Video Encryption Using 7D Hyperchaotic System with Two-Level Discrete Wavelet Transform and Rubik's Cube Scrambling

In the era of the digital revolution and advanced communications, the use of digital visual content has witnessed tremendous growth across various fields, where videos are employed in medical applications, military communications, distance education platforms, and social media, requiring considerable attention to security and privacy issues. This rapid growth has been accompanied by an accelerating increase in security risks, cyber threats, and electronic attacks, as attack methods have evolved to include advanced techniques such as sophisticated statistical analysis, differential attacks, and brute force attacks enhanced by artificial intelligence [1]. This reality necessitates the development of more robust and complex encryption systems capable of confronting these advanced threats while maintaining operational efficiency [2].

Traditional cryptographic methods, such as AES and DES, are computationally demanding even though they offer strong security and can theoretically encrypt video data. When used directly on large-scale, high-resolution video streams, they can result in enormous processing costs and considerable latency. As a result, different strategies have been investigated to better strike a compromise between processing speed and good security [3]. Digital video possesses unique characteristics, including massive data volume, strong correlation between spatially and temporally adjacent pixels, the need for instant processing in live applications, and the necessity of maintaining compression quality to conserve storage space and transmission capacity. These particular challenges require innovative solutions that consider the statistical and structural properties of visual content [4]. Therefore, recent research focuses on developing hybrid systems that combine different techniques to provide advanced encryption solutions that unite superior security with efficiency. Chaotic systems represent one of the most prominent of these techniques, possessing unique properties that make them ideal for generating strong encryption keys. These systems are characterized by exhibiting seemingly random behavior despite their deterministic nature and providing regular distribution of generated values [5]. Nevertheless, the literature on cryptanalysis has revealed recognized flaws in traditional low-dimensional chaos-based cryptosystems. They frequently experience dynamical degradation when implemented on finite-precision digital devices, resulting in short cycle durations and susceptibilities to phase-space reconstruction attacks [6, 7]. This study uses a 7D hyperchaotic system, which offers much more complex dynamics, more positive Lyapunov exponents, and a considerably wider key space, to overcome these intrinsic vulnerabilities and successfully fend off conventional cryptanalytic attacks. Conversely, the Discrete Wavelet Transform (DWT) establishes a rigorous mathematical framework for analyzing and representing visual data in the frequency domain. Through signal decomposition, DWT effectively separates high-frequency components (fine details) from low-frequency components (general structural features). This decomposition facilitates the application of multiple encryption levels specifically designed for each frequency sub-band. Additionally, employing second-level DWT enhances analysis precision and enables a more accurate and robust encryption process [8].

The Rubik's Cube technique has proven its effectiveness as a powerful tool for scrambling data in a complex and organized manner [9], where this technique applies multiple and diverse operations such as row and column rotation, inversion, and diagonal transposition, resulting in a sophisticated scrambling mechanism capable of breaking statistical patterns in a nonlinear fashion while ensuring complete reversibility [10]. This latter characteristic guarantees precise recovery of original data after decryption.

This research presents a novel hybrid video encryption method that combines three advanced techniques: a 7D chaotic system, second-level DWT, and Rubik's Cube algorithm. A sophisticated methodology is developed for processing all color channels in YCbCr space in a parallel and specialized manner, ensuring comprehensive and effective protection of the entire visual content. It also achieves advanced encryption performance that combines high security and efficiency, rendering the method capable of confronting current and future cyber threats.

Main Contributions: (1) Integration of a 7D hyperchaotic system with two-level DWT and Rubik's Cube for comprehensive video encryption, (2) parallel YCbCr channel processing with specialized chaotic keys, (3) dynamic inter-frame key updating, preventing temporal attacks, and (4) fast execution suitable for real-time applications.

Section 2 reviews previous studies. Section 3 presents theoretical foundations. Section 4 describes the proposed method. Section 5 reports experimental results and analysis. Section 6 concludes.

3.1 Chaotic systems

Chaos is a widespread phenomenon in nature that is widely used in diverse applications in many fields of study, such as physics, engineering, mathematics, biology, secure communications, high-performance circuit design for telecommunications, and information processing. It has a behavior similar to noise and is present in nonlinear dynamical systems. The sequences generated by chaotic systems are highly sensitive to their initial conditions, unpredictable, have long periodicities, and have a spread spectrum. Highly chaotic systems tend to have very complex dynamics. A highly chaotic system is often defined as one with more than one positive Lyapunov exponent, meaning the number of propagation directions is greater than one, resulting in the system exhibiting chaotic behavior and high randomness [18]. Therefore, the more chaotic a chaotic system is, the more complex it is and therefore unpredictable for long periods.

