Image secure transmission method based on four-dimensional hyper-chaotic system and DNA coding

Abstract

The invention provides an image secure transmission method based on a four-dimensional hyper-chaotic system and DNA coding, and the method comprises the steps: calculating an initial parameter of the four-dimensional hyper-chaotic system, substituting the initial parameter into the four-dimensional hyper-chaotic system, generating four chaotic sequences, and obtaining an integer chaotic sequence; converting the plaintext image into a binary matrix, and performing DNA coding to obtain a base matrix; performing DNA coding on the integer chaotic sequence F1 to obtain a base matrix J1; constructing a Latin square DNA replacement table, and replacing the basic groups in the basic group matrix according to the Latin square DNA replacement table by using a DNA replacement rule to generate a new basic group matrix; performing index cross scrambling on the base matrix to obtain a base matrix T6; carrying out DNA exclusive-or operation on the base matrix T6 and the base matrix J1, carrying out DNA decoding, and converting into a decimal matrix to obtain a matrix J; and performing forward and reverse random weighted diffusion on the matrix J in sequence to obtain a ciphertext image. The method disclosed by the invention shows good safety and robustness on indexes such as information entropy, NPCR, UACI and the like.

Description

Technical Field

[0001] The present invention relates to the technical field of digital image transmission, and particularly to an image secure transmission method based on a four-dimensional hyperchaotic system and DNA coding. Background Art

[0002] With the rapid development of Internet of Things (IoT) technology, image data is increasingly widely used in scenarios such as intelligent monitoring, telemedicine, and autonomous driving, and the volume of data acquisition and transmission has increased exponentially. Although this trend has brought convenience and intelligent experience, it has also caused serious information security and privacy protection problems. As an important data type carrying a large amount of semantic information, images are extremely vulnerable to attacks during the process of transmission and storage. Therefore, there is an urgent need to design an efficient, lightweight, and highly secure image encryption algorithm.

[0003] Among many encryption technologies, chaotic systems have received extensive attention in the field of image encryption due to their initial value sensitivity, unpredictable trajectories, and pseudo-randomness. Compared with traditional encryption algorithms, chaotic-based image encryption schemes not only have strong robustness and non-linear characteristics but also can meet the requirements of real-time and big data processing capabilities in IoT scenarios. In recent years, researchers have continuously expanded the structure and application methods of chaotic models. For example, Abdurrahim introduced Bessel functions to construct a one-dimensional chaotic map and applied it to bit-level encryption; Feyza combined the Rastrigin function and the visual field function to design a 2D-RG map for color image encryption; Omer proposed an image encryption algorithm combining the MILE map and particle swarm optimization. However, most of the above schemes are based on one-dimensional or two-dimensional low-dimensional chaotic systems, with short periods and limited state spaces, making it difficult to resist high-intensity cryptographic analysis attacks.

[0004] To overcome the limitations of low-dimensional chaotic systems, researchers have introduced high-dimensional chaotic systems. Such as the three-dimensional chaotic system proposed by Lorenz, and more complex dynamic models developed from the Chen system subsequently. In addition, Yahi proposed a 3D-CSC system based on sine and cosine functions and designed a double scrambling and diffusion mechanism; Wang constructed a new four-dimensional chaotic system and proposed a multi-base diffusion strategy based on it. High-dimensional chaos is particularly suitable for encrypting high-redundancy data such as images due to its complex trajectories and diverse states.

[0005] Typical image encryption processes usually include two key steps: "scrambling" and "diffusion". The former is used to disrupt the position distribution of image pixels and weaken the spatial correlation; the latter enhances the sensitivity and irreversibility of the encryption algorithm by modulating pixel values to expand the influence range of the initial perturbation. In recent years, complex encryption structures combining mathematical structures or bio-inspired mechanisms have also emerged continuously. For example, Hua improved the system mixing degree through orthogonal Latin squares and finite field multiplication; Li proposed a combined structure integrating block-level scrambling and weight diffusion; scholars such as Pooja and Wassim introduced the DNA coding mechanism into image encryption to enhance the randomness and unpredictability of ciphertext through base operations. However, most DNA encryption schemes still rely on static coding rules and have potential security risks when facing known plaintext-ciphertext attacks. Summary of the Invention

[0006] Aiming at the technical problem of the security threat faced by image data of Internet of Things terminals during the transmission process, the present invention proposes an image security transmission method based on a four-dimensional hyperchaotic system and DNA coding, integrating a new image encryption scheme that improves the four-dimensional hyperchaotic system and DNA coding to enhance the dynamicity and anti-attack ability of encryption.

[0007] To achieve the above purpose, the technical solution of the present invention is realized as follows: An image security transmission method based on a four-dimensional hyperchaotic system and DNA coding, the steps are as follows:

[0008] Step 1: Use the SHA-256 hash function to calculate the hash value of the input plaintext image P with a size of M×N, and combine with the external key to calculate the initial parameters of the four-dimensional hyperchaotic system;

[0009] Step 2: Substitute the initial parameters into the four-dimensional hyperchaotic system, iterate multiple times and then continue to iterate M×N times to generate four chaotic sequences X', Y', Z', W' respectively, and perform data processing to obtain integer chaotic sequences F1, F2, F3, F4 and random numbers W1, W2, W3 and random sequence G;

[0010] Step 3: Convert the plaintext image P into an eight-bit binary matrix T1, select a DNA coding rule to perform DNA coding on the binary matrix T1 to obtain a base matrix T2; perform DNA coding on the integer chaotic sequence F1 to obtain a base matrix J1;

[0011] Step 4: Construct a Latin square DNA replacement table, and use the DNA replacement rule to replace each base in the base matrix T2 by looking up the Latin square DNA replacement table to generate a new base matrix T3;

[0012] Step 5: Convert the base matrix T3 into a one-dimensional DNA sequence T4 and evenly divide it into 4 segments; use the index sequences I1, I2, I3, I4 generated by the integer chaotic sequences F1, F2, F3, F4 to perform index scrambling on the four DNA sequences respectively, and obtain sequences S1, S2, S3, S4; place the bases at the same positions of sequences S1, S2, S3, S4 together in sequence to obtain a base sequence T5, and restore it to matrix form to obtain a base matrix T6;

[0013] Step 6: Perform DNA XOR operation on the base matrix T6 and the base matrix J1 to obtain a base matrix T7, perform DNA decoding on the base matrix T7 according to the DNA decoding rule to obtain a binary matrix J2, and convert the binary matrix J2 into a decimal number to obtain a matrix J;

[0014] Step 7: Perform forward and reverse random weighted diffusion operations on the matrix J in sequence through random numbers W1, W2, W3 and a random sequence G, and finally obtain the ciphertext image C.

[0015] The method for calculating the initial parameters of the four-dimensional hyperchaotic system is as follows: input the plaintext image P of size M×N into the SHA-256 hash function to obtain a 256-bit binary hash value H; divide the hash value H into 32 groups evenly, with each group having 8 bits, to obtain the hash value H = k1, k2, k3, … k 32 ; calculate the initial parameters x0, y0, z0, w0 of the chaotic system by using the hash value and an external key as follows:

[0016]

[0017]

[0018] where η1, η2, η3, η4, η5, η6, η7, η8 are all intermediate variables, and k1 - k 32 are 32 groups of 8-bit binary hash values, is the XOR operation, mod is the modulo function, and θ(1), θ(2), θ(3) and θ(4) are external keys.

[0019] The four-dimensional hyperchaotic system is an improved four-dimensional chaotic system, and the mathematical model of the improved four-dimensional chaotic system is:

[0020]

[0021] where x, y, z, w are the state variables of the four-dimensional chaotic system, are the derivatives of the state variables x, y, z, w respectively, and a, b, c, d, e are the control parameters of the four-dimensional chaotic system.

