Research on wheat image encryption based on different chaotic systems

Abstract

To study the influence of chaotic systems on wheat image encryption, a method to determine the optimal chaotic system encryption based on a wheat image is proposed. Ten different chaotic system schemes were combined to encrypt the wheat images using 13 common chaotic maps. The best chaotic system scheme was obtained by considering the anti-attack capability of these encryption schemes, which analyzed eight commonly used image encryption performance evaluation indexes. The experimental results show that the new four-dimensional chaotic system has the best encryption effect and is suitable for wheat image encryption. The proposed scheme for wheat image encryption based on chaotic systems provides a reference for other crop image encryption methods.

KEYWORDS:

• Chaotic mapping
• image encryption
• wheat image

1. Introduction

Wheat images contain data on wheat growth, pest monitoring, and harvest prediction, which are important for agricultural research (Ma et al., 2018; Zhang et al., 2022; Zhao et al., 2020). To protect the integrity, confidentiality, and availability of sensitive data, wheat images should be encrypted. At the same time, to ensure the reliability and security of wheat production and research, it is necessary to prevent unauthorized access, tampering and leakage (Man et al., 2021). Wheat images are difficult to encrypt because of the complexity of the appearance of wheat, the variability of the planting environment, and the easy overlap of wheat features with soil or other plant features. Image encryption technology based on chaos was used to protect the wheat images. Chaotic image encryption is an encryption technology based on chaos theory that uses a chaotic system to generate pseudo-random number sequences to encrypt and decrypt images (Pourasad et al., 2021). Chaotic systems have the characteristics of high randomness, complexity, and unpredictability (Zhang & Liu, 2023), and are often used in image encryption (Li et al., 2021a). It can also provide higher anti-aggression and increase the difficulty of cracking the encrypted data. However, it is also necessary to select appropriate encryption schemes and parameters according to different requirements to ensure the security and practicality of the encryption algorithms.

In recent years, many scholars have contributed to the development of image-encryption research. A new 2D hyper-chaotic Schaffer map for image encryption was proposed by Erkan et al. (2023), which has the best ergodicity and uncertainty for applications with high complexity requirements. Lai et al. (2023) proposed a pixel-split image encryption scheme based on a two-dimensional Salomon map, which has strong randomness and good encryption effect compared with traditional maps. An image encryption scheme based on a new dynamic four-dimensional chaotic system, Z-type transformation, and DNA operation was proposed by Zhao et al. (2023), and has the characteristics of good security performance and resistance to various attacks. Biban et al. (2023) proposed an 8D hyper-chaotic system grey-scale image encryption algorithm based on Fibonacci Q-Matrix, which helps protect grey-scale images in the public domain from hacker attacks. To realize the security and efficiency of the encryption algorithm, Alexan et al. (2023) proposed a colour image encryption algorithm combining KAA mapping and multiple chaotic mappings. To solve the shortcomings of one-dimensional mapping and multidimensional mapping, a new image encryption algorithm was proposed by Benaissi et al. (2023), which used three hybrid algorithms of the modified and improved chaotic one-dimensional maps to provide good privacy for digital images. Aiming at the problems of the small key space and weak anti-differential attack capability of existing encryption algorithms, a chaotic image encryption scheme based on an artificial fish swarm algorithm and DNA coding was proposed by Zhu, Wang, et al. (2023), which has better encryption performance and higher security. Combining chaos and gene theory, Gao et al. (2023) propose a colour image encryption scheme based on hyper-chaotic mapping and DNA mutation. Using a finite precision error system, an image encryption algorithm based on new chaotic signals and truly random numbers was proposed by Zhou et al. (2023). An image encryption algorithm based on compressed sensing and integer wavelet transform was proposed by Huang et al. (2023), which has a good hiding effect and can resist known plaintext attacks and selected plaintext attacks. To realize image privacy protection in an artificial intelligence environment, an image encryption algorithm that can be applied to work based on artificial intelligence was proposed by Xu et al. (2023), which combined compressed sensing and Rabinovich hyper-chaos. To solve the problem of excessive computation time, composite crossover technology was introduced by Premkumar et al. (2022), who proposed image encryption technology based on genetic operators. A more efficient and secure quantum image encryption scheme has been proposed by Hu et al. (2023), which uses cross-2D chaotic mapping that combines Sine and Logistic chaotic systems. In general, most of the recent research papers on image encryption technology use the chaotic system, which can effectively improve the security and confidentiality of image content. At the same time, it also has the advantages of high efficiency and flexibility, and has important application value in the field of image encryption.

Chaotic image encryption has become a widely used digital image encryption technology in recent years. According to different application fields, various colour image encryption schemes were compared and analyzed by Ghadirli et al. (2019), and their respective advantages and limitations were summarized. Suneja et al. (2019) discussed the characteristics, advantages, and disadvantages of various chaotic systems used for image encryption. In comparison, it is concluded that the security of image encryption based on the low-dimensional chaotic system proposed earlier is low. In recent years, researchers have proposed various high-dimensional chaotic systems for image encryption. Because traditional encryption algorithms cannot be used on resource-limited IoT devices, a lightweight image encryption technology with lossless, effective, and anti-security attack capabilities was proposed by Roy et al. (2021), which is based on two-dimensional von Neumann cellular automaton. The algorithm is suitable for implementation in real-time and sensitive IoT applications. Khan and Byun (2020) proposed a permission-based private blockchain solution that stores the encrypted pixel value of an image on the blockchain, guaranteeing the privacy and security of the image data. Blockchain technology provides a solution for the encryption of sensitive image data for decentralized devices, which is suitable for the security needs of intelligent industries, such as the Industrial Internet of Things. A symmetric key image encryption system based on piece-wise linear chaotic mapping was proposed by Zhang and Tang (2018), which has the same encryption and decryption process, high encryption and decryption speed, and the ability to resist plaintext attacks and can be applied to actual communication. By combining sinusoidal mapping and fractional arithmetic, a new one-dimensional fractional chaotic mapping was proposed by Zhu, Deng, et al. (2023), which was used to design an image encryption algorithm based on parallel DNA coding. The experimental results show that the algorithm has good encryption performance, less time overhead, and good application potential in secure communication applications. Different chaotic systems have different characteristics and are suitable for different image encryption tasks that must be selected according to specific needs.