In this research, we use a 7D hyperchaotic system [19], which has more complex dynamic properties than low-dimensional chaotic systems. This system is obtained as follows:

$\begin{gathered}\frac{d x}{d t}=-\sigma x+\rho y+\delta w-\eta v \\ \frac{d y}{d t}=\Omega x-\Psi_{x z}-\mu y \\ \frac{d z}{d t}=-\phi z+x y+\eta v \\ \frac{d w}{d t}=-w-\mho y z-\phi v p \\ \frac{d v}{d t}=y p+\beta x p-\eta z-v \\ \frac{d u}{d t}=\omega u-\mu z w p+\Omega x p \\ \frac{d p}{d t}=-\alpha p+x y+\eta v\end{gathered}$           (1)

This system has states x, y, z, u, v, w, p, and t, which are positive numbers. Its parameters include σ, ρ, ẟ, Ψ, ϕ, ℧, β, ω, μ, α, ɳ, and Ω. The system exhibits chaotic behavior and a strange attractor when these parameters are set to specific values: σ = 14, ρ = 11, ẟ = 0.4, ɳ = 0.5, Ω = 30, Ψ = 2.5, ϕ = 5, ℧ = 3, β = 2, ω = 15, μ = 4, and α = 13. This chaotic nature is further demonstrated with the initial conditions: x(0) = 0.1, y(0) = 0.5, z(0) = 3.5, w(0) = 0.6, u(0) = 0.4, v(0) = 0.1, and p(0) = 0.6.

In the present work, the 7D hyperchaotic system (Eq. (1)) is numerically integrated using the fourth-order Runge-Kutta (RK4) method with a fixed step size of h = 0.01. The system exhibits fully developed hyperchaotic behavior from the specified initial conditions, as evidenced by three positive Lyapunov exponents (L1 = 0.929593, L2 = 0.327552, L3 = 0.001271) [17, 18] and confirmed through NIST statistical randomness tests [18]. Accordingly, the chaotic sequences used for key generation are derived directly from these initial conditions without discarding transient samples.

3.2 Discrete Wavelet Transform

DWT is an advanced mathematical technique that analyzes a signal into different frequency components while preserving both spatial and temporal information. This makes it an ideal tool for representing visual data in the frequency domain [20]. DWT offers multi-resolution analysis capabilities, which are particularly useful for various image processing tasks, including denoising, compression, enhancement, and feature extraction. Within the context of encryption, DWT provides unique features that make it an optimal choice for advanced security applications [21].

The 2D DWT can be used in visual processing to decompose an image into four sub-bands. It effectively separates high-frequency information (fine details) from low-frequency information (general image properties). A second-level DWT, which involves applying the transform twice, provides a deeper level of frequency analysis. The first level produces four sub-matrices: LL1 (approximation coefficients), LH1 (horizontal details), HL1 (vertical details), and HH1 (diagonal details). The second level then applies the transform only to the LL1 matrix, yielding more precise coefficients: LL2, LH2, HL2, and HH2 [22].

The primary advantages of using DWT for encryption include: Energy Compaction—Most of the important information is concentrated into a few coefficients, allowing for selective and efficient encryption. Multi-resolution Representation—This provides exceptional flexibility for applying different, graded levels of encryption to various image details. Noise Robustness—Encrypting in the frequency domain is less affected by minor noise and slight distortions [8].

3.3 Rubik's Cube

The Rubik's Cube is a 3D (3 × 3 × 3) puzzle of six faces. Each side has nine stickers, which come in six different base colors spread evenly across the cube. Any of the faces can be rotated 90 degrees, clockwise or counter-clockwise. The cube has three middle layers, hence there are 18 quarter-turns possible in a single step. The step is defined by the Face-Turn Metric, which takes a quarter-turn of one face as a step. In contrast, the Alternative Quarter-Turn Metric takes a half-turn as two steps [23]. Complexity and States: The total possibilities for states of a 3 × 3 × 3 Rubik's Cube are:

$\frac{\left(8!\times 3^7\right) \times\left(12!\times 2^{11}\right)}{2} \approx 4.3 \times 10^{19}$

There are around 4.3 × 10¹⁹ possible combinations, which is about 43 quintillion. The incredible number of permutations demonstrates just how complex the Rubik's Cube is. No wonder then that this puzzle is helping inspire scrambling techniques and encryptions [24].