[0022] The method for obtaining the integer chaotic sequences F1, F2, F3, and F4 is as follows: discretize the four chaotic sequences X', Y', Z', and W' respectively, then

[0023]

[0024] where X′(i), Y′(i), Z′(i), W′(i), F1(i), F2(i), F3(i), and F4(i) are the values of the i-th elements of the chaotic sequences X', Y', Z', W', the integer chaotic sequences F1, F2, F3, and F4 respectively, where i = 1, 2, 3, …… M×N; floor is the floor function;

[0025] The method for obtaining the random numbers W1, W2, W3 and the random sequence G is as follows: take one element from each of the integer chaotic sequences F1, F2, and F3 and combine them with the external secret keys θ(1), θ(2), and θ(3) respectively to generate the random numbers W1, W2, and W3; select the last three elements from the integer sequence F4 to obtain the integer sequence g = [F4(M×N), F4(M×N - 1), F4(M×N - 2)], and obtain the integer sequence G, and

[0026]

[0027] where F1(M×N - 1), F2(M×N - 1), and F3(M×N - 1) are the values of the (M×N - 1)-th elements of the integer chaotic sequences F1, F2, and F3 respectively, and F4(M×N), F4(M×N - 1), and F4(M×N - 2) are the values of the (M×N)-th, (M×N - 1)-th, and (M×N - 2)-th elements of the integer chaotic sequence F4 respectively.

[0028] Convert the pixel values of the plaintext image P into 8-bit binary to obtain an 8-bit binary matrix T1, and map every two bits of each element in the 8-bit binary matrix T1 to a base to form a DNA sequence composed of 4 bases; according to different coding methods, there are 8 common DNA coding rules as follows:

[0029]

[0030] Adopt three basic operations: DNA addition +, DNA subtraction –, and DNA exclusive OR The operation rules are

[0031]

[0032] Combine with the random number L to select the DNA coding rule, and the random number L = mod(W1, 8)+1).

[0033] The method for constructing the Latin square DNA substitution table is as follows: Use a 4th-order Latin square to generate a fourth-order matrix R1. Combine each element in matrix R1 with its column number to form a two-digit integer matrix R2, where the tens digit is the element value in matrix R1 and the units digit is the column number where the element is located. Convert the element values in the tens and units digits of the integer matrix R2 into corresponding bases respectively to generate a base matrix R3. Set the row number and column number as bases A, T, C, and G respectively to construct the Latin square DNA substitution table R. Among them, the element values 1 in the tens and units digits of the integer matrix R2 correspond to base A, 2 corresponds to base T, 3 corresponds to base C, and 4 corresponds to base G.

[0034] The element values 1 in the tens and units digits of the integer matrix R2 correspond to base A, 2 corresponds to base T, 3 corresponds to base C, and 4 corresponds to base G.

[0035] The method for replacing each base in the base matrix T2 with the Latin square DNA substitution table according to the DNA substitution rule is as follows: Select the first 256 out of 576 fourth-order Latin squares to construct a DNA substitution table set. Select the DNA substitution table with the corresponding number according to the element value of the integer chaotic sequence F4, and replace the bases in the base matrix T2 in groups of two. The first digit is used to determine the column number, and the second digit is used to determine the row number. Search for the base at the corresponding position in the selected DNA substitution table and replace the original base with the base at the corresponding position to obtain a new DNA base matrix T3.

[0036] The method for obtaining the base matrix T6 is as follows: Convert the base matrix T3 into a one-dimensional DNA sequence T4 and divide it evenly into four segments: the first segment is T4(1:MN), the second segment is T4(MN + 1:2MN), the third segment is T4(2MN + 1:3MN), and the fourth segment is T4(3MN + 1:4MN). Sort the four integer chaotic sequences F1, F2, F3, and F4 in ascending order respectively to obtain the corresponding index sequences I1, I2, I3, and I4. Rearrange the four divided DNA sequences respectively according to the order of the elements in the index sequences I1, I2, I3, and I4, shuffle their base positions, and obtain new sequences S1, S2, S3, and S4 respectively. Combine the bases at the same positions of the four scrambled sequences in order and crosswise to obtain a base sequence T5, and restore it to matrix form to obtain the base matrix T6.

[0037] The method for the forward and reverse random weighted diffusion operations is as follows: Use the random sequence G to select random numbers W1, W2, W3 as weights. The forward random weighted diffusion is

[0038]

[0039] Among them, G is a random sequence, W G(1), W G(2) , w G(3) They are respectively the random numbers W1, W2, and W3 corresponding to the values of G(1), G(2), and G(3). F1(1) and F1(i) are the values of the 1st and the i-th elements of the integer chaos sequence F1. J(1), J(i), and J(M×N) are respectively the values of the 1st, the i-th, and the M×N-th elements of the matrix J after DNA decoding. E(i - 1) is the value of the (i - 1)-th element of the image E after forward weighted diffusion;

[0040] The reverse random weighted diffusion is as follows:

[0041]

[0042] Among them, C(i) is the value of the i-th element of the image C after forward weighted diffusion. E(M×N) represents the M×N-th element of the image E. F2(M×N) represents the value of the M×N-th element of the integer sequence F2. F2(i) represents the value of the i-th element of the integer sequence F2. C(i - 1) represents the value of the (i - 1)-th element of the image C. The matrix J, the image E, and the image C are all accessed in sequence order.

[0043] Compared with the prior art, the beneficial effects of the present invention are as follows: An image encryption algorithm integrating an improved four-dimensional hyperchaotic system and DNA coding is proposed. First, on the basis of the classical three-dimensional Chen system, a non-linear coupling and feedback mechanism is introduced to construct a four-dimensional hyperchaotic system with stronger chaotic characteristics. Its chaotic behavior is verified through phase trajectory diagrams, bifurcation diagrams, and NIST randomness tests, and its maximum Lyapunov exponent reaches 3.1385. Subsequently, a key derivation structure is constructed by combining the SHA-256 hash function and an external key to realize the adaptive generation of a plaintext-driven key. In the encryption process, a DNA substitution table is randomly generated by a fourth-order Latin square, and the sequence after DNA coding of the plaintext image is replaced and confused; the scrambling operation is completed through ascending sorting and cross-combination guided by the chaos sequence, and image reconstruction is realized by combining DNA operations; finally, a bidirectional weighted diffusion mechanism is introduced to enhance the sensitivity and diffusivity of the system. Experimental results show that the method in this paper exhibits good security and robustness in terms of indicators such as information entropy, NPCR, and UACI, and is suitable for the high-strength encrypted transmission requirements of image data in the Internet of Things environment. Description of the Drawings

[0044] 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 following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0045] Figure 1 This is the flowchart of the present invention.

[0046] Figure 2 This is the comparison diagram of the phase diagrams of the four-dimensional hyperchaotic system of the present invention and traditional systems. Among them, (a) is the x-y phase space of the present invention, (b) is the x-z phase space of the present invention, (c) is the x-w phase space of the present invention, (d) is the y-z phase space of the present invention, (e) is the x-y-z phase space of the present invention, (f) is the x-y-z phase space of the Chen system, (g) is the x-y-z phase space of the Lorenz system, and (h) is the x-y-w phase space of the four-dimensional Chen system.

[0047] Figure 3 This is the comparison diagram of the Lyapunov exponent spectra of the four-dimensional hyperchaotic system of the present invention and traditional systems. Among them, (a) is the four-dimensional hyperchaotic system of the present invention, (b) is the Chen system, (c) is the Lorenz system, and (d) is the four-dimensional Chen system.

[0048] Figure 4 This is the comparison of the bifurcation diagrams of chaotic systems. Among them, (a) shows that the control parameter a of the four-dimensional hyperchaotic system of the present invention is variable, (b) shows that the control parameter b of the four-dimensional hyperchaotic system of the present invention is variable, (c) shows that the control parameter d of the four-dimensional hyperchaotic system of the present invention is variable, (d) is the Chen system, (e) is the Lorenz system, and (f) is the four-dimensional Chen system.

[0049] Figure 5 This is the comparison of the LE diagrams of chaotic systems. Among them, (a) shows that the control parameter a of the four-dimensional hyperchaotic system of the present invention is variable, (b) shows that the control parameter b of the four-dimensional hyperchaotic system of the present invention is variable, (c) shows that the control parameter d of the four-dimensional hyperchaotic system of the present invention is variable, (d) is the Chen system, (e) is the Lorenz system, and (f) is the four-dimensional Chen system.

[0050] Figure 6 This is the double-parameter maximum LE diagram of the four-dimensional hyperchaotic system of the present invention. Among them, (a) is the maximum LE diagram with the control parameters a and b variable, and (b) is the maximum LE diagram with the control parameters d and e variable.

[0051] Figure 7 This is the initial value sensitivity analysis diagram of the four-dimensional hyperchaotic system of the present invention. Among them, (a) is x0 + 10 -14 , (b) is y0 + 10 -14 , (c) is z0 + 10 -14 , (d) is w0 + 10 -14 .

