technical Field

[0001] The present application relates to the field of encrypted communication technology, and in particular to an OFDM three-dimensional constellation encryption method based on Rodriguez rotation. Background Art

[0002] With the rapid development of information technology, the demand for high-speed communications and information security has become increasingly prominent. Developing a solution that can meet both high-speed transmission requirements and ensure information security is crucial. Modulation encryption, a key technical approach for physical layer security (PLS), can eliminate the statistical characteristics of waveforms and effectively disrupt the distribution of signals, thereby achieving low probability of intercept (LPI) communications. It has attracted considerable research attention.

[0003] In related technologies, modulation encryption mainly relies on two-dimensional constellation modulation. However, the encryption algorithm for two-dimensional constellation modulation is simple, the encryption system is relatively traditional, and there is a risk of theft. Therefore, encryption technology is gradually developing towards three-dimensional constellation modulation encryption. Three-dimensional constellation modulation encryption mainly includes the following two methods:

[0004] One method is to rotate the three-dimensional constellation points around the axis to achieve encryption. Since each rotation requires a matrix multiplication operation, a total of three matrix multiplications are required to complete the encryption, which has high computational complexity. At the same time, the decryption algorithm needs to use matrix inversion to achieve the desired result, which further increases the computational complexity.

[0005] Another method is to use quaternions to rotate constellation symbols. Although the space complexity is low, the maximum rotation angle of a single quaternion is 180 degrees, and the rotation encryption performance is limited.

[0006] Furthermore, rotational encryption for each constellation point involves only three data encryption keys: three random rotation angles around a fixed axis. While quaternion rotation appears to be controlled by four random parameters, the axis and angle are inherently related, and only three parameters are involved, limiting the adequacy of rotational encryption. Three-dimensional constellation rotational encryption can be compared to the rotation of an object in three-dimensional space, which is determined by both the axis and angle.

[0007] Therefore, it is necessary to provide a new technical solution to improve one or more problems existing in the above solutions.

[0008] It should be noted that the information disclosed in the above background technology section is only used to enhance the understanding of the background of this application, and therefore may include information that does not constitute prior art known to ordinary technicians in this field. Summary of the Invention

[0009] The purpose of this application is to provide an OFDM three-dimensional constellation encryption method based on Rodriguez rotation, thereby overcoming one or more problems caused by the limitations and defects of related technologies to a certain extent.

[0010] According to an embodiment of the present application, an OFDM three-dimensional constellation encryption method based on Rodriguez rotation is provided, including:

[0011] Using Hénon and Logistic chaotic mapping to generate a chaotic sequence, and using the chaotic sequence to control the parameters of Brownian motion to generate a chaotic Brownian motion sequence;

[0012] Using the chaotic Brownian motion sequence to control the rotation axis and rotation angle of the Rodriguez rotation formula to construct a rotation matrix;

[0013] The transmitting end performs serial-to-parallel conversion and three-dimensional modulation on the plaintext information in sequence to obtain a plaintext signal for transmission, and uses the rotation matrix to randomly rotate the plaintext signal to generate an encrypted signal;

[0014] The encrypted signal is sequentially subjected to I / Q conversion processing, CP addition processing, IFFT processing and parallel-to-serial conversion processing to generate an encrypted transmission signal;

[0015] The receiving end sequentially performs serial-to-parallel conversion, FFT processing, CP removal processing, and inverse I / Q transformation processing on the encrypted transmission signal to obtain a signal to be decrypted;

[0016] Performing random inverse rotation on the signal to be decrypted using the rotation matrix to generate a decrypted signal;

[0017] The decrypted signal is subjected to three-dimensional demodulation processing and parallel-to-serial conversion processing in sequence to obtain decrypted information.

[0018] In an embodiment of the present application, the method of generating a chaotic sequence by using Hénon and Logistic chaotic mapping and controlling the parameters of Brownian motion by using the chaotic sequence to generate a chaotic Brownian motion sequence includes:

[0019] On a two-dimensional plane, the expression for the Brownian motion of a point is:

[0020]

[0021] Where τ∈[0,+∞) is the motion step size, α=p×2π, β=q×2π represents the direction of motion, and p and q represent the control parameters of the motion direction.

[0022] In an embodiment of the present application, the method of generating a chaotic sequence by using Hénon and Logistic chaotic mapping and controlling the parameters of Brownian motion by using the chaotic sequence to generate a chaotic Brownian motion sequence includes:

[0023] The mapping expression of Hénon is:

[0024]

[0025] The mapping expression of the Logistic is:

[0026] x′ n+1 =cx′ n (1-x′ n )(3);

[0027] Where n represents the number of iterations of the chaotic sequence; x n ,y n represents the Hénon mapping sequence; x′ n represents the Hénon mapping sequence; a, b, and c represent the control parameters. When 1.07≤a≤1.4 and b=0.3, the Hénon mapping is in a chaotic state; when 3.57≤c≤4, the Logistic mapping is in a chaotic state.