This section presents a series of experimental tests that demonstrate the efficiency of the proposed method and its resistance to various attacks. The method was applied to a diverse set of color videos of different types and sizes. All experiments were conducted using MATLAB R2025a on an Intel Core i5 processor with 8 GB RAM. Table 1 illustrates a selection of these sample.

Table 1. Sample frames

5.1 Key space analysis

A fundamental requirement for any robust encryption method is its resistance to brute-force attacks. These attacks involve systematically attempting all possible keys within the key space. To ensure a high level of security, a key space of at least 2¹²⁸ is recommended to prevent it from being broken within a reasonable timeframe.

In our proposed method, the encryption key is composed of 19 variables, including the initial conditions and parameters of the chaotic system. Given that the precision for each variable is 10⁻¹⁴, the total size of the key space is estimated to be approximately 10²⁶⁶, which is roughly equivalent to 2⁸⁸⁴. This immense size far exceeds the recommended standard, making our method highly effective against brute-force attacks [18].

5.2 Histogram analysis

A histogram is an essential tool for video analysis, providing a description of the distribution of pixel values in each frame. To mitigate analytical attacks targeting histograms, the pixel distribution in the encrypted video must be completely uniform so that it does not reveal any patterns or statistical information about the original video. This uniformity ensures that no data about the relationship between the two videos is leaked [25]. As shown in Figure 2, the histogram of the encrypted video exhibits a quasi-uniform distribution.

Figure 2. Histogram analysis

This result proves that the encrypted video is completely different from the original video in terms of visual and statistical distribution, as there is no statistical similarity between the two histograms.

5.3 Information entropy analysis

Information entropy is a fundamental mathematical property used to evaluate the unpredictability and randomness of an information source. To ensure an effective encryption system, the entropy of the encrypted video should ideally be 8. A value lower than 8 suggests that the source is not sufficiently random, introducing a degree of predictability that compromises its security [13].

$H(m)=-\sum_{i=0}^{N-1} P\left(s_i\right) \log _2\left[P\left(s_i\right)\right]$           (2)

where, S is the information source, s_i represents intensity levels (0-255) for each color channel, and p(s_i) is the probability of occurrence of s_i.

The information entropy was calculated for several original videos and their corresponding encrypted versions. As shown in Table 2, the obtained entropy values are very close to the theoretical value of 8. This result demonstrates that no statistical information was leaked during the encryption process, confirming that the proposed method is secure against entropy analysis attacks.

Table 2. Entropy values

5.4 Differential attacks

Differential attacks are a technique for cracking an encryption method by analyzing the differences between two encrypted versions of the same material. To measure the impact of a small change in a single pixel within the original video frame on the encrypted video, metrics such as the Number of Pixels Change Rate (NPCR) and the Unified Average Changing Intensity (UACI) are used. These metrics are calculated by encrypting the original video, then randomly changing the value of a single pixel in a given frame and re-encrypting it [26].

$N P C R=\frac{\sum_{i=1}^M \sum_{j=1}^N D(i . j)}{M \times N} \times 100$           (3)

$U A C I=\frac{\sum_{i=1}^M \sum_{J=1}^N \frac{\left|C_1(i . j)-C_2(i . j)\right|}{255}}{M \times N} \times 100$            (4)

where, D(i,j) = 0 if C1(i,j) = C2(i,j), else D(i,j) = 1, and C1, C2 are two encrypted frames of size M × N. Table 3 presents the NPCR and UACI values evaluated across 5 frames from multiple test videos.

Table 3. Number of Pixels Change Rate (NPCR) and Unified Average Changing Intensity (UACI) values

The results confirm that both metrics remain consistently close to their ideal values, with a mean NPCR of 99.6853% (std = 0.0180%) and a mean UACI of 33.4578% (std = 0.0214%), demonstrating stable differential sensitivity independent of frame content or video type.

5.5 Correlation coefficient

A strong correlation between adjacent pixels is a key characteristic of unencrypted video. The correlation coefficient reflects the similarity between adjacent pixel values, whether horizontally, vertically, or diagonally. Therefore, any effective encryption system must significantly reduce this correlation, making the encrypted video appear as random data or "noise," to avoid any statistical attacks [14].