[0052] Figure 8 This is an example diagram of a Latin square.

[0053] Figure 9 Schematic diagram for generating the DNA replacement table of the present invention.

[0054] Figure 10 Diagram generated for the DNA replacement of the present invention.

[0055] Figure 11 Example diagram for DNA segmentation and scrambling of the present invention.

[0056] Figure 12 Diagram of the image encryption model used in IoT secure communication.

[0057] Figure 13 Simulation result diagram of typical test images of the present invention, where (a) is the Baboon plaintext image, (b) is the Baboon ciphertext image, (c) is the Baboon decrypted image, (d) is the Brain plaintext image, (e) is the Brain ciphertext image, (f) is the Brain decrypted image, (g) is the Finger plaintext image, (h) is the Finger ciphertext image, (i) is the Finger decrypted image, (j) is the Cell plaintext image, (k) is the Cell ciphertext image, (l) is the Cell decrypted image, (m) is the Boat plaintext image, (n) is the Boat ciphertext image, (o) is the Boat decrypted image.

[0058] Figure 14 Comparison diagram for key sensitivity analysis of the present invention, where (a) is the Baboon plaintext image, (b) is the image encrypted with the correct key, (c) is the image decrypted with θ(1)+10 -14 decryption, (d) is the image decrypted with θ(2)+10 -14 decryption, (e) is the image decrypted with θ(3)+10 -14 decryption, (f) is the image decrypted with θ(4)+10 -14 decryption.

[0059] Figure 15 Comparison diagram of the gray histograms of the plaintext images and the corresponding ciphertext images of the present invention, where (a) is the Baboon plaintext image, (b) is the Baboon ciphertext image, (c) is the Brain plaintext image, (d) is the Brain ciphertext image, (e) is the Finger plaintext image, (f) is the Finger ciphertext image, (g) is the Cell plaintext image, (h) is the Cell ciphertext image, (i) is the Boat plaintext image, (j) is the Boat ciphertext image.

[0060] Figure 16This is the correlation comparison diagram of the Baboon image of the present invention in various directions. Among them, (a) are adjacent pixels in the horizontal direction of the plaintext image, (b) are adjacent pixels in the horizontal direction of the ciphertext image, (c) are adjacent pixels in the vertical direction of the plaintext image, (d) are adjacent pixels in the vertical direction of the ciphertext image, (e) are adjacent pixels in the diagonal direction of the plaintext image, and (f) are adjacent pixels in the diagonal direction of the ciphertext image.

[0061] Figure 17 These are the ciphertext images and decrypted images of the present invention after being attacked by different degrees of noise. Among them, (a) is the ciphertext image with an intensity of 0.01, (b) is the ciphertext image with an intensity of 0.05, (c) is the ciphertext image with an intensity of 0.1, (d) is the decrypted image with an intensity of 0.01, (e) is the decrypted image with an intensity of 0.05, and (f) is the ciphertext image with an intensity of 0.1.

[0062] Figure 18 These are the ciphertext images and decrypted images of the present invention under different cropping attacks. Among them, (a) is the 1 / 16 cropped ciphertext image, (b) is the 1 / 4 cropped ciphertext image, (c) is the 1 / 2 cropped ciphertext image, (d) is the decrypted image of (a), (e) is the decrypted image of (c), and (f) is the decrypted image of (e). Detailed implementation manners

[0063] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying 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.

[0064] As Figure 1 shown, an image secure transmission method based on a four-dimensional hyperchaotic system and DNA coding, the main innovation points are: (1) constructing a four-dimensional hyperchaotic system with high Lyapunov exponents and complex trajectory characteristics to effectively increase the key space and system dynamic complexity; (2) introducing a joint derivation mechanism of the SHA-256 hash function and an external key to realize plaintext-driven key initialization; (3) generating a dynamic DNA substitution table based on a third-order Latin square, and combining chaotic sorting and cross-combination to realize pixel scrambling; (4) designing a two-way weighted diffusion mechanism based on DNA operations to enhance the nonlinearity and security strength of image encryption. The present invention proposes a highly secure image encryption scheme based on an improved four-dimensional chaotic system, and this scheme mainly includes five stages: key generation, Latin square DNA substitution, DNA sequence index cross-scrambling, DNA logical operation, and two-way random weighted diffusion. Suppose the size of the plaintext image P is M×N, and the specific steps of the encryption process are as follows:

[0065] Input: A plaintext image P of size M×N, initial parameters a, b, c, d, f, m, and external keys θ(1), θ(2), θ(3), θ(4).

[0066] Output: A ciphertext image C.

[0067] Step 1: Use the SHA-256 hash function to calculate the hash value H of the input plaintext image P. Combine the external keys to calculate the initial parameters of the four-dimensional hyperchaotic system: x0, y0, z0, w0.

[0068] First, input the plaintext image P into the SHA-256 hash function, and the output is a 256-bit binary hash value H. Divide the hash value H into 32 groups, each group with 8 bits, i.e., H = k1, k2, k3, … k 32 . To increase the key space, the present invention introduces 4 external keys θ(1), θ(2), θ(3), and θ(4). Substitute the hash value and the external keys into the following formulas (1) and (2) to calculate the initial parameters x0, y0, z0, w0 of the chaotic system.

[0069]

[0070] where η1, η2, η3, η4, η5, η6, η7, η9 are all intermediate variables, and k1 - k 32 are 32 groups of 8-bit binary hash values, is the exclusive OR operation, and mod is the modulo function.

[0071] Step 2: Substitute the initial parameters x0, y0, z0, w0 into the four-dimensional hyperchaotic system, iterate 1000 times to remove the transient effect, and then continue to iterate M×N times to generate four chaotic sequences X', Y', Z', W' respectively. Perform data processing to obtain the integer chaotic sequences F1, F2, F3, F4, random numbers W1, W2, W3, and the random sequence G required for subsequent encryption.

[0072] The four-dimensional hyperchaotic system is an improved four-dimensional chaotic system. The chaotic system is a typical class of nonlinear dynamic systems, whose behavior has high sensitivity and unpredictability, and is widely used in security fields such as image encryption. The Chen system is a classic three-dimensional continuous chaotic system, and its mathematical model is shown in formula (3):

[0073]

[0074] where the parameters a, b, and c control the dynamic behavior of the Chen chaotic system. When the values are a = 35, b = 3, c = 28, the Chen chaotic system presents a typical chaotic state. This Chen chaotic system has good initial value sensitivity and long-period characteristics, and is suitable for generating key streams or scrambling sequences in encryption schemes.

[0075] To improve the performance of the chaotic system and enhance the nonlinear behavior and dynamic complexity of the system, based on the classical Chen chaotic system, this invention introduces additional dimensions and nonlinear terms, and its mathematical model is shown in Equation (4):

[0076]

[0077] where x, y, z, w are the state variables of the four-dimensional chaotic system, are the derivatives of the state variables x, y, z, w respectively, and a, b, c, d, e are the control parameters of the four-dimensional chaotic system.

[0078] The phase diagram is an important means for visualizing the dynamic behavior of a chaotic system. For a system with good chaotic characteristics, its phase diagram usually appears disordered and has a complex structure, reflecting the high nonlinearity and randomness of the system. Under the conditions that the control parameters are set as a = 26, b = 18, c = 1.1, d = 1.2, e = 3, and the initial parameters are x0 = 0.2, y0 = 1, z0 = 0.3, w0 = 0.1, the multi-dimensional phase diagram of the improved system is plotted, and the results are as Figure 2 shown. Compared with the classical three-dimensional Lorenz system and four-dimensional Chen system, the phase diagram of the improved system has a wider coverage range and a more complex structure, further verifying its excellent chaotic performance.

[0079] The corresponding Lyapunov Exponents (LE) results are as Figure 3 and Table 1 show. It can be seen that the improved system has a larger chaotic range, and there are two positive values in the LE spectrum. Its maximum positive LE is significantly higher than that of other systems, indicating that it is a typical hyperchaotic system. In addition, the sum of the LEs is less than zero, meeting the basic criterion of a chaotic system.

