High-fidelity steganography in EEG signals using advanced transform-based methods

Dataset and data preparation

This study utilizes the following three publicly accessible datasets:

Graz A dataset (Brunner et al., 2008): This dataset includes EEG recordings from nine individuals engaged in motor imagery tasks. The signals were captured at a sampling frequency of 250 Hz. For this study, data from the first eight channels (Fz, FT7, FC3, FCz, FC4, FT8, T3, C4) within the training dataset (containing 1,600 samples) for all nine participants were utilized.

DEAP dataset (Koelstra et al., 2011): The DEAP dataset comprises EEG recordings collected from 32 healthy individuals (16 males and 16 females) who were exposed to music videos as stimuli. The EEG signals were originally sampled at 512 Hz but were later preprocessed by down-sampling to 128 Hz and applying a band-pass filter between 4.0 and 45.0 Hz. In this study, EEG signals from the first eight channels (Fp1, AF3, F3, F7, FC5, FC1, C3, T7) were extracted for all 32 participants, each containing 40 trials. The data used corresponds to the initial 12.5 s, resulting in 1,600 samples.

BONN dataset (Andrzejak et al., 2001): The Bonn dataset includes EEG recordings collected from five healthy individuals and five patients diagnosed with epilepsy. The dataset consists of single-channel recordings, organized into five subsets labeled F, S, N, Z, and O. Each subset comprises 100 data segments, sampled at a frequency of 173.61 Hz. For this study, the first 9.216 s of data, corresponding to 1,600 samples, were selected for analysis.

Stationary wavelet transform

The SWT, also known as the translation-invariant wavelet transform, is a redundant, non-decimated variant of the DWT. Unlike the standard DWT, which involves downsampling operations at each decomposition level, the SWT maintains the same signal length across all scales by upsampling the filter coefficients instead of decimating the sub-band signals (Strasser, Muma & Zoubir, 2012). This characteristic ensures shift-invariance, which can be particularly advantageous in signal denoising and feature extraction, as the time shift of a signal does not alter the location of significant features in the transformed domain.

Mathematically, let $x\left[n\right]$ be the discrete-time signal, and let $h\left[n\right]$ and $g\left[n\right]$ denote the low-pass and high-pass filters derived from the chosen wavelet basis function. For the SWT decomposition at level $j$, the approximation coefficients $a_j\left[n\right]$ and the detail coefficients $d_j\left[n\right]$ are computed from the approximation coefficients of the previous level $a_{j-1}\left[n\right]$ as expressed in Eqs. (1) and (2):

(1) $$a_j\left[n\right]=\sum _k{a}_{j-1}\left[k\right]\mathrm{h}\left[2^{j-1}\left(\mathrm{n}-\mathrm{k}\right)\right]$$

(2) $$d_j\left[n\right]=\sum _k{a}_{j-1}\left[k\right]\mathrm{g}\left[2^{j-1}\left(\mathrm{n}-\mathrm{k}\right)\right].$$Here, $a_0\left[n\right]=x\left[n\right]$ represents the original signal. At each decomposition level, the filters are effectively upsampled by a factor of $2^{j-1}$ before the convolution, ensuring that the resulting coefficients have the same length as the original signal. As such, the SWT provides a multi-resolution representation of the signal without losing time-invariance, making it suitable for applications where shift sensitivity is undesirable (Kumar et al., 2021).

Tent map

In this section, we introduce and analyze the tent map, which serves as a well-known piecewise linear chaotic map similar in certain aspects to the logistic map. The tent map provides a simplified yet effective environment for examining nonlinear dynamical behavior and chaos. Furthermore, it allows for a clear comparison with the logistic map in terms of their bifurcation structures and spectral characteristics.

The practical advantages of using the tent map in our proposed method are detailed below. The tent map provides significant advantages, including low computational complexity, straightforward implementation, and a wide range of chaotic behavior, making it particularly suitable for real-time applications. Due to its positive Lyapunov exponent, it exhibits stable chaotic dynamics across a wide parameter range, which has led to its frequent use in data hiding and security applications. Although more complex chaotic systems such as hyperchaotic maps may offer larger key spaces, they often introduce additional computational burdens and implementation complexity. In contrast, the tent map’s one-dimensional, piecewise linear nature enables fast key generation while maintaining predictable and stable output. It is also less prone to undesirable behaviors such as periodic cycles or convergence to zero, which are sometimes observed in logistic maps. Furthermore, the tent map tends to produce output sequences that are more uniformly distributed over the [0, 1] interval, contributing to enhanced statistical randomness.

In addition to its role as a chaotic key generator, the tent map also serves as an efficient lightweight encryption mechanism in our method. The secret data is scrambled using the tent map sequence prior to embedding, aligning with the principles of combined cryptographic and steganographic security. This dual-layer approach is widely recommended in the literature, as it enhances protection: steganography conceals the presence of the data, while encryption ensures its confidentiality even if discovered. Such “encryption-before-embedding” strategies provide defense-in-depth, making attacks significantly harder. While some modern chaotic systems offer larger key spaces, they often suffer from instability and higher computational complexity. The tent map, in contrast, maintains stability, predictable behavior, and fast execution—all essential in real-time EEG-based applications. Therefore, it offers a well-balanced trade-off between simplicity, security, and speed (Hussain et al., 2017; Sharma, Anand & Singh, 2021).

