We define a tensor-dependent orthogonal projection based on Ψ¯t:

(1)y=Ω(ψ¯t)⋅x+b,Ω(ψ¯t)∈O(n)

where Ω(ψ¯t) is an orthogonal matrix dependent on the tensor ψ¯t belonging to the set O(n) of orthogonal matrices of dimension n and b is a bias term.

Here, the matrix Ω(ψ¯t) is regularized to remain orthogonal:

(2)Ω(ψ¯t)T⋅Ω(ψ¯t)=I

The proposed activation function is not fixed but depends on the spectral-directional features:

(3)φHybride(x)=α(ψ¯t)⋅ReLU(x)+(1−α(ψ¯t))⋅σ(x)

where ReLU(x) denotes the Rectified linear function, σ(x) denotes the sigmoid function, and α(ψ¯t)∈(0,1) is a coefficient dynamically computed from ψ¯t.

The input-output Jacobian, as mentioned in [14], factorizes layer by layer:

(4)Jt=∂Kt∂ψ¯t=∏l=1L(Ω(ψ¯t).Diag(φhybride′(zl)))

where Kt denotes the encryption key at frame t, ψ¯t is the normalized spectral-directional tensor, Ω(ψ¯t) is the weight matrix of layer l, φhybride′(zl) is the derivative of the hybrid activation function and zl is the pre-activation at layer l.

(5)zl=Ω(ψ¯t)hl−1+bl

where Ω(ψ¯t) is an adaptive orthogonal matrix dependent on the tensor ψ¯t, hl−1 represents the outputs of the previous layer, and bl is the bias vector of layer l.

Owing to the bounded nature of the hybrid activation derivative, we impose:

(6)0<Sl≤φhybride′(Zl)≤S¯l≤1

Ensuring that the Frobenius norm ‖Jt‖F remains controlled. The lower bound Sl prevents degenerate mapping with vanishing Jacobian norm, while the upper bound S¯l≤1 avoids gradient explosion.

The network was trained using gradient descent with an adaptive learning rate initialized at η₀ = 10⁻³ and dynamically modulated according to the spectral energy of the input tensor. The number of epochs was set to E = 50, as the proposed orthogonally constrained and Jacobian-regularized architecture exhibited rapid convergence. No mini-batching was used, as training was performed sequentially frame by frame. The composite loss weights were empirically fixed as follows: orthogonality penalty λ₁ = 0.1, inter-frame decorrelation λ₂ = 0.5, and Jacobian margin constraint λ₃ = 0.2.

For each frame t, the network parameters θt are locally updated as:

(7)θt+1=θt−ηt∇θℒunsup(ft,Nθ(ψ¯t))

where Nθ denotes the deep neural network parameterized by θ, applied to the normalized tensor ψ¯t denotes the normalized input tensor at time t; ft denotes the frame at time t; ℒunsup(⋅,⋅) denotes the unsupervised loss function; ∇θ denotes the gradient of the loss with respect to the parameters θ; and ηt denotes the learning rate at iteration t.

The learning rate ηt was itself modulated by the spectral energy of the tensor:

(8)ηt=h(Espec(ψ¯t))

where ψ¯t denotes the normalized input tensor at time t, and Espec(ψ¯t) denotes the spectral energy of ψ¯t.

As a result, the update becomes self-adaptive: each frame adjusts the learning rate according to its spectral content.

The unsupervised objective ℒunsup enforces orthogonality, temporal decorrelation, and minimum sensitivity.

In accordance with (1), the dense layer is parameterised by a square orthogonal matrix conditioned on the spectral–directional tensor, denoted Ω(ψ¯t). After each gradient update, we enforce orthogonality by reprojecting the updated matrix onto the orthogonal manifold using a QR factorisation and retaining the Q factor, such that:

(9)Ω(ψ¯t)T⋅Ω(ψ¯t)=I

To further stabilise optimisation, we add a soft orthogonality penalty:

(10)Lorth=‖Ω(ψ¯t)T⋅Ω(ψ¯t)−I‖F2

To enforce inter-frame independence, we minimize the squared Pearson correlation between successive keys:

(11)Ldiv=corr(Kt,Kt+1)2

where corr(·,·) is the Pearson correlation on vectorized keys, averaged over the mini-batch.