[0028] Assume that the initial value of the Hénon map is (x0, y0), and after m1 iterations, a chaotic sequence of length L is generated (x n ,y n ), the initial value of the Logistic map is x′0, and the chaotic sequence x′ of the same length L is generated by iterating m2 times n , then the expression of the chaotic Brownian motion sequence is:

[0029]

[0030] Where, mean(x n ) represents the Hénon mapping sequence x n The average value of .

[0031] In the embodiment of the present application, the rotation matrix is:

[0032] Q(u,θ)I+Usinθ+U 2 (1-cosθ) (5);

[0033] Where θ is the rotation angle, u is the unit rotation axis vector, I is the 3×3 unit matrix, and U is the antisymmetric matrix;

[0034]

[0035] Where u x,u y ,u z are the components of the rotation axis vector u on the x, y, and z coordinate axes respectively; the rotation matrix Q(u,θ) is an orthogonal matrix.

[0036] In an embodiment of the present application, the transmitting end sequentially performs serial-to-parallel conversion and three-dimensional modulation on the plaintext information to obtain a plaintext signal, and uses the rotation matrix to randomly rotate the plaintext signal to generate an encrypted signal, including:

[0037] The N-bit plaintext information is mapped to N / 2 constellation points according to the three-dimensional constellation diagram of the regular tetrahedron, and the i-th constellation point S i The coordinates of (x i ,y i ,z i ), then the plaintext signal is:

[0038]

[0039] Where x N / 2 ,y N / 2 ,z N / 2 Indicates the N / 2th constellation point S N / 2 's coordinates.

[0040] In an embodiment of the present application, the transmitting end sequentially performs serial-to-parallel conversion and three-dimensional modulation on the plaintext information to obtain a plaintext signal, and uses the rotation matrix to randomly rotate the plaintext signal to generate an encrypted signal, including:

[0041] According to the chaotic Brownian motion sequence, N / 2 rotation axes and rotation angles are obtained; wherein, the i-th rotation axis k i The coordinates (k i,x ,k i,y ,k i,z ) and the i-th rotation angle θ i They are:

[0042]

[0043] Where i = 1, 2,…, N / 2.

[0044] In an embodiment of the present application, the transmitting end sequentially performs serial-to-parallel conversion and three-dimensional modulation on the plaintext information to obtain a plaintext signal, and uses the rotation matrix to randomly rotate the plaintext signal to generate an encrypted signal, including:

[0045] Based on the expression of the rotation matrix, the i-th rotation matrix is ​​Q i (k i ,θ i );

[0046] Each constellation point is randomly rotated around the i-th rotation matrix to obtain the encrypted information S′ of the i-th constellation point i , and obtain the encrypted signal S'; wherein the encrypted information S' of the i constellation points i for:

[0047]

[0048] The encrypted signal S′ is:

[0049]

[0050] Where T represents the transpose of the matrix, x′ N / 2 ,y′ N / 2 ,z′ N / 2 Indicates the N / 2th constellation point S′ N / 2 's coordinates.

[0051] In an embodiment of the present application, the encrypted conversion signal is:

[0052]

[0053] Where j represents the imaginary unit.

[0054] In an embodiment of the present application, the to-be-decrypted conversion signal is:

[0055]

[0056] In an embodiment of the present application, performing random inverse rotation on the to-be-decrypted converted signal using the rotation matrix to generate a decrypted signal includes:

[0057] The information to be decrypted of the i-th constellation point is obtained according to the converted signal to be decrypted, and the information to be decrypted of the i-th constellation point is randomly inversely rotated using the i-th rotation matrix to generate the i-th constellation point R i " decrypted signal; wherein the i-th constellation point R i The decryption signal is:

[0058] R″ i =R′ i ×[(Q i (k i ,θ i )) T ] -1 =R′ i ×Q i (k i ,θ i ) (13);

[0059] Where R′ iIndicates the constellation information to be decrypted for the i-th constellation point.

[0060] The technical solutions provided by the embodiments of the present application may have the following beneficial effects:

[0061] In one embodiment of the present application, the above-mentioned method, combined with the randomness of Brownian motion, uses classic Hénon and Logistic chaotic maps to generate a chaotic Brownian motion sequence. This is used to control the rotation axis and rotation angle in the Rodriguez rotation formula, construct a rotation matrix, and randomly rotate the three-dimensional constellation symbols to complete the encryption operation. This encryption method provides more complete encryption and avoids matrix inversion operations during the decryption process, resulting in higher computational efficiency. In addition, this encryption method disrupts the distribution pattern of the original constellation points. The key is extremely sensitive. The receiving end can accurately demodulate with the shared key, while an illegal eavesdropper cannot demodulate without knowing the key, effectively ensuring the security of the communication signal.

[0062] It should be understood that the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the present application. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] The accompanying drawings are incorporated into and constitute a part of the specification, illustrate embodiments consistent with the present application, and together with the specification, are used to explain the principles of the present application. Obviously, the drawings described below are only some embodiments of the present application, and those skilled in the art can derive other drawings based on these drawings without inventive effort.