To protect the security of agricultural information, it is necessary to build a scheme that is suitable for agricultural image encryption. An image-driven multi-feature plant management model based on a feature data encryption scheme was constructed by Santhosh et al. (2022), which used a dynamic scheme and key to encrypt data, thereby improving the security and performance of smart agriculture. Perumal et al. (2023) realized the data security of different smart devices in farmland by using data encryption schemes that use different encryption schemes and keys to encrypt data of farmland devices controlled by users, thus achieving a higher accuracy of low-rate attack detection. A new homomorphic cryptosystem was proposed by Kulalvaimozhi et al. (2020), which combines the compression process to encrypt field-cropped images, reduce the encryption time, and preserve high-quality reconstructed images. A method combining Logistic-Sine and Logistic-Tent chaotic systems was proposed by Padmapriya et al. (2019) to encrypt agricultural image information; this method is effective and robust.

After analyzing the above literature, it was found that a good image encryption scheme is very important for the development of smart agriculture. However, there are too few agricultural image encryption algorithms based on chaotic systems, so it is impossible to determine whether the performance of different chaotic systems in agricultural image encryption is good or bad. To solve the problem of wheat image encryption, in this study, an existing popular chaotic system is applied to the wheat image encryption algorithm, and the common image encryption performance evaluation index is compared and analyzed to measure its encryption effect. The aim was to find the most suitable chaotic system scheme for wheat image encryption and provide a reference for agricultural image encryption schemes based on chaotic systems.

The ten chaotic system schemes proposed in this paper include Piece-wise Linear Chaotic Map (PWLCM) chaotic mapping, Sine mapping, Tent mapping, Logistic mapping, Lorenz system, Rossler system, Chen system, hyper-chaotic Chen system, hyper-chaotic Lorenz system, hyper-chaotic Rossler system, hyper-chaotic Hide-Skeldon-Acheson system, new four-dimensional chaotic system, and hyper-chaotic Lü system.

The main contributions of this paper are as follows:

1. The image encryption algorithm based on chaotic system is applied to wheat image encryption, which solves the problems of complexity, variability and feature overlap of wheat image encryption. 2. A combination scheme was proposed to select appropriate chaotic systems for wheat image encryption by verifying the encryption performance through ablation study. 3. By comparing 8 commonly used image encryption performance evaluation indexes, the new four-dimensional chaotic system that is most suitable for wheat image encryption is selected, which has good encryption effect and good anti-attack ability.

2. Chaotic systems

Chaotic systems are mathematical models that describe chaotic dynamic systems characterized by high nonlinearity, complexity, and unpredictability (Smith et al., 1999). The output sequence of chaotic systems is pseudo-random and highly sensitive to initial conditions (Zhao et al., 2019), which makes chaotic systems have a wide range of applications in the fields of encryption and pseudo-random number generation. In digital image encryption systems, chaotic systems are primarily used to generate chaotic sequences (Wang et al., 2020), which are used to generate encryption keys, pixel replacements, initialize encryption algorithms, and other operations.

Common one-dimensional chaotic systems are as follows:

(1) Sine map (1) ${\text{x}}_{\text{t + 1}}=\mathrm{S}\left(\mathrm{x}\right)=ϵ\sin (\pi {\text{x}}_{\text{t}}\text{)}$(1) where $ϵ\in \left[0,1\right]$ is the control parameter and ${\text{x}}_{\text{t}}\in \left[0,1\right]$ is the status value of the system at time $\text{t}$.

(2) Tent map (2) ${\text{x}}_{\text{t + 1}}\text{ = T(x) = }\{\begin{array}{lc}2ϵ{\text{x}}_{\text{t}},& {\text{x}}_{\text{t}}<0\text{.5}\\ 2ϵ(1-{\text{x}}_{\text{t}}),& {\text{x}}_{\text{t}}\ge \text{0}\text{.5}\end{array}$(2)

(3) Logistic map (3) ${\text{x}}_{\text{t + 1}}=\mathrm{L}\left(\mathrm{x}\right)=4ϵ{\text{x}}_{\text{t}}\text{(1 - }{\text{x}}_{\text{t}}\text{)}$(3)

(4) PWLCM map (4) ${\text{x}}_{\text{t + 1}}\text{ = }\{\begin{array}{lc}{\displaystyle \frac{{\text{x}}_{\text{t}}}{ϵ},}& {\text{x}}_{\text{t}}\in [0,ϵ)\\ {\displaystyle \frac{\text{1 - }{\text{x}}_{\text{t}}}{1-ϵ},}& {\text{x}}_{\text{t}}\in (ϵ,1]\end{array}$(4) Common high-dimensional chaotic systems are as follows:

(5) Chen system (5) $\{\begin{array}{c}\dot{\mathrm{x}}=\mathrm{a}(\mathrm{y}-\mathrm{x})\\ \dot{\mathrm{y}}=(\mathrm{c}-\mathrm{a})\mathrm{x}-\mathrm{xz}+\mathrm{cy}\\ \dot{\mathrm{z}}=\mathrm{xy}-\mathrm{bz}\end{array}$(5) where $\text{a}$ = 35, $\text{b}$ = 3, and $\text{c}\in \text{[20, 28}\text{.4]}$. The $\text{x}$, $\text{y}$, and $\text{z}$ are the status values of the system at time $\text{t}$. The $\dot{\mathrm{x}}$, $\dot{\mathrm{y}}$, and $\dot{\mathrm{z}}$ represent the derivatives of the independent variables $\text{x}$, $\text{y}$, and $\text{z}$ at time $\text{t}$.