In our scheme, the tent map is not used in isolation. The chaotic sequences it generates are integrated into the embedding process via singular value decomposition (SVD). Therefore, the system’s overall security is based not only on the tent map’s initial conditions and control parameter, but also on the embedding positions determined by the SVD matrices. This integration enlarges the effective key space and strengthens resistance against attacks. Recent studies support the continued use of the tent map in secure data hiding applications. Taken together, these factors confirm the tent map’s suitability for real-time secure data hiding tasks where both speed and protection are critical (Daoui et al., 2023; Liu et al., 2024).

The tent map is defined by the iterative Eq. (3):

(3) $$x_{n+1}=f\left(x_n\right)=\{\begin{array}{c}\mu x_{n}if0\le x_n<\frac{1}{2}\\ \mu (1-x_n)if\frac{1}{2}\le x_n\le 1\end{array}$$where $\mu \in \left[0,2\right]$ is the control parameter. Unlike the logistic map, which is nonlinear and defined by $x_{n+1}=r{x}_n\left(1-x_n\right)$, the tent map’s piecewise linearity makes its dynamics more transparent while still exhibiting a rich variety of behaviors, ranging from fixed points to chaotic regimes as $\mu$ increases (Valle, Machicao & Bruno, 2022). As with the logistic map, the bifurcation diagram of the tent map provides insight into the transitions from periodic to chaotic behavior. When $\mu$ is near zero, the system tends to a stable fixed point. Increasing $\mu$ leads to period-doubling cascades, culminating in a chaotic regime similar to that of the logistic map. This pattern can be observed in the bifurcation diagrams provided in Fig. 1 (tent map) and Fig. 2 (logistic map), where the abscissa represents the parameter r and the ordinate indicates the long-term values of $x_n$ (Ćmil et al., 2024).

In addition to bifurcation diagrams, the frequency spectrum of the generated time series can yield valuable information about the underlying dynamics. By applying a suitable Fourier or spectral analysis to the iterates $\left\{x_n\right\}$, one can detect the presence of dominant frequencies and broadband components indicative of chaotic behavior. The frequency spectra for both the tent map and the logistic map are illustrated in Figs. 3 and 4, respectively. These spectra highlight the fundamental differences and similarities in the chaotic dynamics of the two maps, providing a complementary perspective to the bifurcation analysis. Moreover, analyzing the amplitude of the generated signals from the Tent Map is crucial, particularly when considering applications such as steganography. The tent map’s piecewise linear structure tends to produce a uniform amplitude distribution of chaotic values, providing a broader range of embedding options. This uniformity and controlled amplitude range can be leveraged in steganographic methods, as the signal variations introduced by embedding hidden information are less noticeable. Thus, the tent map, due to its predictable amplitude characteristics and chaotic behavior, offers a practical advantage in designing steganographic algorithms that are both secure and robust (van Harten, 2018).

Singular value decomposition

SVD is a fundamental linear algebra technique used to factorize a given rectangular matrix into three distinct matrices. Suppose we have a data matrix X of size $m\times n$, where $\mathrm{m}\ge \mathrm{n}$. The SVD of X is defined in Eq. (4):

(4) $$X=U\mathrm{\Sigma }{\mathrm{V}}^T$$Here, $U$ is an $m\times m$ orthogonal matrix $\left({\mathrm{U}}^{T}U=I_m\right)$, $\mathrm{\Sigma }$ is an $m\times n$ rectangular diagonal matrix with nonnegative real numbers on the diagonal (these values are called the singular values), and $V$ is an $n\times n$ orthogonal matrix ( ${\mathrm{V}}^{T}V=I_n)$.

The diagonal elements of $\mathrm{\Sigma }$, denoted as ${\sigma }_1,{\sigma }_2,\dots ,{\sigma }_r$, are the singular values of X, where $r=\mathrm{m}\mathrm{i}\mathrm{n}\left(m,n\right)$. These singular values are typically arranged in descending order ${\sigma }_1\ge {\sigma }_2\ge \dots \ge {\sigma }_r\ge 0)$. The columns of $U$ are called the left singular vectors, and the columns of $V$ are called the right singular vectors. SVD is particularly useful for dimensionality reduction, noise filtering, and feature extraction. By truncating the SVD at a certain rank $\mathrm{k}(\mathrm{k}<\mathrm{r})$, we can approximate X as Eq. (5):

(5) $$X_k=U_k{\mathrm{\Sigma }}_k{{\mathrm{V}}_k}^T$$where $U_k$ contains the first $k$ columns of $U,{\mathrm{\Sigma }}_k$ is the $k\times k$ upper-left submatrix of $\mathrm{\Sigma }$ containing the top $k$ singular values, and $V_k$ contains the first $k$ columns of $V$. This truncated representation often preserves the most significant structures of the data while reducing computational complexity and noise (Baker, 2005).