[0064] Figure 1 A flowchart schematically illustrates a method for encrypting an OFDM three-dimensional constellation based on Rodriguez rotation in an exemplary embodiment of the present application;

[0065] Figure 2 A schematic diagram schematically illustrating a three-dimensional constellation diagram in an exemplary embodiment of the present application;

[0066] Figure 3 A schematic diagram schematically illustrates an OFDM three-dimensional constellation encryption system model based on Rodriguez rotation in an exemplary embodiment of the present application;

[0067] Figure 4 A schematic diagram schematically illustrating a strange attractor of a Hénon map in an exemplary embodiment of the present application;

[0068] Figure 5 A schematic diagram schematically illustrating a strange attractor of a Logistic map in an exemplary embodiment of the present application;

[0069] Figure 6A schematic diagram schematically illustrates the motion trajectory of chaotic Brownian motion after 100 iterations in an exemplary embodiment of the present application;

[0070] Figure 7 Schematically illustrating positions of constellation points rotated at different angles around the same rotation axis in an exemplary embodiment of the present application;

[0071] Figure 8 Schematically illustrating a position diagram of constellation points rotated by the same angle around different rotation axes in an exemplary embodiment of the present application;

[0072] Figure 9 Schematically showing a top view of constellation points rotated at different angles around the same rotation axis in an exemplary embodiment of the present application;

[0073] Figure 10 Schematically showing a top view of constellation points rotated by the same angle around different rotation axes in an exemplary embodiment of the present application;

[0074] Figure 11 Schematically showing a constellation diagram before encryption in an exemplary embodiment of the present application;

[0075] Figure 12 Schematically illustrating an encrypted constellation diagram in an exemplary embodiment of the present application;

[0076] Figure 13 Schematically illustrates a key sensitivity analysis diagram in an exemplary embodiment of the present application;

[0077] Figure 14 The figure schematically shows a bit error rate analysis diagram in an exemplary embodiment of the present application. DETAILED DESCRIPTION

[0078] Example embodiments will now be described more fully with reference to the accompanying drawings. However, example embodiments can be embodied in many forms and should not be construed as limited to the examples set forth herein; rather, these embodiments are provided so that this disclosure will be thorough and complete and will fully convey the concepts of the example embodiments to those skilled in the art. The described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments.

[0079] In addition, the accompanying drawings are only schematic illustrations of the present application and are not necessarily drawn to scale. The same reference numerals in the drawings represent the same or similar parts, and thus their repeated description will be omitted.

[0080] This example embodiment first provides an OFDM three-dimensional constellation encryption method based on Rodriguez rotation. Figure 1 As shown in , the method may include: steps S101 to S107.

[0081] Wherein, step S101: using Hénon and Logistic chaotic mapping to generate a chaotic sequence, and using the chaotic sequence to control the parameters of Brownian motion to generate a chaotic Brownian motion sequence.

[0082] Step S102: Using the chaotic Brownian motion sequence to control the rotation axis and rotation angle of the Rodriguez rotation formula to construct a rotation matrix.

[0083] Step S103: The transmitting end performs serial-to-parallel conversion and three-dimensional modulation on the plaintext information in sequence to obtain a plaintext signal for transmission, and uses a rotation matrix to randomly rotate the plaintext signal to generate an encrypted signal.

[0084] Step S104: The encrypted signal is sequentially subjected to I / Q conversion processing, CP addition processing, IFFT processing, and parallel-to-serial conversion processing to generate an encrypted transmission signal.

[0085] Step S105: The receiving end sequentially performs serial-to-parallel conversion processing, FFT processing, CP removal processing, and inverse I / Q transform processing on the encrypted converted signal to obtain a signal to be decrypted.

[0086] Step S106: using the rotation matrix to perform random inverse rotation on the signal to be decrypted to generate a decrypted signal.

[0087] Step S107: performing three-dimensional demodulation processing and parallel-to-serial conversion processing on the decrypted signal in sequence to obtain decrypted information.

[0088] In one embodiment of the present application, the above-mentioned method, combined with the randomness of Brownian motion, uses classic Hénon and Logistic chaotic maps to generate a chaotic Brownian motion sequence. This is used to control the rotation axis and rotation angle in the Rodriguez rotation formula, construct a rotation matrix, and randomly rotate the three-dimensional constellation symbols to complete the encryption operation. This encryption method provides more complete encryption and avoids matrix inversion operations during the decryption process, resulting in higher computational efficiency. In addition, this encryption method disrupts the distribution pattern of the original constellation points. The key is extremely sensitive. The receiving end can accurately demodulate with the shared key, while an illegal eavesdropper cannot demodulate without knowing the key, effectively ensuring the security of the communication signal.

[0089] Below, we will refer to Figures 1 to 6 Each step of the above method in this exemplary embodiment is described in more detail.

[0090] Before discussing this method, the following abbreviations are explained.

[0091] OFDM (Orthogonal frequency-division multiplexing) is a multicarrier modulation technology. CP (Cyclic Prefix) refers to the prefix of a symbol. IFFT stands for Inverse Fast Fourier Transform, and FFT stands for Fast Fourier Transform.