(6) hyper-chaotic Chen system (6) $\{\begin{array}{c}\dot{\mathrm{x}}=\mathrm{a}(\mathrm{y}-\mathrm{x})\\ \dot{\mathrm{y}}=-\mathrm{xz}+\mathrm{dx}+\mathrm{cy}-\mathrm{w}\\ \dot{\mathrm{z}}=\mathrm{xy}-\mathrm{bz}\\ \dot{\mathrm{w}}=\mathrm{x}+\mathrm{k}\end{array}$(6) where $\text{a}$ = 36, $\text{b}$ = 3, $\text{c}$ = 28, $\text{d}$ = −16, and $\text{ - 0}\text{.7}\le \text{k}\le \text{0}\text{.7}$. The $\text{w}$ is the status of the system at time $\text{t}$. The $\dot{\mathrm{w}}$ represents the derivative of independent variable $\text{w}$ at time $\text{t}$.

(7) Lorenz system (7) $\{\begin{array}{c}\dot{\mathrm{x}}=-\mathrm{a}(\mathrm{x}-\mathrm{y})\\ \dot{\mathrm{y}}=-\mathrm{xz}+\mathrm{bx}-\mathrm{y}\\ \dot{\mathrm{z}}=\mathrm{xy}-\mathrm{cz}\end{array}$(7) where $\text{a}$ = 10, $\text{b}$ = 28, and $\text{c}$ = 8/3.

(8) hyper-chaotic Lorenz system (8) $\{\begin{array}{c}\dot{\mathrm{x}}=\mathrm{a}(\mathrm{y}-\mathrm{x})+\mathrm{w}\\ \dot{\mathrm{y}}=\mathrm{cx}-\mathrm{y}-\mathrm{xz}\\ \dot{\mathrm{z}}=\mathrm{xy}-\mathrm{bz}\\ \dot{\mathrm{w}}=-\mathrm{yz}+\mathrm{kw}\end{array}$(8) where $\text{a}$ = 10, $\text{b}$ = 8/3, c = 28, $\text{d}$ = −16, and $\text{ - 1}\text{.52}\le \text{k}\le \text{ - 0}\text{.06}$.

(9) Rossler system (9) $\{\begin{array}{c}\dot{\mathrm{x}}=-\mathrm{ay}-\mathrm{z}\\ \dot{\mathrm{y}}=\mathrm{ax}+\mathrm{by}\\ \dot{\mathrm{z}}=\mathrm{c}+\mathrm{z}(\mathrm{x}-\mathrm{d})\end{array}$(9) where $\text{a}$ = 1.0, $\text{b}$ = 0.165, $\text{c}$ = 0.2, and $\text{d}$ = 10.

(10) hyper-chaotic Rossler system (10) $\{\begin{array}{c}\dot{\mathrm{x}}=-\mathrm{y}-\mathrm{z}\\ \dot{\mathrm{y}}=\mathrm{x}+\mathrm{ax}+\mathrm{w}\\ \dot{\mathrm{z}}=\mathrm{b}+\mathrm{xz}\\ \dot{\mathrm{w}}=-\mathrm{cz}+\mathrm{dw}\end{array}$(10) where $\text{a}$ = 0.25, $\text{b}$ = 3, $\text{c}$ = 0.5, and $\text{d}$ = 0.05.

(11) hyper-chaotic Hide-Skeldon-Acheson system (Chen & Bao, 2021) (11) $\{\begin{array}{c}\dot{\mathrm{x}}=-\mathrm{y}-\mathrm{z}\\ \dot{\mathrm{y}}=\mathrm{x}+\mathrm{ax}+\mathrm{w}\\ \dot{\mathrm{z}}=\mathrm{b}+\mathrm{xz}\\ \dot{\mathrm{w}}=-\mathrm{cz}+\mathrm{dw}\end{array}$(11) where $\text{a}$ = 0.01, $\text{b}$ = 30, $\text{c}$ = 0, $\text{d}$ = 2, $\text{e}$ = 0.001, $\text{f}$= 28.5, $\text{g}$ = 1, $\text{k}$= 1.2, $\text{q}$ = 0, and $\text{h}$= 0.

(12) New four-dimensional chaotic system (Wang et al., 2019) (12) $\{\begin{array}{c}\dot{\mathrm{x}}=-\mathrm{y}-\mathrm{z}\\ \dot{\mathrm{y}}=\mathrm{x}+\mathrm{ax}+\mathrm{w}\\ \dot{\mathrm{z}}=\mathrm{b}+\mathrm{xz}\\ \dot{\mathrm{w}}=-\mathrm{cz}+\mathrm{dw}\end{array}$(12) where $\text{a}$ = 25, $\text{b}$ = 3, $\text{c}$ = 18, $\text{d}$ = 19, and $\text{e}$ = 14.

(13) hyper-chaotic Lü system (Yan & Zhang, 2023) (13) $\{\begin{array}{c}\dot{\mathrm{x}}=\mathrm{y}-\mathrm{ax}\\ \dot{\mathrm{y}}=\mathrm{bz}-\mathrm{xz}\\ \dot{\mathrm{z}}=\mathrm{xy}-\mathrm{xw}\\ \dot{\mathrm{w}}=-\mathrm{x}+\mathrm{c}\end{array}$(13) where $\text{a}$ = 10, $\text{b}$ = 3, and $\text{c}$ = 12.