[0080] Table 1 Lyapunov Exponents of the Chaotic System

[0081]

[0082] Among them, the four-dimensional multi-scroll hyper-chaotic system is applied in Reference [1]---[Zhang J, Zuo J, Guo Y, Hou J, Xie Q (2023) Nonlinear analysis, circuit implementation, and application in image encryption of a four-dimensional multi-scroll hyper-chaotic system. Integration, 102126.]; the new hyper-chaotic system with line equilibrium is applied in Reference [2]---[Benkouider K, Bouden T, Halimi M (2019) Dynamical analysis, synchronization and circuit implementation of a new hyperchaotic system with line equilibrium. In 2019 6th international conference on control, decision and information technologies (CoDIT) IEEE, pp, 1717-1722.]; the quadratic autonomous four-dimensional hyper-chaotic system is applied in Reference [3]---[Zarei A, Tavakoli S (2016) Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system. Applied Mathematics and Computation, 291, 323-339.]; the Lorenz hyper-chaotic system is applied in Reference [4]---[Jia Q (2007) Hyperchaos generated from the Lorenz chaotic system and its control. Physics Letters A, 366, 3, 217-222)].

[0083] Bifurcation diagrams are often used to study the dynamic evolution process of chaotic systems under parameter variations. A good bifurcation diagram should clearly show the transition path of the system from stability to chaos. Set the initial conditions as x0 = 0.2, y0 = 1, z0 = 0.3, w0 = 0.1, and the parameters as a = 26, b = 18, c = 1.1, d = 1.2, e = 3. With only one parameter changed, the following three-parameter bifurcation analyses were conducted on the improved four-dimensional hyperchaotic system: a ∈ (20, 45), with the other parameters unchanged. When a < 23.12, it shows an alternating state of periodicity and chaos; when a > 23.12, the system enters the hyperchaotic state, as shown in Figure 4 (a) of Figure 4 ; b ∈ (5, 20), with the other parameters unchanged. The system is in a chaotic state when b < 10.13; it is in a hyperchaotic state when b > 10.13, as shown in (b) of

[0084] ; when d ∈ (0, 50), with the other parameters unchanged, the system is always in a hyperchaotic state. Figure 4

[0085] LE is one of the key indicators for identifying the chaotic characteristics of dynamic systems. For high-order dynamic systems, due to different directions of the initial separation vectors, there may be multiple LEs. When the LEs of the system are four negative numbers, it indicates that the system is in a fixed-point state; if the LE is one positive number and three negative numbers, the system is in a chaotic orbit; and if there are at least two positive numbers or more for the LEs, it indicates that the system is in a hyperchaotic state. The improved system is a four-dimensional chaotic system, so it has four LEs.

[0086] For the improved system, when setting the initial conditions as x0 = 0.2, y0 = 1, z0 = 0.3, w0 = 0.1, and the parameters as a ∈ (20, 45), b = 18, c = 1.1, d = 1.2, e = 3, the LE spectrum is as shown in Figure 5 (a) of Figure 5 . When the control parameter a is in the range of a ∈ (20, 23.12), there is one positive LE, indicating that the system is in a chaotic state; when the control parameter a ∈ (23.12, 45), the LEs are two positive values, indicating that the system enters the hyperchaotic state. (b) of Figure 5Figure (c) shows the LE spectrum when the initial value remains unchanged, with parameters a = 26, b = 18, c = 1.1, d ∈ (0, 50), and e = 3. When d is within the range of (0, 50), there are two positive LEs, indicating that the system is in a hyperchaotic state. Figure 5 Figures (d), (e), and (f) respectively show the LE spectra of the three-dimensional Chen, Lorenz systems, and the four-dimensional Chen system. The comparison results show that the chaotic range of the improved four-dimensional hyperchaotic system exceeds that of other systems, and its maximum LE value is also higher. In summary, the improved system exhibits more excellent chaotic characteristics compared to the three-dimensional Chen, Lorenz, and four-dimensional Chen systems, and is more suitable for the field of Internet of Things encryption.

[0087] The complexity of the chaotic system makes it highly sensitive to parameters, and small changes may directly affect the chaotic state of the system. The dynamic characteristics of the system are usually significantly reflected by the maximum LE. When the initial conditions are set as x0 = 0.2, y0 = 1, z0 = 0.3, w0 = 0.1, and the control parameters are a ∈ (20, 45), b ∈ (5, 20), c = 1.1, d = 1.2, e = 3, its two-parameter LE diagram is as shown in Figure 6 Figure (a); when the initial conditions are set as x0 = 0.2, y0 = 1, z0 = 0.3, w0 = 0.1, and the parameters are a = 26, b = 18, c = 1.1, d ∈ (0, 50), e ∈ (-10, 20), its two-parameter LE diagram is as shown in Figure 6 Figure (b). The hue in the figure represents the magnitude of the LE, with warm colors representing larger values and dark colors representing smaller values. It can be seen that the improved system has a wide chaotic range.

[0088] The high sensitivity of the chaotic system to the initial conditions is one of its core characteristics. To verify this characteristic, the change amount of each initial value is set to 10 -14 . As shown in Figure 7 , any small perturbation of a variable causes the system trajectory to quickly deviate from the original orbit, verifying that the system has typical chaotic initial value sensitivity.

[0089] To further verify the randomness of the sequence of the improved system, the present invention uses the NIST SP800-22 standard for testing. The test results are shown in Table 2. All state variables pass the judgment in each statistical test, indicating that the generated sequence has good statistical characteristics and pseudo-randomness.

[0090] Table 2 NIST Test Results

[0091]

[0092]

[0093] In a chaotic system, an equilibrium point is defined as a point or trajectory where the system state remains unchanged. For a dynamical system, the equilibrium point can be expressed as a certain value \(x\) of the state vector \(x\) * , which satisfies the condition: \(f(x\) * ) = 0, where \(f(x)\) represents the evolution function or iterative equation of the chaotic system. When the system is at the equilibrium point \(x\) * , the state will remain unchanged.

[0094] By solving, one equilibrium point \(E(0, 0, 0, 0)\) of the four-dimensional hyperchaotic system is obtained. The Jacobian matrix \(J\) of the four-dimensional hyperchaotic system is shown in formula (5):

[0095]

[0096] Set the parameters as \(a = 26\), \(b = 18\), \(c = 1.1\), \(d = 1.2\), \(e = 3\). The eigenvalues of the three equilibrium points are calculated through the Jacobian matrix, and the results are shown in Table 3.

[0097] Table 3 The characteristic roots of each equilibrium point

[0098]

[0099] As can be seen from Table 3, the real parts of the characteristic roots corresponding to the equilibrium points are not all negative. Therefore, this equilibrium point is an unstable equilibrium point, which indicates that the improved system is highly sensitive and responsive to small changes in the initial conditions, and the system behavior has a high degree of uncertainty, thus being able to generate highly complex and unpredictable sequences. The divergence of the improved system is:

[0100]

[0101] Set the parameters as \(a = 26\), \(b = 18\), \(c = 1.1\), \(d = 1.2\), \(e = 3\), and the initial values as \(x0 = 0.2\), \(y0 = 1\), \(z0 = 0.3\), \(w0 = 0.1\). It can be obtained that \(\tau=-10.7678\lt0\). The divergence is negative, so the improved system is a dissipative system.

[0102] Substitute the initial parameters \(x0\), \(y0\), \(z0\), \(w0\) into the improved four-dimensional chaotic system. First, iterate 1000 times to eliminate the transient state, and then iterate \(M\times N\) times to obtain four chaotic sequences: \(X'=[x'1,x'2,x'3,\cdots x' M×N , Y'=[y'1,y'2,y'3,\cdots y' M×N , Z'=[z'z1,z'2,z'3,\cdots z' M×N , W'=[w'1,w'2,w'3,\cdots w' M×N, the four chaotic sequences are brought into formula (7) for discretization to obtain integer chaotic sequences F1, F2, F3, and F4. One element is taken from each of the integer chaotic sequences F1, F2, and F3 and combined with the external secret keys θ(1), θ(2), and θ(3) respectively to generate random numbers W1, W2, and W3. Finally, three elements at the tail are selected from the integer sequence F4 to obtain the integer sequence g = [F4(M×N), F4(M×N - 1), F4(M×N - 2)], and the integer sequence G is obtained according to formula (8).

[0103]

[0104] Among them, floor is the floor function, and θ(1), θ(2), and θ(3) are external secret keys.