[0092] In step S101, Brownian motion is a common phenomenon in nature, and its motion trajectory has the natural advantages of irregularity and unpredictability. Using a chaotic sequence to control the parameters of the Brownian motion can further disrupt the distribution of the chaotic sequence, thereby effectively enhancing the randomness of the sequence.

[0093] In step S1011, the Brownian motion of a point on a two-dimensional plane is expressed as:

[0094]

[0095] Where τ∈[0,+∞] is the motion step, α=p×2π, β=q×2π represents the direction of motion, and p, q represent the control parameters representing the motion direction.

[0096] In sub-step S1012, the classical Hénon and Logistic chaotic mappings are used to design the chaotic Brownian motion sequence. The Hénon mapping expression is:

[0097]

[0098] The mapping expression of Logistic is:

[0099] x′ n+1 =cx′ n (1-x′ n )(3);

[0100] Where n represents the number of iterations of the chaotic sequence; x n ,y n represents the Hénon mapping sequence; x′ n The Hénon mapping sequence a, b, and c represent the control parameters. When 1.07≤a≤1.4 and b=0.3, the Hénon mapping is in a chaotic state. When 3.57≤c≤4, the Logistic mapping is in a chaotic state.

[0101] Assume that the initial value of the Hénon map is (x0, y0), and after m1 iterations, a chaotic sequence of length L is generated (x n ,yn ), the initial value of the Logistic map is x′0, and the chaotic sequence x′ of the same length L is generated by iterating m2 times n , then the expression of chaotic Brownian motion sequence is:

[0102]

[0103] Where, mean(x n ) represents the Hénon mapping sequence x n The average value of .

[0104] The chaotic Brownian motion sequence is iterated 100 times to draw its trajectory and compared with the strange attractors of Hénon map and Logistic map, as shown in Figures 4 to 6 As shown in Table 1, it is not difficult to see that the chaotic Brownian motion sequence is more random and chaotic. Spectral entropy (SE) can quantitatively characterize the structural complexity of a chaotic system. Further comparison of the spectral entropy values ​​of the original chaotic sequence and its Brownian motion sequence shows that the spectral entropy value of the chaotic Brownian motion sequence is higher than that of the original sequence, indicating that the chaotic Brownian motion sequence is more random, making it an excellent choice for encryption operations.

[0105] Table 1 Comparison of spectral entropy between original chaotic sequence and Brownian motion sequence

[0106] <![CDATA[x n ]]> <![CDATA[y n ]]> <![CDATA[x′ n ]]> <![CDATA[dx n ]]> <![CDATA[dy n ]]> SE 0.8792 0.8596 0.8654 0.9208 0.9097

[0107] In step S102, the rotation of an object in three-dimensional space can be represented by mathematical formulas such as Euler angles, quaternions, and Rodriguez rotation matrices. Each formula has its own advantages and disadvantages. The following analyzes the Euler angle representation, quaternion representation, and Rodriguez rotation matrix representation to determine which representation method is used to construct the rotation matrix required in this application.

[0108] (1) Euler angle representation

[0109] Euler angles use three angle values ​​to represent three-dimensional rotations. These three angles correspond to rotations around the x, y, and z axes, respectively. Their expressions are:

[0110]

[0111] Among them, (x, y, z) is the coordinate position of the object before rotation, (x′, y′, z′) is the coordinate position after rotation, and α, β, and γ are the rotation angles around the x, y, and z axes respectively.

[0112] (2) Quaternion representation

[0113] Quaternion is a type of hypercomplex number consisting of a real part and three imaginary parts. When representing the rotation of a space object, it can be regarded as a combination of a rotation vector and a rotation angle. Its expression is:

[0114] r′=q·r·q -1 (15);

[0115] Where r = (x, y, z) is the coordinate position of the object before rotation, r′ = (x′, y′, z′) is the coordinate position after rotation, and q = (q0, q1, q2, q3) is the quaternion. Rotation axis n and rotation angle They are:

[0116]

[0117]

[0118] Among them, q -1 =q * / ||q * || is the inverse of the quaternion q, q * =(q0,-q1,-q2,-q3) is the common orbital of the quaternion q, The modulus of the quaternion q.

[0119] (3) Rotation matrix representation

[0120] In three-dimensional space, a vector rotates around an axis by a certain angle, which can also be represented by multiplying the rotation matrix by the vector. The expression is:

[0121] v rot =Q(u,θ)v(18);

[0122] Among them, v rot is the rotated vector, v is the original vector, θ is the rotation angle, and u is the unit rotation axis vector. The rotation matrix Q(u,θ) is given by the Rodrigues' rotation formula, which is expressed as:

[0123] Q(u,θ)=I+Usinθ+U 2 (1-cosθ) (5);

[0124] Where I is the 3×3 identity matrix and U is the antisymmetric matrix (U T =-U),

[0125]

[0126] U T represents the transpose of the matrix, u x ,u y ,uz are the components of the rotation axis vector u on the x, y, and z coordinate axes respectively. The rotation matrix Q(u,θ) is an orthogonal matrix, that is, Q -1 (u,θ)=Q T (u,θ) and |Q(u,θ)|=1.