3. Algorithm description

3.1. Original image encryption algorithm

An algorithm to modify the initial conditions and control parameters of PWLCM, Lorenz, and hyper-chaotic Chen systems using the SHA-256 hash function was proposed by Rehman et al. (2018) and selected for colour image encryption. This scheme consists of four stages: two scrambling operations, DNA encoding, and DNA decoding. In the first scrambling process, the red, green, and blue channels of the original colour image are arranged into one-dimensional vectors, and the chaotic sequence obtained by the PWLCM is used to reorder them. The sorted pixel list is then re-divided into red, green, and blue channels. In the second scrambling process, the Lorenz system was used to generate three chaotic sequences, and the red, green, and blue channels obtained after the first scrambling were rearranged. After two scrambling operations, the hyper-chaotic Chen system was used to generate four chaotic sequences, namely, U, V, W, and X. Sequence U is divided into three sub-vectors, which select the DNA coding rules according to Table 1. The value of each pixel in the red, green, and blue channels was converted to a DNA base. The XOR rules based on DNA bases in the pixel-level confusion stage are listed in Table 2. Xor operations are performed on the red, green, and blue channels using the DNA replacement rules shown in Table 3. In addition, sequence V was used to select DNA rules, and sequence W was used to determine the number of repeated iterations of the XOR operation. Sequence X was then divided into three sub-vectors, which selected the DNA decoding rules according to Table 1. Realize the conversion of DNA bases to values for each pixel in the red, green, and blue channels. Finally, the three decoded red, green, and blue channels are combined to obtain the encrypted image.

Table 1. Encoding and decoding rules of DNA.

Rule	Chaotic intervals	Encoding	Decoding
1	0.01–0.05, 0.20–0.25, 0.40–0.45, 0.50–0.55, 0.95–0.99	AGCT	GTAC
2	0.05–0.10, 0.30–0.35, 0.60–0.65, 0.70–0.75, 0.85–0.90	ACGT	TGCA
3	0.10–0.15, 0.35–0.40, 0.55–0.60, 0.65–0.70, 0.80–0.85	GATC	CTAG
4	0.15–0.20, 0.25–0.30, 0.45–0.50, 0.75–0.80, 0.90–0.95	CATG	TCGA

Table 2. XOR operations for DNA Bases.

XOR	A	G	C	T
A	A	G	C	T
G	G	A	T	C
C	C	T	A	G
T	T	C	G	A

Table 3. DNA rules for Substitution.

Value	0	1	2	3	4	5	6	7
DNA Rules for XOR	AGCT	GATC	CTAG	TGCA	GTAC	ACGT	TCGA	CATG

3.2. Image encryption schemes based on different chaotic systems

Based on Section 3.1, it can be observed that three chaotic systems, PWLCM, Lorenz, and Chen, are used in the colour image encryption algorithm proposed by Rehman et al. (2018). We applied this algorithm to wheat image encryption to compare the effects of different chaotic systems on wheat image encryption. The above three chaotic systems were successively replaced to find the most suitable chaotic system scheme for wheat image encryption.

Based on the original image encryption scheme, the specific replacement scheme is as follows:

1. Scheme 1: Replace the PWLCM with the Sine mapping.

2. Scheme 2: Replace the PWLCM with the Tent mapping.

3. Scheme 3: Replace the PWLCM with the Logistic mapping.

4. Scheme 4: Replace the Lorenz system with the Rossler system.

5. Scheme 5: Replace the Lorenz system with the Chen system.

6. Scheme 6: Replace the hyper-chaotic Chen system with the hyper-chaotic Lorenz system.

7. Scheme 7: Replace the hyper-chaotic Chen system with the hyper-chaotic Rossler system.

8. Scheme 8: Replace the hyper-chaotic Chen system with the hyper-chaotic Hide-Skeldon-Acheson system.

9. Scheme 9: Replace the hyper-chaotic Chen system with the new four-dimensional chaotic system.

10. Scheme 10: Replace the hyper-chaotic Chen system with the hyper-chaotic Lü system.

In general, the ablation study was performed by the proposed algorithm to verify the encryption performance. The PWLCM chaotic mapping was replaced by the Sine mapping, Tent mapping, and Logistic mapping in turn. The Lorenz chaotic system was replaced by the Rossler system and Chen system in turn. The hyper-chaotic Chen system was replaced by the Lorenz system, Rossler system, Hide-Skeldon-Acheson system, new four-dimensional chaotic system, and hyper-chaotic Lü system in turn.

4. Results and discussion

The experimental platform was a PC with a 13th Gen Intel(R) Core(TM) i5-13400F @ 2.50 GHz CPU, 32.0 GB memory, NVIDIA GeForce RTX 4060 graphics, and Windows 11 operating system. Based on the encryption scheme mentioned in Section 3, using the wheat image of Gaoping City, Shanxi Province, as an example, the encryption and decryption of the wheat encryption system is shown in Figure 1.

Figure 1. Image of wheat encryption and decryption processes. (a) The original image; (b) The encrypted image; (c) The decrypted image.

4.1. Histogram analysis

The histogram refers to the graph drawn by the frequency of each grey value in the statistical image, reflecting the most basic statistical characteristics of the image (Zhang & Hu, 2021). To resist statistical analysis attacks, the histogram of the encrypted image should be distributed as evenly as possible. Variance was used to measure the frequency distribution of the histogram, and the calculation formula is shown in Equation (16(16) $\begin{array}{rl}\text{D}\left(\text{u}\right)& =\frac{1}{\text{N}}\sum _{\text{n}=1}^{\text{N}}({\text{u}}_{\text{n}}-\text{E}\left(\text{u}\right){)}^2\end{array}$(16) ). The smaller the variance, the more uniform the pixel distribution. The arithmetic square root of the variance is called the standard deviation, which reflects the degree of dispersion in the dataset.