[0105] Step 3: Convert the plaintext image P into an 8-bit binary matrix T1, select a DNA coding rule to perform DNA coding on the binary matrix T1 to obtain a base matrix T2. Similarly, perform DNA coding on the integer chaotic sequence F1 to obtain a base matrix J1.

[0106] DNA coding is an image data encryption technology that simulates the characteristics of biomolecules. Its core idea is to convert pixel data into an analog DNA sequence to achieve complex representation and processing of data.

[0107] In the present invention, the pixel values of the plaintext image P are first converted into 8-bit binary, and then every two bits of binary are mapped to a base (A, T, C, G), thereby forming a DNA sequence composed of 4 bases. According to different coding methods, there are 8 common DNA coding rules, and their definitions are shown in Table 4. For example, for the pixel value 222, its binary is 11011110. If rule 2 is used for coding, its corresponding DNA sequence is TCTG. DNA decoding is the reverse operation of the above process. Based on the currently selected coding rule, it is restored to the corresponding binary string by looking up the table one by one according to the base sequence, and then restored to the image pixel value.

[0108] Table 4 DNA Coding Rules

[0109]

[0110] DNA operation is a special data operation method based on the base sequence, which is suitable for constructing the diffusion process in image encryption. The present invention adopts three basic operations: DNA addition (+), DNA subtraction (–), and DNA exclusive OR (⊕), and the operation rules are shown in Table 5. By flexibly selecting coding rules and operation methods, the nonlinearity and complexity of the encryption scheme can be effectively improved, and the ability to resist statistical attacks and differential attacks can be enhanced.

[0111] Table 5 Three DNA Operation Rules

[0112]

[0113] Step 4: Construct a Latin Square DNA substitution table, and use the DNA substitution rule to replace each base in the base matrix T2 by looking up the Latin Square DNA substitution table, generating a new base matrix T3.

[0114] A Latin Square is a classic combinatorial design structure that can be traced back to the 18th century and has extensive engineering and scientific applications. It exhibits good randomness and balance, especially in image encryption. An n-order Latin square is an n×n square matrix in which the rows and columns both contain the numbers from 1 to n, and each number appears only once in each row and each column, with no fixed order. Its main features include: each symbol is evenly distributed in different rows and columns, which helps the uniform diffusion of information during the encryption process; multiple Latin squares can be constructed by randomly generating different permutations, enhancing the diversity and unpredictability of the scrambling operation. Figure 8 Examples of 3-order and 4-order Latin squares are shown to illustrate their permutation characteristics and construction methods.

[0115] The present invention proposes a DNA substitution method based on a Latin square, which mainly includes two parts: the construction of a Latin Square DNA substitution table and the design of a DNA substitution rule.

[0116] Construction of the Latin Square DNA substitution table: First, use a 4-order Latin square to generate a fourth-order matrix R1. Then, combine each element in matrix R1 with its column number to form a two-digit integer matrix R2, where the tens digit is the element value in matrix R1 and the units digit is the column number of the element. Subsequently, according to the element values of the integer matrix R2, convert them into corresponding bases (where 1 corresponds to A, 2 corresponds to T, 3 corresponds to C, and 4 corresponds to G), generating a base matrix R3. Finally, set the row numbers and column numbers to A, T, C, and G respectively to construct the DNA substitution table R. The detailed process is as Figure 9 shown.

[0117] Design of DNA replacement rules: The total number of fourth-order Latin squares is known to be 576. The present invention generates all 576 fourth-order Latin squares according to a specific construction method, and selects the first 256 of them to construct a corresponding set of DNA replacement tables. During the encryption process, for each pixel to be encrypted, the DNA replacement table with the corresponding number is selected according to the element value of the integer chaos sequence F4, further increasing the complexity of the pixel value transformation. The original plaintext image P is first converted into a binary matrix, and DNA encoding is performed in combination with the DNA encoding rule selected by another random number L (L = mod(W1, 8)+1). Then, every two bits in the DNA encoding result, i.e., the base matrix T2, are used as a group to look up and replace in the Latin square DNA replacement table, where the first bit is used to determine the column number and the second bit is used to determine the row number. Look up the base at the corresponding position in the selected DNA replacement table R and replace the original base with it (for example, if the first bit is A and the second bit is G, then select the element AA in the first row and second column of the replacement table for replacement), and finally obtain a new DNA base matrix. The replacement process is as Figure 10 shown.

[0118] Step 5: Convert the base matrix T3 into a one-dimensional DNA sequence T4 and evenly divide it into 4 segments. Use the index sequences I1, I2, I3, I4 generated by the integer chaos sequences F1, F2, F3, F4 to perform index scrambling on the four DNA sequences respectively, and obtain sequences S1, S2, S3, S4 respectively; cross the bases at the same positions of the four sequences S1, S2, S3, S4 in order and put them together to obtain a base sequence T5, and restore it to the form of a matrix to obtain a base matrix T6.

[0119] To enhance the randomness of the DNA sequence and effectively weaken the high correlation between adjacent pixels in the plaintext image, a DNA sequence index cross-scrambling technique is proposed. First, convert the base matrix T3 into a one-dimensional DNA sequence T4 and evenly divide it into four segments: the first segment is T4(1:MN), the second segment is T4(MN + 1:2MN), the third segment is T4(2MN + 1:3MN), and the fourth segment is T4(3MN + 1:4MN). Sort the four integer chaos sequences F1, F2, F3, and F4 in ascending order respectively to obtain the corresponding index sequences I1, I2, I3, and I4 at the corresponding positions. Use the order of the elements in the index sequences I1, I2, I3, and I4 to rearrange the four divided DNA sequences respectively, and shuffle their base positions to obtain new sequences S1, S2, S3, S4 respectively. Then, to further shuffle the base positions, cross-combine the bases at the same positions of the four scrambled sequences in order and put them together to obtain a base sequence T5, and restore it to the form of a matrix to obtain a base matrix T6. The detailed process is as Figure 11 shown.

[0120] Step 6: Randomly select a DNA coding rule to perform DNA coding on the integer sequence F1 to obtain the base matrix J1. Perform a DNA exclusive OR operation on the base matrix T6 and the base matrix J1 to obtain the base matrix T7. Perform DNA decoding on the base matrix T7 according to the DNA decoding rule to obtain the binary matrix J2, and convert the binary matrix J2 to decimal to obtain the matrix J.

[0121] Step 7: Use the random numbers W1, W2, W3 and the random sequence G to perform forward and reverse random weighted diffusion operations on the matrix J in sequence, and finally obtain the ciphertext image C.

[0122] To further improve the security of the encrypted image, a method of bidirectional random weighted diffusion is proposed. The random sequence G is used to select the weight W to perform double diffusion on the image pixels in the forward and reverse directions, which can not only completely change the pixel values but also achieve the global diffusion effect. The length of the random sequence G is 3, and the values are from 1 to 3, which is used to select the random numbers W1, W2, W3. For example, if G(1)=1, then W G(1) =W1; if G(1)=2, then W G(1) =W2. The forward random weighted diffusion is carried out according to formula (9).

[0123]

[0124] Among them, G is the random sequence, W is the weight, F1 is the chaotic sequence, J is the matrix after DNA decoding, and E is the image matrix after forward weighted diffusion. The reverse weighted diffusion is carried out according to formula (10), where the weight W is randomly selected according to the random sequence G.

[0125]

[0126] Among them, F2 is the chaotic sequence, and C is the image after forward weighted diffusion. E(M×N) represents the M×Nth element of the image matrix E, F2(M×N) represents the value of the M×Nth element of the integer sequence F2, F2(i) represents the value of the ith element of the integer sequence F2, C(i - 1) represents the (i - 1)th element of C, and C(i) represents the ith element of the image C. The matrices J, the images E, and the image C in the sequence processing process are all accessed according to the sequence.

[0127] The decryption process is the inverse process of the encryption algorithm, Figure 1 which is the flowchart of the encryption algorithm.

[0128] The image encryption scheme proposed by the present invention has good versatility and can be widely applied to security communication scenarios in the Internet of Things environment. Such as Figure 12As shown, in a typical Internet of Things (IoT) communication model, the sender and receiver are usually embedded devices with the capabilities of image acquisition, storage, display, and transmission. To ensure the data confidentiality and integrity of images during network transmission, the sender first encrypts the images so that even if the communication process is illegally eavesdropped, attackers cannot recover the valid information. On the premise of legally obtaining the key, the receiver can use the decryption algorithm to restore the original plaintext image.