Algorithm for private information hiding

In the proposed method, encryption and steganography are integrated in a complementary way to provide multi-layered data protection. Specifically, chaotic encryption based on the Tent map is applied to ensure the confidentiality and randomness of the secret data, while the SVD-based embedding strategy minimizes signal distortion and preserves the main characteristics of the EEG signal. This combined approach not only enhances security against potential attacks but also maintains computational efficiency and high signal quality.

In this section, we present the detailed steps of our proposed private information hiding algorithm, as outlined in Algorithm 1 and depicted in Fig. 5. Our approach starts by encoding the private information and then carefully embedding it into the transformed coefficients of the host signal, ensuring minimal distortion and preserving perceptual quality. Specifically, the algorithm proceeds as follows, as detailed in Algorithm 1.

Step 1. Encoding private information:

We begin by encoding the private information, denoted as Q, into its binary form Bin, where each bit corresponds to a specific segment of the signal. The private information in this study is composed of both letters and digits, randomly generated, with letters constituting approximately two-thirds of the entire dataset.

Step 2. Chaotic key generation using tent map:

To guarantee security and unpredictability, we employ a chaotic map. Here, we use the tent map with parameters $\mathrm{r}=2$ and the initial condition $x_0=0.123456789$. Specifically, to remove transient effects and focus on the steady-state chaotic behavior, we skip the first 100 iterations. This chaotic sequence $P$ is generated to serve as a key stream that modulates the encoded bits. Each bit in Bin is multiplied by a corresponding value $P_i$ from the chaotic key stream to produce a spread-out bit sequence Bin*, thereby enhancing security.

Step 3. Transforming the host signal:

The host signal, denoted as S, undergoes a one-level SWT using the ‘haar’ wavelet. This decomposition generates the approximation coefficients cA1 and the detail coefficients cD1. The length of cA1 is recorded as L.

Step 4. Matrix reshaping and SVD:

The approximation coefficients cA1 are reshaped into an $N\times N$ matrix B, where $N=\sqrt{L}$. If necessary, zero-padding is applied to ensure a perfect square dimension. Subsequently, we perform SVD on B, obtaining U₁, Σ₁, and V₁^T.

Step 5. Embedding the information:

The off-diagonal elements of Σ₁ are replaced with the spread-out bits Bin*. By embedding information into these off-diagonal positions, we minimize changes to the matrix structure and preserve the dominant features of the host signal.

Step 6. Second SVD and reconstruction:

After inserting the private bits, a second SVD is performed on the modified Σ₁ (denoted as Σ₁^*) to produce U₂, Σ₂, and V₂^T. This step ensures that the introduced modifications are well-distributed throughout the singular values. The modified matrix B* is then reconstructed using U₁, Σ₂, and V₁^T.

Step 7. Inverse transform and final output:

The modified approximation coefficients cA1* are reshaped back into a vector and truncated to the original length L. These updated approximation coefficients replace the original ones, while the detail coefficients cD1 remain unchanged. Finally, the inverse SWT (ISWT) is applied to yield the modified signal S*. The outputs of the algorithm are the watermarked signal S*, the chaotic key P, and the matrices U₂ and V₂^T.

Algorithm for private information extraction

In this section, we detail the steps involved in the private information extraction process, as outlined in Algorithm 2. Figure 6 illustrates the overall workflow of the extraction procedure. The detailed steps of the extraction process are presented in Algorithm 2. Given the modified signal S*, the chaotic key P, and the matrices U₂ and V₂^T used during the embedding phase, this algorithm recovers the original private information Q_extracted. Similar to the embedding procedure, we begin by applying a wavelet transform and utilize the singular value decomposition to retrieve the embedded bits.

Step 1. Wavelet decomposition of the modified signal:

Starting with the modified signal S*, we apply a one-level SWT using the same ‘haar’ wavelet. This step yields the approximation coefficients cA1* and the detail coefficients cD1*. We record the length L = length(cA1*).

Step 2. Matrix formation and SVD:

Similar to the embedding phase, we reshape cA1* into an N × N matrix B*, where $N=\sqrt{L}$. If necessary, zero-padding is applied. We then perform SVD on B*, obtaining U₁, Σ₂_extracted, and V₁^T.

Step 3. Reconstruction of Σ₁^*:

Using the stored matrices U₂ and V₂^T from the embedding stage, we reconstruct Σ₁^* by Σ₁^* = U₂ × Σ₂_extracted × V₂^T. This step effectively reverses the second SVD operation performed during the embedding, allowing us to retrieve the embedded off-diagonal elements.

Step 4. Extracting the embedded bits:

From the reconstructed Σ₁^*, we identify all off-diagonal indices (i, j) where i ≠ j. These off-diagonal entries contain the embedded information bits. We extract the sequence Bin* from these positions.