[0127] From the mathematical formulas of the above three rotation forms, it can be seen that the Euler angle representation uses three angle values ​​to represent the rotation of a spatial object, which requires three matrix multiplication operations and is computationally complex. The quaternion representation occupies less space when describing and calculating the rotation of an object in three-dimensional space, but can only represent the rotation of an object in any direction, and a single quaternion cannot represent a rotation of more than 180 degrees in that direction. The Rodriguez rotation matrix can represent the rotation of an object in any direction and at any angle, and is orthogonal. To restore a rotated object to its original position, it only requires transposing the rotation matrix, and no additional computational overhead is added due to matrix inversion operations.

[0128] Therefore, after subjecting the chaotic sequence to Brownian motion to generate a more random chaotic Brownian motion sequence, the present application uses the chaotic Brownian motion sequence to control the rotation axis and rotation angle of the three-dimensional modulation constellation point to construct the rotation matrix in the present application.

[0129] In step S103, three-dimensional modulation is to map the plaintext information to constellation points in three-dimensional space. Each constellation point has specific characteristics such as amplitude, phase and frequency to distinguish different signal states. The efficient modulation of the plaintext information is achieved through the designed constellation diagram. s =2 d The constellation points are evenly distributed on the sphere, where d is the number of bits of coded information. Figure 2 The constellation is a three-dimensional space mapping of a regular tetrahedron. Assuming the radius of the sphere is 1, the coordinates of the four constellation points S(0), S(1), S(2), and S(3) are S(0) = (0, 0, 1). Correspondingly, d = 2, which means that every 2 bits of plaintext information are mapped to one constellation point. The mapping relationship between the transmitted information and the constellation point is:

[0130] It is understandable that the encryption system model in this application is as follows Figure 3 As shown, at the transmitter, plaintext information (i.e., the plaintext bit stream) is mapped to a three-dimensional constellation symbol through serial-to-parallel conversion. Each constellation point is rotated according to a random rotation matrix constructed using the Rodriguez rotation formula. The rotated constellation symbol undergoes I / Q transformation, inverse fast Fourier transform (IFFT), and parallel-to-serial conversion before being transmitted over the wireless channel. The legitimate receiver possesses the shared key and can accurately recover the received encrypted signal.

[0131] In sub-step S1031, assuming that the three-dimensional constellation diagram is a regular tetrahedron, the N-bit plaintext information is mapped into N / 2 constellation points according to the three-dimensional constellation diagram of the regular tetrahedron, and the i-th constellation point S i The coordinates of (x i ,y i ,z i ), then the plaintext signal is:

[0132]

[0133] Where x N / 2 ,y N / 2 ,z N / 2 Indicates the N / 2th constellation point S N / 2 's coordinates.

[0134] Generate a chaotic Brownian motion sequence of length N according to formulas (1) to (4).

[0135] In sub-step S1032, N / 2 rotation axes and rotation angles are obtained according to the chaotic Brownian motion sequence; wherein the i-th rotation axis k i The coordinates (k i,x ,k i,y ,k i,z ) and the i-th rotation angle θ i They are:

[0136]

[0137] Where i = 1, 2,…, N / 2.

[0138] In sub-step S1033, based on the expression of the rotation matrix, the i-th rotation matrix is ​​obtained as Q i (k i ,θ i );

[0139] Each constellation point is randomly rotated around the i-th rotation matrix to obtain the encrypted information S′ of the i-th constellation point i , and get the encrypted signal S'; where the encrypted information S' of the i constellation point i for:

[0140]

[0141] The encrypted signal S′ is:

[0142]

[0143] Where T represents the transpose of the matrix, x′ N / 2 ,y′ N / 2 ,z′ N / 2Indicates the N / 2th constellation point S′ N2 's coordinates.

[0144] In step S104, the encrypted transmission signal is:

[0145]

[0146] In the formula, j represents an imaginary unit. After obtaining the encrypted transmission signal, the present application converts the encrypted signal S′ into an I / Q two-way transmission according to formula (11).

[0147] In step S105, after the receiving end receives the encrypted transmission signal, the encrypted transmission signal is sequentially subjected to serial-to-parallel conversion, FFT processing, CP removal processing, and inverse I / Q transform processing. The real and imaginary parts of the encrypted transmission signal are separated and combined into constellation points to be decrypted, i.e., the signal to be decrypted. The signal to be decrypted is:

[0148]

[0149] In step S106, the legal receiving end (ie, the legal receiver) shares a key with the sending end in advance, and generates a rotation matrix in the same way as the sending end.

[0150] In sub-step S1061, the constellation information to be decrypted of the i-th constellation point is obtained according to the signal to be decrypted, and the constellation information to be decrypted of the i-th constellation point is randomly inversely rotated using the i-th rotation matrix to generate the i-th constellation point R″ i The decrypted signal of the i-th constellation point R″ i The decryption signal is:

[0151] R″ i =R′ i ×[(Q i (k i ,θ i )) T ] -1 =R′ i ×Q i (k i ,θ i )(13);

[0152] Where R′ i represents the information to be decrypted at the i-th constellation point. Formula (11) uses the orthogonality of the rotation matrix.