A histogram of the wheat encryption image of Gaoping City based on Scheme 1 is shown in Figure 2. Table 4 compares the values of variance and standard deviation for each scheme. The results show that the variance and standard deviation of scheme 7 are the best, while the variance and standard deviation of scheme 9 are the worst.

Figure 2. The histogram of the encrypted image. (a) Encrypted image; (b) Channel R; (c) Channel G; (d) Channel B.

Table 4. Variance and standard deviation of wheat encrypted images.

Chaotic System Scheme	1	2	3	4	5	6	7	8	9	10
Variance	5458.30	5453.56	5446.62	5450.37	5459.35	5448.77	5443.85	5446.41	6356.91	5897.04
Standard Deviation	73.88	73.85	73.80	73.83	73.89	73.82	73.78	73.80	79.73	76.79

4.2. Correlation analysis of adjacent pixels

The correlation of adjacent pixels refers to the degree of correlation of pixel values in adjacent positions of an image (Li et al., 2021b), and the calculation formula is as follows: (14) $\begin{array}{rl}{\text{R}}_{\text{uv}}& =\frac{\text{cov}\left(\text{u},\text{v}\right)}{\sqrt{\text{D}\left(\text{u}\right)}\sqrt{\text{D}\left(\text{v}\right)}}\end{array}$(14) (15) $\begin{array}{rl}\text{cov}\left(\text{u},\text{v}\right)& =\frac{1}{\text{N}}\sum _{\text{n}=1}^{\text{N}}\left({\text{u}}_{\text{n}}-\text{E}\left(\text{u}\right)\right)\left({\text{v}}_{\text{n}}-\text{E}\left(\text{v}\right)\right)\end{array}$(15) (16) $\begin{array}{rl}\text{D}\left(\text{u}\right)& =\frac{1}{\text{N}}\sum _{\text{n}=1}^{\text{N}}({\text{u}}_{\text{n}}-\text{E}\left(\text{u}\right){)}^2\end{array}$(16) (17) $\begin{array}{rl}\text{E}\left(\text{u}\right)& =\frac{1}{\text{N}}\sum _{\text{n}=1}^{\text{N}}{\text{u}}_{\text{n}}\end{array}$(17) where $\text{v}$ is the adjacent pixel of $\text{u}$, and $\text{cov}\left(\text{u},\text{v}\right)$ is the covariance of pixels $\text{u}$ and $\text{v}$, respectively. The $\text{N}$ is the total number of pixels in an image. $\text{E}\left(\text{u}\right)$ is the mean, $\text{D}\left(\text{u}\right)$ is the variance, and $\sqrt{\text{D}\left(\text{u}\right)}$ is the standard deviation. The ${\text{R}}_{\text{uv}}$ is the correlation between the adjacent pixels.

In general, there is a strong correlation between adjacent pixels in the horizontal, vertical, and diagonal directions in plaintext images. This feature is often used by attackers to infer the values of adjacent pixels from the values of known pixels, thus cracking the encrypted images. A good image encryption algorithm should reduce the correlation between adjacent pixels and achieve zero correlation as much as possible.

The correlation of adjacent pixels of the wheat encryption image of Gaoping City based on Scheme 1 is shown in Figure 3. Tables 5–7 lists the values of each scheme for the correlation of adjacent pixels in the horizontal, vertical, and diagonal directions, respectively. In general, Scheme 9 shows a good correlation, with the value closest to 0. Moreover, it was found that the correlations on the R, G, and B channels of the same scheme will also show significant differences. Therefore, the difference in colour of the original image also affects the choice of the best scheme.

Figure 3. The correlation of encrypted image. (a) Channel R; (b) Channel G; (c) Channel B.

Table 5. Horizontal correlation of wheat encrypted images.

Chaotic System Scheme		1	2	3	4	5	6	7	8	9	10
Correlation	R	0.0080	0.0018	−0.0251	0.0315	−0.0329	0.0050	−0.0202	−0.0186	−0.0000	0.0276
	G	−0.0309	−0.0137	0.0117	−0.0149	0.0173	−0.0126	0.0093	0.0068	−0.0019	0.0166
	B	−0.0194	−0.0081	−0.0281	0.0243	0.0230	−0.0023	0.0318	0.0399	0.0111	0.0095

Table 6. Vertical correlation of wheat encrypted images.

Chaotic System Scheme		1	2	3	4	5	6	7	8	9	10
Correlation	R	0.0113	0.0095	−0.0029	0.0037	0.0178	0.0177	0.0127	−0.0094	0.0159	0.0114
	G	0.0158	0.0153	0.0499	0.0079	0.0082	0.0284	0.0087	−0.0098	−0.0121	−0.0066
	B	0.0351	0.0372	0.0168	0.0244	0.0350	−0.0189	0.0036	0.0032	0.0223	0.0093

Table 7. Diagonal correlation of wheat encrypted images.

Chaotic System Scheme		1	2	3	4	5	6	7	8	9	10
Correlation	R	0.0319	0.0145	−0.0087	0.0293	−0.0105	−0.0023	−0.0130	−0.0057	0.0018	−0.0208
	G	0.0160	0.0004	0.0021	0.0123	−0.0073	0.0054	−0.0203	0.0230	−0.0156	−0.0363
	B	−0.0004	0.0154	−0.0265	0.0097	−0.0234	−0.0026	0.0263	0.0265	−0.0003	−0.0099