Step 5. Recovering the original bits using the chaotic key:

Given the chaotic key P (the same sequence used in embedding), we revert the spreading process. Each bit in Bin* is divided by the corresponding P_i and then rounded to recover the original binary bits Bin. Clipping ensures that values remain in {0, 1}.

Step 6. Decoding the binary sequence:

Finally, the binary sequence Bin is decoded back into the original private information Q_extracted. Since we originally encoded letters and digits (with letters constituting about two-thirds of the content), this decoding step will restore the original textual and numeric information.

Performance evaluation metric

• I)

  Peak signal-to-noise ratio (PSNR): PSNR is a metric used to measure the quality of a reconstructed or processed signal compared to an original signal. It is commonly used in image and video compression to quantify how much the signal has degraded. A higher PSNR value generally indicates better reconstruction quality. The formula for PSNR is shown in Eq. (6). (6) $$PSNR=10{\log }_{10}\left(\frac{MA{X}^2}{MSE}\right).$$

  Here, MAX is the maximum possible value of the signal and MSE is the mean square error between the original signal $x\left(n\right)$ and the reconstructed (or processed) signal $\hat{x}\left(n\right)$.

• II)

  Percent residual deviation (PRD): PRD is a measure often used in biomedical signal processing (e.g., EEG signal compression) to quantify the relative error between the original and the reconstructed signal in percentage terms. It gives an idea of how much the reconstructed signal deviates from the original one. The formula for PRD is shown in Eq. (7). (7) $$PRD\left(\mathrm{\%}\right)=\frac{\sqrt{{\sum }_{n=1}^N{\left(x\left(n\right)-\hat{x}\left(n\right)\right)}^2}}{\sqrt{{\sum }_{n=1}^N{\left(x\left(n\right)\right)}^2}}x100$$

• III)

  Structural similarity index (SSIM): SSIM is a widely used metric for measuring the similarity between two signals or images. Unlike MSE, which only considers pixel-wise differences, SSIM evaluates the structural information, luminance, and contrast of the signals, providing a more perceptually relevant assessment of quality. SSIM values range from 0 to 1, where a value closer to 1 indicates higher similarity and better quality between the original and processed signals. The formula for SSIM is given in Eq. (8). (8) $$SSIM\left(x,\hat{x}\right)=\frac{\left(2{\mathrm{\mu }}_x{\mathrm{\mu }}_{\hat{x}}+C_1\right)\left(2{\mathrm{\sigma }}_{x\hat{x}}+C_2\right)}{\left({\mathrm{\mu }}_x^2+{\mathrm{\mu }}_{\hat{x}}^2+C_1\right)\left({\mathrm{\sigma }}_x^2+{\mathrm{\sigma }}_{\hat{x}}^2+C_2\right)}$$

  where $x$ and $\hat{x}$ represent the original and processed (or stego) signals, respectively; ${\mathrm{\mu }}_x$ and ${\mathrm{\mu }}_{\hat{x}}$ denote their mean values; ${\mathrm{\sigma }}_x^2$ and ${\mathrm{\sigma }}_{\hat{x}}^2$ are their variances; ${\mathrm{\sigma }}_{x\hat{x}}$ is the covariance between them; and $C_1$ and $C_2$ are small constants introduced to avoid division instability.

• IV)

  Normalized correlation coefficient (NCC): NCC measures the similarity between two signals. It is often used for pattern matching and similarity assessments. NCC values range from −1 to 1, where 1 indicates perfect correlation, 0 indicates no correlation, and −1 indicates perfect negative correlation. The formula for NCC is shown in Eq. (9). (9) $$NCC=\frac{{\sum }_{n=1}^N\left(x\left(n\right)-\overline{x}\right)\left(\hat{x}\left(n\right)-\overline{\hat{x}}\right)}{\sqrt{{\sum }_{n=1}^N{\left(x\left(n\right)-\overline{x}\right)}^2}\sqrt{{\sum }_{n=1}^N{\left(\hat{x}\left(n\right)-\overline{\hat{x}}\right)}^2}}.$$

  Here, $\overline{x}$ and $\overline{\hat{x}}$ are the mean values of $x\left(n\right)$ and $\hat{x}\left(n\right)$ respectively.

• V)

  Bit error rate (BER): BER is a fundamental metric in communication systems, used to assess the reliability and quality of data transmission. It is defined as the ratio of the number of bit errors to the total number of bits transmitted over a communication channel. A lower BER signifies a more dependable communication channel, ensuring that the transmitted data is received with minimal corruption. This metric is particularly crucial in applications requiring high data integrity, such as medical signal processing or secure data transfer, where even minor errors can have significant implications. The formula for BER is shown in Eq. (10). (10) $$BER=\frac{{\displaystyle Numberofbiterrors}}{Totalnumberoftransmittedbits}.$$

• VI)

  Pearson correlation coefficient (PCC): PCC measures the linear correlation between two variables/signals. It has a value between −1 and 1, where 1 indicates total positive linear correlation, 0 indicates no linear correlation, and −1 indicates total negative linear correlation. The formula for PCC is shown in Eq. (11). (11) $$r=\frac{{\sum }_{n=1}^N\left(x\left(n\right)-\overline{x}\right)\left(y\left(n\right)-\overline{y}\right)}{\sqrt{{\sum }_{n=1}^N{\left(x\left(n\right)-\overline{x}\right)}^2}\sqrt{{\sum }_{n=1}^N{\left(y\left(n\right)-\overline{y}\right)}^2}}.$$

  Here, $x\left(n\right)$ and $y\left(n\right)$ are the two signals or sets of values being compared, and $\overline{x}$, $\overline{y}$ are their mean values.