[0129] In summary, the image encryption algorithm proposed by the present invention can effectively resist the security threats that images may face during transmission, and has broad application prospects and promotion value in various IoT application scenarios such as smart home, telemedicine, intelligent transportation, and industrial control.

[0130] To verify the feasibility and encryption performance of the algorithm proposed by the present invention, a simulation experiment was conducted on this solution on the MATLAB 2023a platform. During the experiment, the external key parameters were set as: θ(1)=1, θ(2)=15, θ(3)=10, θ(4)=20.

[0131] Figure 13 The encryption effects of typical test images Baboon, Brain, Finger, Cell, and Boat are shown, including the comparison of the plaintext image, ciphertext image, and decrypted image. The test images Baboon, Brain, Finger, Cell, and Boat are from the websites https: / / ccia.ugr.es / cvg / dbimagenes / g256.php and https: / / sipi.usc.edu / database / database.php respectively. It can be clearly observed from the visual effects of the images that: the ciphertext of the encrypted image is completely different from the original image visually, and the information is severely disrupted; the decrypted image is exactly the same as the original image in terms of details, textures, contours, etc., without information loss or distortion. This indicates that the proposed image encryption scheme not only performs well in the non-readability of images, but also has good reversibility and fidelity, and can effectively guarantee the security and integrity of image information during IoT data transmission.

[0132] When evaluating the security and reliability of an image encryption algorithm, a comprehensive analysis needs to be carried out from multiple dimensions. The present invention will conduct a systematic analysis from aspects such as key space, key sensitivity, image histogram, variance analysis, pixel correlation, differential attack, information entropy, local information entropy, PSNR, noise attack, and cropping attack to verify the robustness and anti-attack ability of the proposed encryption scheme.

[0133] The key space is the set of all possible keys in an encryption system, and its size is directly related to the algorithm's ability to resist brute-force attacks. Generally speaking, if the key space is not less than 2 100 , the algorithm can be considered to have sufficient security.

[0134] The key structure of the encryption algorithm proposed in the present invention consists of the following parts: a group of 256-bit image hash values (extracted by SHA-256); the initial values x0, y0, z0, w0 of the chaotic system, with a calculation precision of 10 -15 ; and four external key parameters. Considering all the above key elements comprehensively, the size of the overall key space can be expressed as: 2 256 *10 120 , which is much larger than 2 100 , so it is sufficient to effectively resist exhaustive search and brute-force attacks and has extremely high security.

[0135] A secure image encryption algorithm not only needs to have a large key space but also should exhibit a high degree of key sensitivity. When conducting the key sensitivity test, a small change can be made to the original key, and then the encrypted image can be decrypted with this key. If there are significant differences between the decrypted image and the original plaintext image, it indicates that the key has good sensitivity.

[0136] In the sensitivity test of the present invention, taking the Baboon image as an example, after encrypting with the initial key, a small perturbation (such as changing by 10 -14 ) is made to one of the external keys θ(1)=1, θ(2)=15, θ(3)=10, θ(4)=20, and then the ciphertext image is decrypted. Figure 14 Shows the decrypted images under the original key and the perturbed key. It can be clearly observed that even if only a very small change is made to one key parameter, the resulting decrypted image is completely inconsistent with the original plaintext image, with a large distortion and no valuable image information can be obtained. This experimental result fully demonstrates that the proposed image encryption scheme has extremely strong key sensitivity and can effectively prevent attack strategies of key approximate matching.

[0137] Histogram analysis is a common and effective method for evaluating the performance of image encryption, which is used to measure the distribution characteristics of image pixel values in the gray space. Generally speaking, the pixel values of the plaintext image usually show a highly concentrated or structural distribution and are extremely vulnerable to being exploited for statistical analysis attacks. An ideal ciphertext image should have a uniformly distributed histogram, that is, the pixel frequencies of all gray levels are roughly equal, so as to enhance the ability to resist statistical attacks.

[0138] Figure 15It shows the comparison of the grayscale histograms of images such as Baboon, Brain, Finger, Cell, and Boat before and after encryption. The results indicate that: the histogram of the ciphertext image has a more uniform pixel distribution at each gray level, basically presenting a flat trend, while the histogram of the corresponding plaintext image has obvious fluctuations and peak distributions. This phenomenon further proves that the proposed image encryption scheme can effectively destroy the original pixel statistical characteristics of the image, thereby enhancing the security of the system.

[0139] To further verify the uniformity of the pixel value distribution of the ciphertext image from a quantitative perspective, the present invention introduces the histogram variance index for analysis. The smaller the variance, the more uniform the pixel values are distributed in the gray space, and the more difficult it is for the ciphertext image to be cracked by statistical analysis attacks. The variance calculation formula is as follows:

[0140]

[0141] In the formula, f i represents the frequency of the gray level i appearing in the ciphertext image, g = (M×N) / 256 is the expected frequency of each gray value under the theoretical uniform distribution, and M and N are the number of rows and columns of the image respectively. At the significance level of 0.05, the corresponding critical value is When the histogram variance of the image is satisfied, it can be determined that the pixel distribution of the image tends to be uniform. Table 6 shows the variance values of typical images before and after encryption. It can be observed from Table 6 that the histogram variances of all ciphertext images are significantly lower than those of the plaintext images, and the corresponding pixel distributions are more uniform, meeting Thereby further verifying the effectiveness of the encryption algorithm and its ability to resist statistical attacks.

[0142] Table 6 Variance results of plaintext images and ciphertext images

[0143]

[0144] In the field of image encryption, adjacent pixels of plaintext images usually have strong correlations in the horizontal, vertical, and diagonal directions, and this correlation may be exploited by attackers to obtain the statistical characteristics of the image, thereby threatening the security of the encrypted image. Therefore, an efficient image encryption algorithm should be able to significantly reduce the correlation between adjacent pixels in the ciphertext image. The correlation between adjacent pixels can be calculated by the following formula:

[0145]

[0146] The definitions of each parameter are as shown in formula (13).

[0147]

[0148] Among them, x and y are a pair of pixel values, E(x) is the expectation of x, D(x) is the variance of x, cov(x, y) represents the covariance of x and y, and N is the total number of pixels in the image. Figure 16 shows the results of the correlation analysis of 10,000 pairs of pixel points selected from the Baboon image in three directions (horizontal, vertical, and diagonal). As Figure 16 can be seen, the correlation of the plaintext image is relatively high, while the correlation of the ciphertext image has been effectively destroyed. Table 7 shows the correlation coefficients of the Baboon image and four other images in three directions. The correlation coefficients of the ciphertext image in each direction are close to 0, indicating that the encryption algorithm can effectively reduce the correlation of the image and enhance the anti-statistical analysis ability of image encryption. Table 8 further shows the comparison of the present invention with other mainstream algorithms in terms of destroying pixel correlation. The results show that the present invention has better decorrelation ability.

[0149] Table 7 Correlation coefficients of the plaintext image and the ciphertext image in each direction

[0150]

[0151] Table 8 Comparison of the correlation of the ciphertext image with other schemes

[0152]

[0153] Among them, reference [5] is from [C Cao, K Sun, W Liu (2018) A novel bit-level image encryption algorithm based on 2D-LICM hyperchaotic map. Signal Process, 143, 122-133.]; reference [6] is from [Xu M, Tian Z (2019) A novel image cipher based on 3D bit matrix and latin cubes. Information Sciences, 478, 1-14.]; reference [7] is from [Wu Y, Zhang L, Liu X, Zhang H (2024) A novel image encryption scheme with adaptive Fourier decomposition. Journal of the Franklin Institute, 361, 4, 106630.]; reference [8] is from [Li M, Wang M, Fan H, An K, Liu G (2022) A novel plaintext-related chaotic image encryption scheme with no additional plaintext information. Chaos, Solitons & Fractals, 158, 111989.]; reference [9] is from [Wei D, Jiang M, Deng Y (2023) A secure image encryption algorithm based on hyper-chaotic and bit-level permutation. Expert Systems with Applications, 213, 119074.]; reference

[10] is from [Mishra P, Bhaya C, Pal A K, Singh A K (2023) A medical image cryptosystem using bit-level diffusion with DNA coding. Journal of Ambient Intelligence and Humanized Computing, 14, 3, 1731-1752.; Reference

[11] is from [Benaissi S, Chikouche N, Hamza R(2023) A novel image encryption algorithm based on hybrid chaotic maps using a key image. Optik, 272, 170316]; Reference

[12] is from [Tamba V K, Biamou A L M, Tagne F K, Takougang A C N, Fotsin H B(2024) Hidden extreme multistability in a smooth flux-controlled memristor based four-dimensional chaotic system and its application in image encryption. Physica Scripta, 99, 2, 025210.].