[0153] In step S107, the i-th constellation point R″ is obtained iThe decrypted signal is obtained by calculating the Euclidean distance between the decrypted signal of each constellation point and the four constellation points in the original constellation diagram. The constellation point corresponding to the decrypted signal of each constellation point is determined to be the constellation point with the minimum Euclidean distance to the original constellation diagram. Then, based on the mapping relationship between the constellation point and the decrypted signal, the final decrypted information is demodulated.

[0154] In order to verify the OFDM three-dimensional constellation encryption method based on Rodriguez rotation of the present application, the present application conducted the following simulation experiments.

[0155] The security and effectiveness analysis of the OFDM three-dimensional constellation encryption method based on Rodriguez rotation in this application plays an important role in the encryption design process. This application simulates and analyzes the constellation encryption characteristics, anti-exhaustive attack, bit error rate, etc., and evaluates its performance.

[0156] The simulation experiments were performed on the following platform: CPU Intel Core i3-5005U 4G, GPU NVDIA GeForce920A 4G, Windows 10 Home Edition, and Matlab R2016. The simulation parameters are shown in Table 2.

[0157] Table 2 Simulation parameters

[0158] category parameter Channel Category AWGN Plaintext information length N 512 <![CDATA[Initial values (x0, y0) of the Hénon map]]> (0.5,0.5) Hénon map control parameters (a, b) (1.4,0.3) <![CDATA[Pre-iteration times m1 of the Hénon map]]> 100 <![CDATA[Initial value x′0 of the Logistic map]]> 0.3 Logistic mapping control parameter c 3.96 <![CDATA[Pre-iteration times m2 of Logistic mapping]]> 100

[0159] (1) Analysis of constellation encryption characteristics

[0160] Taking the constellation point S = (0,0,1) as an example, the feasibility of the OFDM three-dimensional constellation encryption method based on Rodriguez rotation is elaborated in detail. Figures 7 to 10 When the constellation point S rotates around the same rotation axis (1,1,1) by different angles of 1.2π and 0.8π, we get Figure 7 and Figure 9 The two points S1 and S2 shown in the figure are rotated by the same angle 0.8π around different rotation axes (1,1,1) and (0,1,1). Figure 8 and Figure 10 Points S1′, S2′ are shown. Figures 7 to 10 It can be seen that after rotation, points S1 and S2 are different, and points S1′ and S2′ are also different. This fully demonstrates that whether the rotation axis or the rotation angle is used as the encryption method, it is feasible.

[0161] In the application, the rotation axis and rotation angle are used together as a constellation rotation encryption method. This method has four independent rotation factors, making the encryption process more flexible and versatile, and can better protect signal security. In addition, this application avoids matrix inversion during the decryption process. Both encryption and decryption methods only require a single multiplication operation of the rotation matrix and the vector. This not only reduces computational complexity but also improves encryption and decryption efficiency while ensuring security.

[0162] After all the constellation points of the transmitted information (i.e. plain text information) are encrypted, the constellation diagram is as follows: Figure 11 and Figure 12 As shown in the figure, the originally regularly arranged constellation points become chaotically scattered across the sphere's surface after encryption. This makes it difficult for interference signals that rely on signal distribution patterns to specifically disrupt the encrypted signal, significantly enhancing the signal's anti-interference capabilities. Furthermore, unauthorized eavesdroppers cannot recover the original information without knowing the key, greatly improving communication confidentiality.

[0163] (2) Anti-exhaustive attack performance

[0164] As the core component of the encryption method, the importance of the key is self-evident. An excellent encryption method must not only have a large enough key space to resist potential brute force cracking, but also be highly sensitive to slight changes in the key. The Rodriguez rotation OFDM three-dimensional constellation encryption method designed in this application, its key system is composed of chaotic initial values ​​and control parameters, including the initial value (x0, y0) of the Hénon map, control parameters a, b and the initial value x′0 of the Logistic map, control parameter c, and the key value range is -1<x0<1, -1<y0<1, 1.07≤a≤1.4, 0vx′0<1, 3.57≤c≤4. In order to further verify the sensitivity of the encryption method to key changes, an experimental analysis was carried out using the control variable method. In the experiment, each key parameter was adjusted one by one, and the correct parameters were used for the remaining keys to obtain the information bit error rate curve after decryption as shown below. Figure 13 As shown, even if the keys x0, y0, a, b, c each have a small 10 -15 Change, the key x′0 has 10 -16 The bit error rate of the decrypted information remains at around 0.5, indicating that the encryption method can ensure the security of information even if it faces a small risk of key leakage. Therefore, without considering the number of chaotic pre-iterations, the key space of the rotation encryption method proposed in this application is 10 15 ×10 15 ×10 15 ×10 15 ×10 16 ×10 15=10 91 ≈2 304 , indicating that any attempt to crack the key through a brute force attack is extremely difficult.