• VII)

  Euclidean distance: Euclidean distance is a measure of the straight-line distance between two points (or signals) in an N-dimensional space. When comparing two signals, it can be used as a measure of their dissimilarity. The formula for Euclidean distance is shown in Eq. (12). (12) $$d=\sqrt{\sum _{n=1}^N{\left(x\left(n\right)-\hat{x}\left(n\right)\right)}^2}.$$

• VIII)

  The Lyapunov exponent: The Lyapunov exponent is a metric used to assess the sensitivity of a dynamical system or time series to initial conditions, indicating whether the system exhibits chaotic behavior. It is frequently used in biomedical signal processing (e.g., EEG, ECG), chaos theory, and the study of complex systems. A positive Lyapunov exponent suggests chaotic tendencies, whereas a negative or near-zero value indicates a more stable and predictable system. For a one-dimensional map or time series $x\left(n\right)$, the Lyapunov exponent can be approximated as shown in Eq. (13). (13) $$\lambda \approx \underset{N\to \mathrm{\infty }}{lim}\frac{1}{N}\sum _{n=1}^N\ln \left|\frac{df\left(x\left(n\right)\right)}{dx\left(n\right)}\right|$$where:

  • $x(n$): The n-th sample of the time series,

  • $f(\cdot )$: The function or map defining the dynamical system,

  • $\frac{df}{dx}$: The derivative of the system, representing local expansion/contraction,

  • $\lambda$: The Lyapunov exponent, indicating the logarithmic measure of the system’s average exponential expansion/contraction rate.

• IX)

  Autocorrelation analysis: Autocorrelation analysis examines the correlation of a signal with its delayed copies, enabling the detection of periodic or repeating components, cyclical structures, and regularities in the signal. In biomedical signal processing (EEG, ECG, etc.), it is useful for identifying fundamental frequency components, periodic behaviors, and noise levels. For a time series $x(n)$, the autocorrelation function $R(\tau )$ with respect to the lag $\tau$ is defined as in Eq. (14). (14) $$R\left(\tau \right)=\sum _{n=1}^{N-\tau }x\left(n\right)x\left(n+\tau \right)$$where:

  • $x(n$): The n-th sample of the time series,

  • $\tau$: The lag (delay) value,

  • $N$: The total number of samples in the time series,

  • $\overline{x}$: The mean of the time series.

Experimental studies

The performance of the experiments was evaluated using various metrics including PSNR, SSIM, PRD, NCC, BER, r, and ED on datasets including DEAP, Graz A, and Bonn, and the impact of private information length on the results was analyzed during this process. The evaluation encompassed several critical analyses, including comparative performance analysis, embedding capacity assessment, robustness testing, and computational efficiency evaluation. All experiments were conducted on a system with an Intel Xeon E5-2630 processor (2.3 GHz) and 12 GB of RAM.

Comparative performance analysis

To rigorously evaluate the effectiveness of the proposed SWT-SVD-tent map-based steganography algorithm, experiments were performed on EEG signals from multiple subjects and channels using the Graz A, DEAP, and Bonn datasets. For consistency, each EEG signal segment was limited to a length of 1,600 samples. Private information, comprising 45 bytes of randomly generated alphanumeric characters (including uppercase, lowercase letters, and digits), was embedded into the low-frequency coefficients of the signal. After embedding, the signals were reconstructed, and the hidden information was successfully extracted to validate the accuracy and robustness of the approach.

The performance evaluation results are presented in two groups: signal quality metrics after embedding and private information extraction metrics. For signal quality after embedding:

• PSNR values exceeded 60 dB in all channels, ensuring excellent perceptual transparency.

• SSIM values were very close to 1, indicating high structural similarity between the original and processed signals.

• PRD values were consistently less than 1, reflecting minimal distortion in the reconstructed signals.

• The correlation coefficient (r) and NCC (signal) were close to 1, confirming a high similarity between the original and watermarked signals.

• ED values were small, supporting the low level of distortion.

• BER (signal) values were very low, indicating negligible bit-level errors in the watermarked signal.

Comprehensive average results for all channels and the detailed performance values for each dataset are presented in Tables 1–3.