4.3. Key sensitivity analysis

Key sensitivity analysis aims to analyze the difference between two ciphertext images obtained from the same plaintext image encrypted when the key changes slightly (Chai et al., 2021). If the two ciphertext images are significantly different, the key sensitivity of the image cryptosystem is strong. If the difference between two ciphertext images is small, the key sensitivity is poor. A good image encryption system should have strong key sensitivity. The Normalized Pixel Contrast Ratio (NPCR) and Unified Average Changing Intensity (UACI) are commonly used to measure the differences between two images of the same size. The NPCR is used to compare the values of pixels in the corresponding positions of two images and record the proportion of the number of different pixels in all pixels. The calculation formula is as follows: (18) $\begin{array}{rl}\text{NPCR}\left({\text{Q}}_1,{\text{Q}}_2\right)& =\frac{1}{\text{WH}}\sum _{\text{i}=1}^{\text{W}}\sum _{\text{j}=1}^{\text{H}}|\text{Sign}\left({\text{Q}}_1\left(\text{i},\text{j}\right)-{\text{Q}}_2\left(\text{i},\text{j}\right)\right)|\\ & \times 100\mathrm{\%}\end{array}$(18) $\text{Sign}(\cdot )$ is a symbolic function, defined as follows: (19) $\text{Sign}\left(\tau \right)=\{\begin{array}{lc}1,& \tau >0\\ 0,& \tau =0\\ -1,& \tau <0\end{array}$(19) where $\text{W}$ and $\text{H}$ denote the width and height of the image, respectively. ${\text{Q}}_1\left(\text{i},\text{j}\right)$ and ${\text{Q}}_2\left(\text{i},\text{j}\right)$ are the pixel values at the positions in the first and second images, respectively.

UACI was used to record the difference in pixels in the corresponding positions of the two images. The average value of the ratio between the difference and the maximum difference (255) of all pixels at the corresponding position is then calculated. The calculation formula is as follows: (20) $\begin{array}{r}\text{UACI}\left({\text{Q}}_1,{\text{Q}}_2\right)=\frac{1}{\text{WH}}\sum _{\text{i}}^{\text{W}}\sum _{\text{j}}^{\text{H}}\frac{|{\text{Q}}_1\left(\text{i},\text{j}\right)-{\text{Q}}_2\left(\text{i},\text{j}\right)|}{255-0}\times 100\text{\%}\end{array}$(20) Owing to the randomness of the location, the theoretical expected values of the NPCR and UACI for the two random images were 99.6094% and 33.4635%, respectively.

The NPCR and UACI values of the wheat encryption images in Gaoping City for each scheme are listed in Table 8. In general, the NPCR value of Scheme 1 is closest to the theoretical expected value of 99.6094%, and the UACI value of Scheme 9 is closest to the theoretical expected value of 33.4635%. In addition, the NPCR and UACI values of the different R, G, and B channels may have significant differences.

Table 8. NPCR and UACI values of wheat encrypted images.

Chaotic System Scheme		1	2	3	4	5	6	7	8	9	10
NPCR (%)	R	99.6056	99.6231	99.6059	99.6059	99.6078	99.5899	99.6040	99.6113	99.6223	99.6098
	G	99.6094	99.6243	99.5979	99.6193	99.6208	99.6349	99.6189	99.6017	99.6269	99.6204
	B	99.6094	99.6078	99.6113	99.6082	99.6223	99.5975	99.6124	99.6113	99.6368	99.6353
UACI (%)	R	31.4796	31.6308	31.3929	31.3833	31.1587	31.4013	31.4481	31.4044	33.0811	32.1839
	G	31.4107	31.5910	31.4502	31.4147	29.9502	31.4897	31.3795	31.4747	33.0621	32.1632
	B	31.3855	31.5985	31.4365	31.4664	33.2818	31.4660	31.3408	31.4391	33.1365	32.2219

4.4. Key space analysis

The key space refers to the set of all legal keys (Zheng & Zeng, 2022). The key space of an image encryption system should be sufficiently large to combat exhaustive attacks effectively. According to the current computer level, the key space of the encryption algorithm should generally be greater than $2^{128}$. The key space for each scheme is listed in Table 9. The key space of scheme 6 and scheme 10 is the smallest, which is ${10}^{84}\cong 2^{280}>2^{128}$. The key space of Scheme 8 is the largest at ${10}^{154}$. The key space of schemes 4 and 9 is the second largest at ${10}^{104}$.

Table 9. Key space.

Chaotic System Scheme	1	2	3	4	5	6	7	8	9	10
Key Space	${10}^{94}$	${10}^{94}$	${10}^{94}$	${10}^{104}$	${10}^{94}$	${10}^{84}$	${10}^{94}$	${10}^{154}$	${10}^{104}$	${10}^{84}$

4.5. Information entropy

Information entropy is used to measure the uniformity of the grey-value distribution in images, reflecting the uncertainty of the image information (Khalil et al., 2021). Generally, the greater the information entropy of an image, the greater the uncertainty. Then, the more uniform the distribution of grey values of pixels, the less visible the information, and the stronger the resistance to entropy attacks. The formula for calculating information entropy is as follows: (21) $\text{I}=-\sum _{\text{m}=0}^{\text{L}}\text{p}\left(\text{m}\right)\text{lo}{\text{g}}_2\text{p}\left(\text{m}\right)$(21) where $\text{L}$ is the number of grey levels in the image. $\text{p}\left(\text{m}\right)$ is the probability that a grey value $\text{m}$ occurs. For a grey random image with $\text{L}=256$, the theoretical value of information entropy $\text{I}$ was 8.

The information entropy values of the wheat encryption images in Gaoping City for each scheme are listed in Table 10. The entropy value of schemes 1–8 is 7.999, which is closest to 8 and has good anti-entropy attack ability. Scheme 9 has the lowest entropy, smaller uncertainty, and the worst anti-entropy attack effect.

Table 10. Information entropy of wheat encryption images.