[0165] (3) Bit Error Rate Analysis

[0166] During the process of spatial propagation, the signal will inevitably be interfered by noise. Compared with the two-dimensional constellation diagram, the three-dimensional modulation constellation diagram has a larger minimum Euclidean distance, which means that under the condition of transmitting the same energy, the bit error rate at the receiving end can be effectively reduced. Comparing the encryption method of this application with the theoretical bit error rate of QPSK, Figure 14 As shown, when the bit error rate reaches 10 -3 When , the signal-to-noise ratio of 3D mapping is reduced by about 2.4dB compared with QPSK.

[0167] (4) Complexity analysis

[0168] The complexity of general encryption methods involves time complexity and space complexity. The OFDM three-dimensional constellation encryption method based on Rodriguez rotation in this application includes the encryption process and decryption process. Its performance in these two aspects is analyzed and compared with the existing encryption methods around the x, y, and z axes and quaternion rotation encryption methods, as shown in Table 3.

[0169] For N bits of plaintext information, it is mapped to N / 2 constellation points according to the 4-ray three-dimensional constellation diagram. In the OFDM three-dimensional constellation encryption method based on Rodriguez rotation of this application, each constellation point needs to be multiplied by the transpose of 1 three-dimensional rotation matrix to complete the encryption. Only multiplication operations are considered. The time complexity of the OFDM three-dimensional constellation encryption method based on Rodriguez rotation of this application is The space complexity is also For encryption around the x, y, and z axes, each constellation point needs to be multiplied by three three-dimensional matrices to complete the encryption, so the time complexity is The space complexity is Quaternion rotation encryption: Each constellation point needs to be encrypted by multiplying it with a quaternion and the inverse of a quaternion. According to the quaternion operation rules, the amount of operation required to complete the encryption of a constellation point is 28, so the time complexity of quaternion rotation encryption is The space complexity is

[0170] For the decryption process, this application uses the orthogonality of the rotation matrix. The received signal only needs to be multiplied by a three-dimensional matrix. Therefore, the time complexity of the decryption method is also 4.5N, and the space complexity is also 4.5N. In the encryption method of rotating around the x, y, and z axes, it is necessary to invert the three three-dimensional matrices separately and then multiply them. When using the adjoint matrix to calculate the inverse of a three-dimensional matrix, 60 multiplications and 9 divisions are required. Therefore, the time complexity of the decryption method is The space complexity is 13.5N. The quaternion decryption method is consistent with the encryption method, with a time complexity of 14N and a space complexity of 4N.

[0171] In summary, the OFDM three-dimensional constellation encryption method based on Rodriguez rotation proposed in this application is efficient in time complexity. In particular, the computational efficiency of the decryption method in the decryption process is improved by 96.15% compared with the decryption method rotating around the x, y, and z axes. The spatial complexity is relatively reasonable, basically consistent with the quaternion, but less than the storage space required for the encryption method rotating around the x, y, and z axes.

[0172] Table 3 Complexity comparison

[0173]

[0174] In summary, after encryption by this method, the originally regularly arranged constellation points are randomly and chaotically distributed over the entire sphere. The key space is large enough and extremely sensitive, which greatly improves the security of the communication signal.

[0175] Those skilled in the art will readily appreciate other embodiments of the present invention after considering the specification and practicing the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of the present invention that follow the general principles of this application and include common knowledge or customary techniques in the art that are not disclosed herein.

Claims

1. An OFDM three-dimensional constellation encryption method based on Rodriguez rotation, characterized in that: include: Using Hénon and Logistic chaotic mapping to generate a chaotic sequence, and using the chaotic sequence to control the parameters of Brownian motion to generate a chaotic Brownian motion sequence; Using the chaotic Brownian motion sequence to control the rotation axis and rotation angle of the Rodriguez rotation formula to construct a rotation matrix; The transmitting end performs serial-to-parallel conversion and three-dimensional modulation on the plaintext information in sequence to obtain a plaintext signal for transmission, and uses the rotation matrix to randomly rotate the plaintext signal to generate an encrypted signal; The encrypted signal is sequentially subjected to I / Q conversion processing, CP addition processing, IFFT processing and parallel-to-serial conversion processing to generate an encrypted transmission signal; The receiving end sequentially performs serial-to-parallel conversion, FFT processing, CP removal processing, and inverse I / Q transformation processing on the encrypted transmission signal to obtain a signal to be decrypted; Performing random inverse rotation on the signal to be decrypted using the rotation matrix to generate a decrypted signal; The decrypted signal is subjected to three-dimensional demodulation processing and parallel-to-serial conversion processing in sequence to obtain decrypted information.

2. The OFDM three-dimensional constellation encryption method based on Rodriguez rotation according to claim 1, characterized in that: The method of generating a chaotic sequence by using Hénon and Logistic chaotic mapping and controlling the parameters of Brownian motion by using the chaotic sequence to generate a chaotic Brownian motion sequence comprises: On a two-dimensional plane, the expression for the Brownian motion of a point is: Where τ∈[0,+∞) is the motion step size, α=p×2π, β=q×2π represents the direction of motion, and p and q represent the control parameters of the motion direction.