To further validate the reliability and stability of the proposed method, statistical analyses were performed on the best-performing channels across all datasets. Table 4 presents the mean, standard deviation, and 95% confidence interval (CI) values for key performance metrics, including PSNR, PRD, r, ED, NCC, and BER.

Undetectability analysis

The undetectability analysis aims to verify whether the embedded private information can remain concealed under standard examination or analytical procedures. In this study, EEG signals sourced from the DEAP, Graz A, and Bonn datasets were enriched with 45 bytes of secret data to evaluate the method’s ability to blend seamlessly into the original signal. Figure 7 presents a side-by-side comparison of an untouched EEG signal and one embedded with private information. Across all examined datasets, visual inspection revealed no perceptible differences between the original EEG signals and the EEG signals with embedded private data.

Robustness analysis

In line with scenarios commonly explored in similar studies (Duy, Tran & Ma, 2017; Nguyen, Pham & Nguyen, 2020; Wen et al., 2024a), we evaluated the robustness of our proposed method by subjecting EEG signals containing embedded private information to various forms of interference and attacks that could occur during transmission. Specifically, five robustness scenarios were evaluated: additive noise, random cropping, low-pass filtering, salt-and-pepper noise, and signal shifting. These experiments were conducted on the Graz A, Bonn, and DEAP datasets, and the corresponding results are presented in Tables 5–7, respectively.

• Noise addition: Additive White Gaussian noise (AWGN) was introduced with different SNR levels (10, 20, and 30 dB) to simulate environmental interference. The results showed that NCC values remained consistently high (above 0.998), and BER values were minimal across all datasets, demonstrating the proposed method’s resilience to additive noise interference.

• Random cropping: To assess resilience against data loss and intentional attacks, 10% of the samples at random positions (front, middle, and back) of the EEG signals were removed. Even under these conditions, NCC values remained 1, and BER values were 0 across all datasets.

• Low-pass filtering: The EEG signals were subjected to a first-order Butterworth low-pass filter with cutoff frequencies of 20, 40, and 60 Hz to simulate different preprocessing scenarios. Despite these filtering operations, the method maintained high signal fidelity, with NCC values exceeding 0.994 and very low BER values across all datasets.

• Salt-and-pepper noise: To simulate impulsive noise that may occur during transmission, different noise densities (0.01 and 0.05) were applied to the signals. While BER values slightly increased in these challenging conditions, NCC values remained acceptable (above 0.992).

• Signal shifting: Finally, the robustness of the method against temporal shifting was tested by shifting the signals by 10 and 50 samples. The results confirmed the stability of the method, as NCC values remained above 0.995 and BER values were low.

In addition, to evaluate our method’s performance under chaotic conditions, we conducted a test using the tent map. The computed Lyapunov exponent was 0.69, indicating a strongly chaotic dynamic highly sensitive to initial conditions. Furthermore, the autocorrelation plot of the samples generated from the tent map is presented in Fig. 8. As seen from the figure, there is no distinct periodicity, consistent with the properties of chaotic processes.

The impact of private information length on the performance of the proposed method

To investigate the influence of varying the length of embedded private information on EEG signal quality, we incorporated up to 200 bytes of private data into the DEAP, Graz-A, and Bonn datasets. As illustrated in Fig. 9 using the results from the Graz-A dataset, the average PSNR, SSIM, and PRD values for both the original EEG signals and those containing embedded private information were examined under different private information lengths. The findings reveal that although the PSNR decreases with increasing embedded private information, its average values for all three datasets remain above 40 dB. Conversely, the PRD values increase with greater amounts of private information, reaching their maximum at 200 bytes, yet still remaining below 3. Moreover, as shown in Fig. 10, the BER values remained very close to zero for all message lengths, confirming the high accuracy and reliability of the extraction process.

Analysis of computational complexity

Computational complexity serves as a crucial indicator for evaluating the performance of the proposed method. In this study, the computational complexity analysis included evaluations of time complexity, space complexity, and execution times for private information embedding and extraction processes. Table 8 presents the results of this analysis. For an m × m square matrix, the time complexity of the algorithm’s SVD operation is $O\left(m^3\right)$, while its space complexity is $O\left(m^2\right)$. When employing the SWT scheme, the one-dimensional EEG signal of length n is converted into a $\sqrt{n}x\sqrt{n}$ square matrix, resulting in a time complexity of $O\left(n^{\frac{3}{2}}\right)$ and a space complexity of $O\left(n\right)$. In a specific scenario, 45 bytes of private information were embedded into a 1,600-length EEG signal. Under these conditions, the SWT scheme required an average of 0.000258 s for information hiding and 0.001356 s for information extraction. Therefore, the total execution time for embedding and extraction processes was approximately 1.97 ms, suitable for real-time EEG applications.