Finally, we enforce a minimum Jacobian norm to promote the avalanche effect:

(12)Ljac=max(0,ε−‖Jt‖F)2

where Jt=∂Kt∂ψ¯t denotes the Jacobian of the key with respect to the normalised input tensor.

The overall loss is then given by the weighted sum:

(13)ℒunsup=λorthLorth+λdivLdiv+λjacLjac

Fig. 1 shows the cumulative distribution of key entropy, illustrating the keys’ uniformity.

The generated keys exhibited a mean entropy of 7.67 bits per byte, with a 95% confidence interval that remained above weak-randomness thresholds, thereby confirming strong statistical randomness and cryptographic suitability [20].

Fig. 2 shows the avalanche effect, where small input changes greatly alter the generated key [21].

The mean Hamming distance of 129.62 bits with a 95% confidence interval from 129.11 to 130.13 confirmed a balanced avalanche effect, while a one-sample t-test against 128 bits yielded p < 0.001, and the range 116–143 bits demonstrated strong diffusion and resistance to differential attacks [22], [23].

The following analysis, as Fig. 3 illustrates, evaluates the independence of keys across frames and color channels to ensure high variability and prevent redundancy [24].

Inter-frame correlations averaged 0.002 over 300 frames with a 95% confidence interval from −0.013 to 0.017 and a p-value of 0.77, consistent with [25], confirming strong temporal independence between successive keys [26].

Fig. 4 shows the correlations between the R, G, and B channels of the generated keys, which are near zero, ranging from minus 0.04 to 0.01, indicating minimal redundancy and strong statistical independence.

According to [27], as shown in Table I, the inter-channel correlations were weak, with averages of 0.0129 for RG, −0.045 for RB, and 0.0129 for GB over 20 frames, corresponding to 95% confidence intervals of [−0.0648, 0.0906], [−0.122, 0.032], and [−0.0648, 0.0906], and p-values of 0.73, 0.23, and 0.73, respectively. These results confirm statistical independence and strong, non-redundant key variability [28].

Original and decrypted frames are visually compared in Fig. 5 to assess encryption fidelity [29].

The Akiyo sequence (a) is unreadable after encryption (b) and fully restored after decryption (c), showing the effectiveness of the key generation method [30].

In accordance with [31], the proposed encryption, as depicted in Fig. 6, significantly reduced adjacent-pixel correlation to a negligible level, confirming key effectiveness.

The original videos exhibited a pixel correlation of approximately 0.9, which dropped close to zero after encryption, confirming the effective removal of spatial redundancies.

Tables II and III presents the entropy and correlations before and after encryption to validate the effectiveness of the generated keys.

According to Ghouate [30], entropy near 8 bits ensures strong randomness, and our encrypted video reached between 7.96 and 7.99 bits with directional correlations averaging 0.006 over 5 frames with a 95% confidence interval from −0.008 to 0.020, demonstrating effective key generation.

Robustness was tested via PSNR after decrypting videos affected by noise, compression, or data loss.

Our keys provided robust encryption, Table IV highlights that we achieved a PSNR of 33.8 dB over 300 frames with a 95% confidence interval from 33.76 to 33.84 dB, with PSNR of 35.7 dB for JPEG compression and 32 dB for data loss, ensuring reliable visual recovery [32].

As Fig. 7 reveals, the proposed method achieved a stable PSNR of 42 dB over 300 frames, with a 95% confidence interval of [41.94, 42.06] dB, thus ensuring high visual quality according to [33]. In contrast, Chaotic Maps, Scalable Video Coding (SVC), and Selective Video Encryption (H.264) exhibited lower PSNR values of 34.02 dB, 37.95 dB, and 30.09 dB, respectively, indicating greater visual loss [34]–[37].