3. The OFDM three-dimensional constellation encryption method based on Rodriguez rotation according to claim 2, characterized in that: The method of generating a chaotic sequence by using Hénon and Logistic chaotic mapping and controlling the parameters of Brownian motion by using the chaotic sequence to generate a chaotic Brownian motion sequence comprises: The mapping expression of Hénon is: The mapping expression of the Logistic is: x′ n+1 =cx′ n (1-x′ n ) (3); Where n represents the number of iterations of the chaotic sequence; x n ,y n represents the Hénon mapping sequence; x′ n represents the Logistic mapping sequence; a, b, and c represent the control parameters. When 1.07≤a≤1.4, b=0.3, the Hénon mapping is in a chaotic state; when 3.57≤c≤4, the Logistic mapping is in a chaotic state. Assume that the initial value of the Hénon map is (x0, y0), and after m1 iterations, a chaotic sequence of length L is generated (x n ,y n ), the initial value of the Logistic map is x′0, and the chaotic sequence x′ of the same length L is generated by iterating m2 times n , then the expression of the chaotic Brownian motion sequence is: Where, mean(x n ) represents the Hénon mapping sequence x n The average value of .

4. The OFDM three-dimensional constellation encryption method based on Rodriguez rotation according to claim 1, characterized in that: The rotation matrix is: Q(u,θ)=I+U sinθ+U 2 (1-cosθ) (5); Where θ is the rotation angle, u is the unit rotation axis vector, I is the 3×3 unit matrix, and U is the antisymmetric matrix; Where u x ,u y ,u z are the components of the rotation axis vector u on the x, y, and z coordinate axes respectively; the rotation matrix Q(u,θ) is an orthogonal matrix.

5. According to the OFDM three-dimensional constellation encryption method based on Rodriguez rotation in claim 1, the transmitting end sequentially performs serial-to-parallel conversion and three-dimensional modulation on the plaintext information to obtain a plaintext signal, and randomly rotates the plaintext signal using the rotation matrix to generate an encrypted signal, comprising: The N-bit plaintext information is mapped to N / 2 constellation points according to the three-dimensional constellation diagram of the regular tetrahedron, and the i-th constellation point S i The coordinates of (x i ,y i ,z i ), then the plaintext signal is: Where x N / 2 ,y N / 2 ,z N / 2 Indicates the N / 2th constellation point S N / 2 's coordinates.

6. The OFDM three-dimensional constellation encryption method based on Rodriguez rotation according to claim 5, characterized in that: The transmitting end sequentially performs serial-to-parallel conversion and three-dimensional modulation on the plaintext information to obtain a plaintext signal, and uses the rotation matrix to randomly rotate the plaintext signal to generate an encrypted signal, including: According to the chaotic Brownian motion sequence, N / 2 rotation axes and rotation angles are obtained; wherein, the i-th rotation axis k i The coordinates (k i,x ,k i,y ,k i,z ) and the i-th rotation angle θ i They are: Where i = 1, 2,…, N / 2.

7. The OFDM three-dimensional constellation encryption method based on Rodriguez rotation according to claim 6, characterized in that: The transmitting end sequentially performs serial-to-parallel conversion and three-dimensional modulation on the plaintext information to obtain a plaintext signal, and uses the rotation matrix to randomly rotate the plaintext signal to generate an encrypted signal, including: Based on the expression of the rotation matrix, the i-th rotation matrix is ​​Q i (k i ,θ i ); Each constellation point is randomly rotated around the i-th rotation matrix to obtain the encrypted information S′ of the i-th constellation point i , and obtain the encrypted signal S'; wherein the encrypted information S' of the i constellation points i for: The encrypted signal S′ is: Where T represents the transpose of the matrix, x′ N / 2 ,y′ N / 2 ,z′ N / 2 Indicates the N / 2th constellation point S′ N / 2 's coordinates.

8. The OFDM three-dimensional constellation encryption method based on Rodriguez rotation according to claim 7, characterized in that: The encrypted transmission signal is: Where j represents the imaginary unit.

9. The OFDM three-dimensional constellation encryption method based on Rodriguez rotation according to claim 8, characterized in that: The signal to be decrypted is:

10. The OFDM three-dimensional constellation encryption method based on Rodriguez rotation according to claim 9, characterized in that: The method of performing random inverse rotation on the signal to be decrypted by using the rotation matrix to generate a decrypted signal includes: Obtain the constellation information to be decrypted of the i-th constellation point according to the signal to be decrypted, and use the i-th rotation matrix to randomly inversely rotate the constellation information to be decrypted of the i-th constellation point to generate the i-th constellation point R″ i The decrypted signal; wherein the i-th constellation point R" i The decryption signal is: R″ i =R′ i ×[(Q i (k i ,i i )) T ] -1 =R′ i ×Q i (k i ,i i ) (13); Where R′ i Indicates the constellation information to be decrypted for the i-th constellation point.

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