[0154] The information entropy is an effective index to measure the randomness of the pixel distribution in an image. The closer its value is to the ideal value of 8, the more uniform the pixel distribution in the image, the more difficult the information is to predict, and thus the better the image encryption effect. The calculation formula of the information entropy is as follows:

[0155]

[0156] where L represents the number of gray levels of the image, and P(m i ) represents the probability that the gray value m i appears. After encrypting five different images in the present invention, the information entropy of each of them was calculated and compared with the encryption results of other algorithms. The results are shown in Table 9. It can be seen that the information entropy of the ciphertext images in the present invention is all close to 8 and higher than that of other reference algorithms, verifying the effectiveness of the algorithm in resisting entropy attacks.

[0157] In addition, the Local Shannon Entropy is also used to evaluate the randomness of the ciphertext image in the local area. Its calculation formula is as follows:

[0158]

[0159] where S i is the entropy value of the i-th group of local pixels, k is the number of groups, and T B is the number of pixels in each group. When k = 3 and T B= 1936, when the significance level α = 0.05, the confidence interval of the theoretical local entropy is [7.901515698, 7.903422936]. Table 10 lists the local information entropy values of each image ciphertext, all of which fall within this interval, indicating that the algorithm in this paper has good local randomness.

[0160] Table 9 Comparison of the global information entropy of the plaintext image and the ciphertext image with other algorithms

[0161]

[0162]

[0163] Table 10 Results of the local information entropy of the ciphertext image

[0164]

[0165] Among them, reference

[13] is from the reference [Chen R, Li X, Teng L, Wang X (2024) An image encryption algorithm based on the LSCMM chaotic map and bidirectional dynamic diffusion. Multimedia Tools and Applications, 83, 2, 3681 - 3706.], and reference

[14] is from the reference [Enayatifar R, Abdullah AH, Isnin I F, Altameem A, Lee M (2017) Image encryption using a synchronous permutation - diffusion technique. Optics and Lasers in Engineering, 90, 146 - 154.].

[0166] Peak signal - to - noise ratio (PSNR) and mean square error (MSE) are important indicators for evaluating the difference between the encrypted image and the original image. In an ideal state, the encrypted image should have a large difference from the original image. Therefore, the lower the PSNR and the larger the MSE, the better the encryption effect. The calculation formulas are as follows:

[0167]

[0168] Among them, M and N are the image sizes, P is the plaintext image, C is the encrypted image, and R is the maximum value of the pixels (usually 255). Table 11 shows the comparison results of the present invention and other algorithms in terms of PSNR and MSE metrics. It can be seen that the PSNR value of the image encrypted by the present invention is smaller and the MSE value is larger, which is better than the comparative algorithms, further proving a higher encryption strength.

[0169] Table 11 PSNR and MSE of the ciphertext image and comparison with other algorithms

[0170]

[0171] Among them, reference

[15] is from [Zhu S, Deng X, Zhang W, Zhu C (2023) Image encryption scheme based on newly designed chaotic map and parallel DNA coding. Mathematics, 11, 1, 231.].

[0172] Differential attack is a common way of image encryption attack. Its principle is to make minor modifications to the plaintext image and then observe the changes between the ciphertext images to infer the encryption rule or key. Therefore, an excellent image encryption algorithm should have strong resistance to differential attacks.

[0173] Common anti-differential attack metrics include the number of pixel change rate (NPCR) and unified average change intensity (UACI), and their calculation methods are as follows:

[0174]

[0175] Among them, M and N are the image sizes, and C1 and C2 are the ciphertext images obtained by encrypting the original image and the fine-tuned image respectively. Theoretically, the ideal values of NPCR and UACI are 99.6049% and 33.4635% respectively. Tables 12 and 13 show the NPCR and UACI results of the present invention under differential attack and compare them with other algorithms. The results show that the ciphertext images generated by the present invention have higher NPCR and UACI values, approaching the theoretical expected values, indicating that the present invention has strong resistance to differential attacks.

[0176] Table 12 NPCR and UACI values of the ciphertext image

[0177]

[0178] Table 13 NPCR and UACI values of the ciphertext image and comparison with other algorithms

[0179]

[0180] During the image transmission process, noise interference may cause varying degrees of damage to the ciphertext image, thereby affecting the visibility and recoverability of the image. To evaluate the robustness of the image encryption algorithm proposed in the present invention against noise attacks, the present invention adds salt-and-pepper noise with different intensities (0.01, 0.05, 0.1) to the ciphertext of the Baboon image and decrypts it while keeping the key unchanged. Figure 17 The ciphertext image after adding noise and its corresponding decrypted image are shown.

[0181] From Figure 17 it can be observed that even under strong noise interference, the decrypted image still retains a high level of recognizability. This indicates that the encryption algorithm proposed in the present invention has a certain degree of strong anti-noise attack ability and is applicable to practical application scenarios with channel interference or transmission errors.

[0182] In image encryption applications, partial data loss or malicious cropping of the image may occur during transmission or storage. If the encryption algorithm has strong anti-cropping attack ability, it should be able to recover a recognizable decrypted image even when part of the ciphertext is lost. To evaluate the effect of the encryption algorithm of the present invention against cropping attacks, cropping operations with different ratios of 1 / 16, 1 / 4, and 1 / 2 are respectively performed on the ciphertext of the Baboon image and decryption is carried out. Figure 18 The decryption results under different cropping degrees are shown.

[0183] The experimental results show that even when up to half of the pixels of the ciphertext image are lost, the decrypted image still has good recognizability. This shows that the image encryption algorithm proposed in the present invention has strong fault tolerance and anti-cropping attack ability, and can effectively ensure the recoverability of the image under incomplete transmission conditions.

[0184] The present invention proposes an image encryption scheme based on an improved four-dimensional hyperchaotic system, aiming to effectively protect sensitive image data and provide reliable guarantee for secure communication between Internet of Things devices. Through theoretical analysis and simulation verification of the constructed improved hyperchaotic system, the results show that it has a wider parameter space, higher complexity and better unpredictability compared with traditional low-dimensional chaotic systems, and thus is more suitable for Internet of Things application scenarios with higher security requirements. The present invention uses the SHA-256 hash function in combination with an external key to generate the initial value of the chaotic system, effectively enhancing the randomness and sensitivity of the key. At the same time, combined with the dynamic DNA substitution table generated by the fourth-order Latin square structure, the DNA encoding process has stronger nonlinearity and anti-attack ability. By uniformly dividing the plaintext image, ascendingly rearranging the chaotic sequence, cross-combining and performing DNA operations, the confusion and diffusion effects of the algorithm are effectively improved. In addition, the designed bidirectional weighted diffusion mechanism can fully destroy the correlation between image pixels, further enhancing the overall security of the encryption system.

[0185] Experiments and analysis show that the encryption algorithm of the present invention exhibits good robustness and security in the face of common attack means (such as statistical analysis, differential attack, cropping and noise interference, etc.), and is suitable for popularization and application in various Internet of Things scenarios.