Rationality and superiority

To substantiate the rationality and superiority of our proposed method, we conducted comprehensive evaluations using a variety of performance metrics—including PSNR, PRD, NCC, BER, r, and ED—across three distinct datasets. We then benchmarked these results against previously reported methods. Among the comparative studies, Akgul et al. (2020) integrated LSB-based embedding with chaotic encryption to enhance the protection of EEG signals. Pham, Tran & Ma (2015) employed DWT and SVD to achieve blind watermarking, ensuring the recovery of watermarks without access to the original signals. Expanding on this approach, Duy, Tran & Ma (2017) incorporated modifications in the wavelet domain and utilized machine learning to bolster the robustness of watermark embedding.

Other notable works leveraged optimization algorithms for refined watermark embedding. For instance, Gupta, Chakraborty & Gupta (2021) and Nguyen, Pham & Nguyen (2020) aimed to minimize EEG signal degradation while enhancing data security. Similarly, Gupta, Chakraborty & Gupta (2021) proposed a method based on prediction error techniques and security blocks to improve both robustness and security. The approach by Wen et al. (2024a) utilized a two-level WPT to decompose the EEG signal into four sub-band signals, followed by SVD for embedding private data. By further incorporating the logistic map, this method improved the security and perceptual fidelity of the embedded information.

The comparative results presented in Table 9 clearly illustrate the competitive performance of our proposed method compared to existing approaches. In our method, all results were obtained under a fixed payload size of 45 bytes, which was used for embedding private information into EEG signals. As PSNR values above 40 are generally considered an indicator of good signal quality (Chen, Chang & Hwang, 1998), our method’s results demonstrate its ability to achieve reliable performance across multiple datasets. Additionally, performance evaluation metrics such as PRD, BER, r, and ED were used to provide a more comprehensive quality assessment. For the DEAP dataset, while Wen et al. (2024a) achieved the highest PSNR (79.16), our proposed method obtained a PSNR of 63.84, with a PRD of 0.22, a BER of 0.0004, and a correlation coefficient (r) of 0.999997. These values indicate minimal signal distortion and a high similarity between the original and processed signals. On the Bonn dataset, our method achieved a PSNR of 67.01, a PRD of 0.16, a BER of 0, and r of 0.999998. While Wen et al. (2024a) obtained a higher PSNR (101.87), our results remain competitive and demonstrate robustness in embedding private information. For the Graz A dataset, the proposed method achieved a PSNR of 65.48, a PRD of 0.22, a BER of 0, and r of 0.999997. These results confirm that signal fidelity was effectively maintained. While some previous methods, such as Wen et al.’s (2024a), achieved higher PSNR values, our proposed method demonstrates a consistent balance of signal quality, robustness, and minimal embedding error. Its performance across multiple datasets confirms its competitiveness and reliability as a viable alternative for embedding private information into EEG signals while ensuring minimal signal degradation.

Furthermore, to empirically verify the effectiveness of the tent map used in the proposed method, additional comparative experiments were performed using several well-known chaotic maps (e.g., Logistic, Sine, Gauss, Henon, Fractional Lorenz, Coupled Map Lattice, Cubic, and Bernoulli). These experiments were conducted under the same embedding framework and experimental conditions across all datasets. The results revealed that the tent map consistently outperformed other chaotic maps in terms of PSNR and SSIM metrics. Especially in the Graz A dataset, while the PSNR values obtained with alternative chaotic maps ranged from approximately 20 to 33 dB, the proposed method achieved a PSNR of 65.48 dB. These findings support the effectiveness of the tent map for secure and robust information hiding in EEG signals.

Robustness analysis

To assess the robustness of the proposed method, a comparative robustness analysis was performed against the approaches presented by Duy, Tran & Ma (2017), Nguyen, Pham & Nguyen (2020), and Wen et al. (2024a). All methods were subjected to identical test procedures, including noise addition, random cropping, and low-pass filtering. The results of these evaluations are summarized in Table 10. On the DEAP dataset, the proposed method outperformed the methods of Duy, Tran & Ma (2017), Nguyen, Pham & Nguyen (2020), and Wen et al. (2024a) in both the noise addition and random cropping tests, consistently achieving the lowest BER values. However, in the low-pass filtering test, while the proposed method’s performance remained competitive, Wen et al.’s (2024a) approach achieved slightly better results. On the Graz A dataset, the proposed algorithm delivered the best results across all three robustness tests, demonstrating superior adaptability and resilience under challenging conditions. For the Bonn dataset, a trend similar to the DEAP dataset emerged. The proposed method excelled in noise addition and random cropping tests, outperforming the competing approaches. Although Wen et al.’s (2024a) method slightly surpassed our method in the low-pass filtering test, the proposed algorithm remained highly competitive. In conclusion, the proposed method exhibits strong robustness across multiple datasets and challenging test scenarios. While Wen et al.’s (2024a) approach achieved marginally better results under low-pass filtering conditions, the proposed method consistently excelled in noise addition and random cropping tests, confirming its capacity to maintain stable and reliable performance under various robustness challenges.