Chaotic System Scheme		1	2	3	4	5	6	7	8	9	10
Entropy	R	7.999	7.999	7.999	7.999	7.999	7.999	7.999	7.999	7.934	7.987
	G	7.999	7.999	7.999	7.999	7.999	7.999	7.999	7.999	7.934	7.986
	B	7.999	7.999	7.999	7.999	7.999	7.999	7.999	7.999	7.933	7.986

4.6. Grey difference degree

The grey difference degree is used to measure the grey level difference between the original image and encrypted image (Murali et al., 2023), and the calculation formula is as follows: (22) $\begin{array}{rl}\text{GVD}& =\frac{\mathrm{A}{\mathrm{N}}^{\mathrm{\prime }}\left[\text{GN}\left(\alpha ,\text{β}\right)\right]-\text{AN}\left[\text{GN}\left(\alpha ,\text{β}\right)\right]}{\mathrm{A}{\mathrm{N}}^{\mathrm{\prime }}\left[\text{GN}\left(\alpha ,\text{β}\right)\right]+\text{AN}\left[\text{GN}\left(\alpha ,\text{β}\right)\right]}\end{array}$(22) (23) $\begin{array}{rl}\text{GN}\left(\alpha ,\text{β}\right)& =\frac{\sum {\left[\text{G}\left(\alpha ,\text{β}\right)-\text{G}\left({\alpha }^{\mathrm{\prime }},{\text{β}}^{\mathrm{\prime }}\right)\right]}^2}{4}\end{array}$(23) where $({\alpha }^{\mathrm{\prime }},{\text{β}}^{\mathrm{\prime }})=\{\begin{array}{c}(\alpha -1,\text{ β})\\ (\alpha +1,\text{ β})\\ (\alpha ,\text{ β}+1)\\ (\alpha ,\text{ β}-1)\end{array}$. (24) $\text{AN}\left[\text{GN}\left(\alpha ,\text{β}\right)\right]=\frac{{\sum }_{\alpha =2}^{\text{W}-1}{\sum }_{\text{β}=2}^{\text{H}-1}\text{GN}\left(\alpha ,\text{β}\right)}{\left(\text{W}-2\right)\left(\text{H}-2\right)}$(24) where $\text{G}\left(\alpha ,\text{β}\right)$ is the grey-scale value at position $\left(\alpha ,\text{β}\right)$ position. $\text{GN}$ is the degree of grey difference. $\text{AN}$ is the average neighbourhood grey difference in the original image. $\mathrm{A}{\mathrm{N}}^{\mathrm{\prime }}$ is the average neighbourhood grey difference of the encrypted image. The GVD value ranged between 0 and 1. 0 indicates two images that are exactly the same, and 1 indicates two images that are completely different.

Each pixel in an image has a grey-scale value that represents its brightness or colour intensity. The GVD quantifies the degree of variation between images by calculating the grey-scale difference between the original and encrypted images.

The GVD values of the wheat encryption images in Gaoping City for each scheme are listed in Table 11. Scheme 9 exhibits a large GVD value that is closest to 1 and has the best effect.

Table 11. GVD value of wheat encryption image.

Chaotic System Scheme		1	2	3	4	5	6	7	8	9	10
GVD	R	0.8909	0.8905	0.8904	0.8905	0.891	0.8903	0.8905	0.8908	0.9054	0.8983
	G	0.8887	0.8894	0.8891	0.889	0.889	0.8891	0.8889	0.8888	0.9039	0.8969
	B	0.8966	0.8961	0.8967	0.8964	0.8963	0.8962	0.8967	0.8964	0.9105	0.9039

4.7. Peak signal to noise ratio

The peak signal-to-noise ratio (PSNR) is the calculation of the error between the corresponding pixels and is based on error-sensitive image quality evaluation (Wang et al., 2021). The calculation formula is as follows: (25) $\begin{array}{rl}\text{MSE}& =\frac{1}{\text{H}\times \text{W}}\sum _{\text{s}=1}^{\text{H}}\sum _{\gamma =1}^{\text{W}}(\text{X}\left(\text{s},\gamma \right)-\text{Y}\left(\text{s},\gamma \right){)}^2\end{array}$(25) (26) $\begin{array}{rl}\text{PSNR}& =10\text{lo}{\text{g}}_{10}\left(\frac{{\left(2^{\text{B}}-1\right)}^2}{\text{MSE}}\right)\end{array}$(26) where $\text{MSE}$ is the Mean Square Error of the encrypted image X and the original image Y. $\text{H}$ is the height of the image, and $\text{W}$ is the width of the image. $\text{B}$ is the number of pixel bits, generally valued at 8, and the grey level of the pixels is 256. The larger the value of PSNR, the smaller the distortion, the smaller the gap between the original image and encrypted image, and the worse the encryption effect.

The PSNR values of the wheat encryption images in Gaoping City for each scheme are listed in Table 12. It can be seen that the PSNR value of scheme 9 is the smallest, which indicates that the greater the gap between the original image and the encrypted image, the better the encryption effect.

Table 12. PSNR value of wheat encryption image.

Chaotic System Scheme		1	2	3	4	5	6	7	8	9	10
PSNR	R	7.79	7.757	7.809	7.803	7.79	7.806	7.791	7.81	7.37	7.645
	G	8.788	8.738	8.789	8.782	8.776	8.781	8.809	8.783	8.323	8.556
	B	8.399	8.373	8.399	8.412	8.398	8.424	8.397	8.429	8.079	8.122

4.8. Robustness analysis

The ability of encrypted images to resist noise and blocking attacks (Hosny et al., 2022). Encrypted images may be affected by various factors in the transmission process of the channel, such as blur, distortion, and partial information loss. This affects the image decryption effect and increases the difficulty of image decryption.

The 60 × 50 pixels in the encrypted image, 80 × 80 pixels in the R channel, and 50 × 80 pixels in the G channel are randomly lost. The pixel loss of the wheat encryption image in Gaoping City based on Scheme 1 is shown in Figure 4(a), and the decrypted result is shown in Figure 4(b). The anti-blocking ability of the encryption algorithm in each scheme was measured according to the values of the NPCR, UACI, and PSNR, as shown in Table 13. Scheme 9 shows the good encryption effect on the UACI and PSNR values.