$r_{x y}=\frac{\operatorname{cov}(x, y)}{\sqrt{D(y)} \sqrt{D(x)}}$           (5)

$\operatorname{cov}(x, y)=\frac{1}{N} \sum_{i=1}^N\left(x_i-E(x)\right)\left(y_i-E(y)\right)$           (6)

$E(x)=\frac{1}{N} \sum_{i=1}^N x_i, D(x)=\frac{1}{N} \sum_{i=1}^N\left(x_i-E(x)\right)^2$            (7)

where, x and y represent intensity values of two adjacent pixels (horizontal, vertical, or diagonal); N is the total number of pixel pairs sampled; cov(x,y) is the covariance; D(x) is the variance; and E(x) is the mean. The results in Table 4 show that the correlation coefficients in the original video are very high (approximately 1).

Table 4. Correlation coefficients

However, these values decrease significantly after encryption, becoming close to zero. This decrease indicates that the encryption method has successfully removed statistical relationships between pixels, which enhances the resistance of the encrypted video to analytical attacks and makes it more difficult to exploit.

5.6 Mean Squared Error, Peak Signal to Noise Ratio, and Structural Similarity Index analysis

The Peak Signal to Noise Ratio (PSNR) is a commonly used metric for measuring the quality of video after decryption. In addition to this matter, Mean Squared Error (MSE) is also another useful metric for assessing the decoded video quality. MSE determines the average difference between the original video frames and the frames after processing, while PSNR indicates the quality of reconstruction. Seeing as the decrypted frames are identical to the original frames, the MSE becomes zero, so we can say that the PSNR becomes infinite. The pioneer measure of assessing the quality of video is the Structural Similarity Index (SSIM) [27]. This assessment process compares the structural features of video samples. The parameters, such as brightness, contrast, etc., are compared in this assessment between the original video and the decrypted video.

$M S E=\frac{1}{M \times N} \Sigma_i \quad \Sigma_j[I(i, j)-K(i, j)]^2$            (8)

$P S N R=20 \times \log _{10}\left(\frac{255}{\sqrt{M S E}}\right) d B$           (9)

$\operatorname{SSIM}(X, Y)=\frac{\left(2 \mu_x \mu_y+C_1\right)\left(2 \delta_{x y}+C_2\right)}{\left(\mu_X^2 \mu_Y^2+C_1\right)\left(\delta_x^2 \delta_y^2+C_2\right)}$           (10)

where, μ_x, μ_y are mean intensities, σ_x², σ_y² are variances, σ_xy is covariance, and C₁, C₂ are stability constants. According to our test results, the SSIM value is 1. It should be mentioned that obtaining a PSNR of ∞ and an SSIM of 1 for our symmetric encryption strategy are expected outcomes that are presented only to verify the precise and lossless nature of the decryption process, not as cryptographic security metrics. The proposed encryption system is capable of full decryption; the video is recovered without the loss of any information.

5.7 Sensitivity analysis

A robust encryption system must exhibit high sensitivity to any perturbation in the secret keys, whereby even infinitesimal modifications to the initial conditions of the chaotic system yield entirely different key sequences. To evaluate this key sensitivity, one of the initial condition parameters was altered, z(0), from 3.5 to 3.50000000000001, and the subsequently generated keys were employed in an attempt to decrypt the video frame. The decryption results using the altered key are presented in Table 5.

Table 5. Key sensitivity test

The decryption attempt using the slightly perturbed key failed completely, and the original frame could not be reconstructed. This result confirms that even minor deviations in the secret key lead to complete decryption failure, thereby demonstrating the method's strong resistance against brute-force and key-guessing attacks.

5.8 Security model and resistance analysis

Threat model: We consider an adversary with the following capabilities: (1) Known-plaintext attack: access to plaintext-ciphertext pairs. (2) Chosen-plaintext attack: ability to encrypt chosen frames. (3) Temporal analysis: exploiting inter-frame correlations. (4) Statistical analysis: histogram, entropy, and correlation attacks. (5) Brute-force attack: exhaustive key search.

Resistance analysis: (1) Brute-force: Key space of 10²⁶⁶ ≈ 2⁸⁸⁴ makes exhaustive search computationally infeasible. (2) Known/Chosen-plaintext: Dynamic key updating between frames prevents pattern extraction. (3) Temporal attacks: plaintext-dependent key updating ensures unique chaotic keys per frame, completely eliminating inter-frame temporal correlation. (4) Statistical attacks: Quasi-uniform histogram distribution, entropy ≈ 8, and near-zero correlation coefficients demonstrate strong resistance to analytical attacks. (5) Differential attacks: NPCR = 99.69% and UACI = 33.46% confirm complete diffusion satisfying the avalanche criterion.