The superior robustness exhibited by the proposed SWT-SVD-tent map method primarily stems from three key mathematical characteristics, each contributing significantly to its resistance against common signal processing attacks. Firstly, SWT preserves signal length and ensures shift-invariance, making the embedding resilient to typical signal-processing attacks such as random cropping and shifting. Secondly, the SVD-based embedding into off-diagonal singular values reduces distortion of dominant signal features, maintaining high structural integrity under conditions like noise addition and filtering. Finally, the tent map’s chaotic encryption exhibits strong sensitivity to initial conditions and produces highly uniform chaotic sequences, significantly complicating attempts at unauthorized extraction or tampering. The combination of these features results in a mathematically robust embedding scheme capable of superior performance compared to alternative methods.

Comparative analysis of computational complexity

Time and space complexities constitute crucial metrics for evaluating the computational efficiency and practicality of steganographic algorithms. To assess the complexity of the proposed method, a comparative analysis was performed against the approaches presented by Akgul et al. (2020), Pham, Tran & Ma (2015), Gupta, Chakraborty & Gupta (2021), Nguyen, Pham & Nguyen (2020), and Wen et al. (2024a). Since explicit time and space complexity measures were not provided in these prior studies, approximate complexity estimates were inferred based on their described methodologies. The summarized results are given in Table 11. Gupta, Chakraborty & Gupta (2021) first applied the short-time Fourier transform (STFT) to the EEG signal, converting it into a two-dimensional representation, and then performed a two-dimensional discrete cosine transform (DCT) for information hiding. Their method exhibited a time complexity of $O\left(n^{2}logn\right)$ and space complexity of $O\left(n^2\right)$. By contrast, Akgul et al. (2020) utilized a combination of least significant bit (LSB) embedding and chaotic mapping. Since these operations are essentially linear, their algorithm achieved time and space complexities of $O\left(n\right)$. The methods of Nguyen, Pham & Nguyen (2020), Pham, Tran & Ma (2015), and the proposed approach all employed SVD in their embedding processes. Pham, Tran & Ma (2015) converted the one-dimensional EEG signal into an $nxn$ image, resulting in a time complexity of $O\left(n^3\right)$ and a space complexity of $O\left(n^2\right)$. In contrast, both Nguyen, Pham & Nguyen (2020) and the proposed method transformed the EEG signal into a $\sqrt{n}x\sqrt{n}$ matrix, which reduced computational demands. Under these conditions, both approaches achieved a time complexity of $O\left(n^{\frac{3}{2}}\right)$ and a space complexity of $O\left(n\right)$. Wen et al. (2024a) also employed a Wavelet Packet Transform (WPT) followed by SVD on a $\sqrt{n}x\sqrt{n}$ matrix, resulting in similar computational complexities of $O\left(n^{\frac{3}{2}}\right)$ for time and $O\left(n\right)$ for space. In summary, the proposed method exhibits computational complexities comparable to those of Nguyen, Pham & Nguyen (2020), and Wen et al. (2024a), which are relatively lower than the complexities associated with converting the signal into an $nxn$ image, as seen in Pham, Tran & Ma (2015). Consequently, the proposed method remains a highly competitive and computationally efficient approach for embedding private information into EEG signals, offering advantages in practical real-time scenarios.

To further analyze the efficiency of the proposed method, a comparison was conducted against (Wen et al., 2024a), examining time complexity, space complexity, and actual execution times—including hiding, extraction, and total processing durations. The summarized results are presented in Table 12. Both the proposed method and Wen et al.’s (2024a) approach exhibit comparable time complexity $O\left(n^{\frac{3}{2}}\right)$ and space complexity $O\left(n\right)$, indicating a similar level of computational efficiency from a theoretical standpoint. However, their execution times differ notably. Specifically, the proposed method achieves a hiding time of 0.000258 s, which is considerably faster than Wen et al.’s (2024a) 0.001090 s. Moreover, the extraction time for the proposed method is 0.001356 s, outperforming Wen et al.’s (2024a) extraction time of 0.001710 s. Consequently, the total execution time of the proposed method is 0.001968 s, significantly lower than the 0.002800 s reported by Wen et al. (2024a). Since execution time data from other comparable studies are not available, direct comparisons were limited to Wen et al.’s (2024a) method. In conclusion, while offering similar theoretical time and space complexities, the proposed method demonstrates a distinct advantage in execution speed, making it highly suitable for real-time applications where rapid performance is essential.

Practical feasibility considerations

While the primary objective of this research was to design and validate an effective EEG signal steganography algorithm, practical feasibility considerations were also thoroughly examined. The proposed method was evaluated on a system equipped with an Intel Xeon E5-2630 processor and 12 GB RAM, demonstrating low computational cost and fast execution times suitable for real-time applications. Moreover, the storage requirement of the proposed method remains minimal since only private information of limited size (up to 200 bytes) is embedded per EEG segment. The method operates on a segment-by-segment basis without altering the overall signal length, ensuring compatibility with storage and transmission systems used in telemedicine platforms. From a power consumption viewpoint, the use of lightweight encryption (tent map) combined with efficient signal processing methods (SWT and SVD) ensures compatibility with low-power devices typically employed in wearable or portable EEG monitoring systems.