Figure 4. Wheat image encryption with missing pixels. (a) Encrypted image; (b) Decrypted image.

Table 13. The ability to resist blocking attacks.

Chaotic System Scheme		1	2	3	4	5	6	7	8	9	10
NPCR (%)	R	99.6059	99.6239	99.6075	99.6132	99.6147	99.5975	99.6113	99.6140	99.6437	99.6185
	G	99.6105	99.6269	99.6025	99.6208	99.6246	99.6380	99.6250	99.6140	99.6441	99.6258
	B	99.6105	99.6117	99.6193	99.6101	99.6300	99.6029	99.6174	99.6124	99.6445	99.6399
UACI (%)	R	31.5418	32.2438	32.2125	32.1892	31.4231	32.1904	32.2455	32.2100	33.7230	32.8882
	G	31.4715	32.2282	32.0488	31.9774	30.8427	32.0953	32.0015	32.0887	33.5209	32.6257
	B	31.4595	32.0328	31.7358	31.7522	33.6116	31.7633	31.6484	31.7250	33.2831	32.3870
PSNR	R	7.568	7.546	7.385	7.577	7.466	7.362	7.472	7.587	7.091	7.227
	G	8.465	8.511	8.693	8.601	8.542	8.488	8.648	8.658	8.171	8.471
	B	8.324	8.332	8.406	8.472	8.418	8.432	8.409	8.462	8.117	8.106

Gaussian white noise with the mean of 0, standard deviation of 1, and variance of 1 was added to the encrypted image. The wheat encryption image with white Gaussian noise in Gaoping City based on Scheme 1 is shown in Figure 5(a), and the decrypted result is shown in Figure 5(b). According to the values of NPCR, UACI, and PSNR, the anti-noise ability of the encryption schemes in each scheme was measured, as shown in Table 14. Scheme 8 shows good performance on the NPCR values, and Scheme 9 shows a good encryption effect on the UACI and PSNR values.

Figure 5. Wheat image encryption with Gaussian white noise. (a) Encrypted image; (b) Decrypted image.

Table 14. The ability to resist noise attacks.

Chaotic System Scheme		1	2	3	4	5	6	7	8	9	10
NPCR (%)	R	99.6059	99.6010	99.6075	99.5918	99.6140	99.6166	99.5987	99.6120	99.6204	99.6212
	G	99.6113	99.6140	99.5956	99.6082	99.5995	99.6277	99.6170	99.6090	99.6258	99.6162
	B	99.6155	99.6162	99.6113	99.6262	99.6288	99.6174	99.6334	99.6113	99.6452	99.6296
UACI (%)	R	31.5531	31.6761	31.4419	31.4250	31.1376	31.4580	31.4930	31.4583	33.1435	32.2366
	G	31.4826	31.6383	31.5071	31.4782	30.0468	31.5326	31.4354	31.5303	33.1200	32.2096
	B	31.4702	31.6361	31.4787	31.5228	33.3605	31.5210	31.4026	31.4969	33.2069	32.2705
PSNR	R	7.772	7.74	7.791	7.785	7.773	7.789	7.774	7.793	7.353	7.629
	G	8.781	8.731	8.782	8.775	8.769	8.774	8.802	8.776	8.315	8.55
	B	8.416	8.389	8.416	8.428	8.415	8.44	8.413	8.445	8.094	8.138

4.9. Experimental summary

In summary, Scheme 9 shows good performance in correlation, UACI value, GVD value, PSNR value, anti-blocking attack ability, and anti-noise attack ability. Its key-space size is second only to scheme 8, which has the largest key-space size, but its variance, standard deviation, and anti-entropy attack ability are the worst. Scheme 8 shows good performance in key-space size, entropy value, and anti-noise attack ability. The variance, standard deviation, and entropy of Scheme 7 were the best. Scheme 1 is closest to the theoretical expected value of the NPCR (99.6094%) and has good entropy. Although the key space of schemes 6 and 10 is the smallest, it is much larger than the value that can resist brute-force attacks.

5. Conclusions

In this study, the encryption effects of several common chaotic systems are discussed based on wheat images from Gaoping City, Shanxi Province. The eight common image encryption performance evaluation indices are histogram analysis, correlation analysis of adjacent pixels, key sensitivity analysis, key space analysis, information entropy analysis, GVD analysis, PSNR analysis, and robustness analysis. Through the above analysis, it was found that different chaotic system schemes have different effects on different evaluation indexes and are suitable for different application scenarios. Among the chaotic systems proposed in recent years, such as the new four-dimensional chaotic system represented by Scheme 9, the overall encryption effect of wheat encrypted images is good. Owing to the high similarity of agricultural images, the new four-dimensional chaotic system can be extended to encrypt other agricultural images. The data presented in Section 4 can also provide references for the selection of encryption schemes for agricultural images.

Author contributions statement

Yi Shao and Hua Yang were involved in the conception and design, or analysis and interpretation of the data; Yi Shao and Hua Yang were involved in the drafting of the paper, revising it critically for intellectual content; and Yi Shao, Hua Yang, Huiru Zhu and Xuefeng Deng were involved in the final approval of the version to be published; and that all authors agree to be accountable for all aspects of the work.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data that support the findings of this study are available on request from the corresponding author.

Additional information

Funding

This work was supported by the Postgraduate Scientific Research Innovation Project of Shanxi Province in 2023 [grant no 2023KY323], the Shanxi Province Basic Research Program Project (Free Exploration) [grant no 20210302123408], and the Open Project Foundation of Intelligent Information Processing Key Laboratory of Shanxi Province [grant no CICIP2023002].