[0186] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. An image secure transmission method based on a four-dimensional hyperchaotic system and DNA coding, characterized in that The steps are as follows: Step 1: Use the SHA-256 hash function to calculate the hash value of the input plaintext image P with size M×N, and combine with the external key to calculate the initial parameters of the four-dimensional hyperchaotic system; Step 2: Substitute the initial parameters into the four-dimensional hyperchaotic system, iterate multiple times and then continue to iterate M×N times to generate four chaotic sequences X', Y', Z', W' respectively, and perform data processing to obtain integer chaotic sequences F1, F2, F3, F4 and random numbers W1, W2, W3 and random sequence G; Step 3: Convert the plaintext image P into an eight-bit binary matrix T1, select the DNA coding rule to perform DNA coding on the binary matrix T1 to obtain a base matrix T2; perform DNA coding on the integer chaotic sequence F1 to obtain a base matrix J1; Step 4: Construct a Latin square DNA substitution table, and use the DNA substitution rule to replace each base in the base matrix T2 by looking up the Latin square DNA substitution table to generate a new base matrix T3; Step 5: Convert the base matrix T3 into a one-dimensional DNA sequence T4, and evenly divide it into 4 segments; use the index sequences I1, I2, I3, I4 generated by the integer chaotic sequences F1, F2, F3, F4 to perform index scrambling on the four DNA sequences respectively to obtain sequences S1, S2, S3, S4; place the bases at the same positions of the sequences S1, S2, S3, S4 together in order to obtain a base sequence T5, and restore it to the matrix form to obtain a base matrix T6; Step 6: Perform DNA exclusive OR operation on the base matrix T6 and the base matrix J1 to obtain a base matrix T7, perform DNA decoding on the base matrix T7 according to the DNA decoding rule to obtain a binary matrix J2, and convert the binary matrix J2 into decimal to obtain a matrix J; Step 7: Perform forward and reverse random weighted diffusion operations on the matrix J in turn through the random numbers W1, W2, W3 and the random sequence G to finally obtain the ciphertext image C.

2. The image secure transmission method based on a four-dimensional hyperchaotic system and DNA coding according to claim 1, wherein The method for calculating the initial parameters of the four-dimensional hyperchaotic system is: Input the plaintext image P of size M×N into the SHA-256 hash function to obtain a 256-bit binary hash value H; divide the hash value H into 32 groups, each group with 8 bits, to obtain the hash value H = k1, k2, k3, … k 32 ; calculate the initial parameters x0, y0, z0, w0 of the chaotic system using the hash value and the external key as follows: Among them, η1, η2, η3, η4, η5, η6, η7, and η8 are all intermediate variables, and j1-k 32 is a hash value of 32 groups of 8-bit binary numbers, is an exclusive OR operation, mod is a modulo function, and θ(1), θ(2), θ(3), and θ(4) are external keys.

3. The image security transmission method based on a four-dimensional hyperchaotic system and DNA coding according to claim 1 or 2, characterized in that, The four-dimensional hyperchaotic system is an improved four-dimensional chaotic system, and the mathematical model of the improved four-dimensional chaotic system is: Among them, x, y, z, and w are the state variables of a four-dimensional chaotic system, which are the derivatives of the state variables x, y, z, and w respectively, and a, b, c, d, and e are the control parameters of the four-dimensional chaotic system.

4. The image security transmission method based on the four-dimensional hyperchaotic system and DNA coding according to claim 3, characterized in that, The method for obtaining the integer chaotic sequences F1, F2, F3, F4 is: discretize the four chaotic sequences X', Y', Z', W' respectively, then where X′(i), Y′(i), Z′(i), W′(i), F1(i), F2(i), F3(i), F4(i) are the values of the i-th elements of the chaotic sequences X', Y', Z', W', integer chaotic sequences F1, F2, F3, F4 respectively, where i = 1, 2, 3, …… M×N; floor is the floor function; The method for obtaining random numbers W1, W2, W3 and random sequence G is as follows: Take one element from integer chaotic sequences F1, F2, F3 respectively and combine them with external secret keys θ(1), θ(2), θ(3) to generate random numbers W1, W2, W3 respectively; Select the last three elements from integer sequence F4 to obtain integer sequence g = [F4(M×N), F4(M×N - 1), F4(M×N - 2)], and obtain integer sequence G, and where F1(M×N - 1), F2(M×N - 1), F3(M×N - 1) are the values of the (M×N - 1)-th elements of integer chaotic sequences F1, F2, F3 respectively, and F4(M×N), F4(M×N - 1), F4(M×N - 2) are the values of the (M×N)-th, (M×N - 1)-th, (M×N - 2)-th elements of integer chaotic sequence F4 respectively.

5. The image security transmission method based on a four-dimensional hyperchaotic system and DNA coding according to claim 3, wherein Convert the pixel values of the plaintext image P into 8-bit binary to obtain an 8-bit binary matrix T1, and map every two bits of each element in the 8-bit binary matrix T1 to a base to form a DNA sequence composed of 4 bases; According to different coding methods, there are 8 common DNA coding rules as follows: Adopt three basic operations: DNA addition +, DNA subtraction –, and DNA exclusive OR ⊕, and the operation rules are Combine random number L to select the DNA coding rule, and random number L = mod(W1, 8)+1).

6. The image secure transmission method based on a four-dimensional hyperchaotic system and DNA coding according to claim 4 or 5, characterized in that, The method for constructing the Latin square DNA substitution table is as follows: Use a 4th-order Latin square to generate a 4th-order matrix R1, combine each element in matrix R1 with its column number to form a two-digit integer matrix R2, where the tens digit is the element value in matrix R1 and the units digit is the column number where the element is located; Convert the element values in the tens and units digits of integer matrix R2 into corresponding bases respectively to generate a base matrix R3; Set the row number and column number as bases A, T, C, G respectively to construct the Latin square DNA substitution table R; Among them, the element values 1 in the tens and units digits of integer matrix R2 correspond to base A, 2 corresponds to base T, 3 corresponds to base C, and 4 corresponds to base G.

7. The image secure transmission method based on a four-dimensional hyperchaotic system and DNA coding according to claim 6, wherein The element values 1 in the tens and units digits of the integer matrix R2 correspond to base A, 2 corresponds to base T, 3 corresponds to base C, and 4 corresponds to base G.

8. The image security transmission method based on a four-dimensional hyperchaotic system and DNA coding according to claim 6, characterized in that, The method for replacing each base in the base matrix T2 with reference to the Latin square DNA substitution table using the DNA substitution rule is as follows: Select the first 256 out of 576 4th-order Latin squares to construct a DNA substitution table set; Select the DNA substitution table with the corresponding number according to the element value of the integer chaotic sequence F4, and replace every two bases in the base matrix T2 as a group by referring to the DNA substitution table, where the first digit is used to determine the column number and the second digit is used to determine the row number; Search for the base at the corresponding position in the selected DNA substitution table and replace the original base with the base at the corresponding position to obtain a new DNA base matrix T3.

9. The image security transmission method based on a four-dimensional hyperchaotic system and DNA coding according to claim 8, characterized in that, The method for obtaining the base matrix T6 is as follows: Convert the base matrix T3 into a one-dimensional DNA sequence T4, and evenly divide it into four segments: the first segment is T4(1:MN), the second segment is T4(MN + 1:2MN), the third segment is T4(2MN + 1:3MN), and the fourth segment is T4(3MN + 1:4MN); Ascendingly sort the four integer chaotic sequences F1, F2, F3, and F4 respectively to obtain the corresponding index sequences I1, I2, I3, and I4; Rearrange the four segmented DNA sequences respectively according to the order of the elements in the index sequences I1, I2, I3, and I4, and after scrambling their base positions, obtain new sequences S1, S2, S3, and S4; Combine the bases at the same positions of the four scrambled sequences in order and crosswise to obtain a base sequence T5, and restore it to a matrix form to obtain the base matrix T6.

10. The image security transmission method based on a four-dimensional hyperchaotic system and DNA coding according to any one of claims 7-9, characterized in that, The method for the forward and reverse random weighted diffusion operations is as follows: Use the random sequence G to select random numbers W1, W2, W3 as weights. The forward random weighted diffusion is Among them, G is a random sequence, W G(1) , W G(2) , W G(3) are the random numbers W1, W2, W3 corresponding to the values of G(1), G(2), G(3) respectively. F1(1) and F1(i) are the values of the first and the i-th elements of the integer chaotic sequence F1. J(1), J(i), and J(M×N) are the values of the first, the i-th, and the M×N-th elements of the matrix J after DNA decoding respectively. E(i - 1) is the value of the (i - 1)-th element of the image E after forward weighted diffusion; The reverse random weighted diffusion is: Where, C(i) is the value of the i-th element of the image C after forward weighted diffusion, E(M×N) represents the M×N-th element of the image E, F2(M×N) represents the value of the M×N-th element of the integer sequence F2, F2(i) represents the value of the i-th element of the integer sequence F2, C(i - 1) represents the value of the i - 1-th element of the image C, and the matrix J, the image E, and the image C are all accessed in sequence order.

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