A novel cryptographic framework and mathematical modeling for secure transmission of Parkinson’s disease data using RSA and block-based secret sharing

Abstract

The secure transmission of medical data is an essential requirement in modern telemedicine systems, particularly for chronic neurological disorders such as Parkinson’s disease. This paper proposes a novel hybrid cryptographic framework that combines RSA encryption with block-based secret sharing enhanced by a Hilbert matrix-driven mathematical model. The framework introduces dynamic block-wise key generation and adaptive sharing to strengthen data confidentiality and robustness against cryptanalytic attacks. Mathematical modeling is employed to analyze encryption stability, numerical conditioning of the Hilbert matrix, and the diffusion properties of the key space. The proposed method is validated using publicly available Parkinson’s EEG and spiral drawing datasets, with quantitative analysis including encryption/decryption time, computational overhead, and image quality metrics (PSNR, SSIM). The framework is further benchmarked against AES-Shamir and ECC-based hybrid models. Experimental results indicate that the proposed system achieves higher security entropy and lower computational cost, making it suitable for deployment in resource-constrained medical IoT environments.

Introduction

Parkinson’s Disease (PD) is a progressive neurodegenerative disorder that primarily affects the motor system and is characterized by tremors, rigidity, bradykinesia, and postural instability as shown in Fig. 1. It results from the gradual degeneration of dopaminergic neurons in the substantia nigra pars compacta, leading to dopamine deficiency in the basal ganglia. Beyond motor symptoms, patients often exhibit non-motor manifestations such as cognitive decline, depression, and sleep disturbances, which significantly affect quality of life¹. According to recent epidemiological studies, PD is the second most common neurodegenerative disorder after Alzheimer’s disease, affecting over 10 million people worldwide². The early and accurate diagnosis of PD remains a major clinical challenge, as initial symptoms often overlap with other movement disorders. Recent advances in biomedical signal processing and artificial intelligence have enabled the analysis of physiological data such as EEG, EMG, and spiral or wave drawings to identify PD-specific patterns^3,4. The secure transmission and processing of such sensitive medical data are crucial for the development of reliable telemedicine and Internet-of-Medical-Things (IoMT) applications. Hence, designing cryptographically secure and computationally efficient frameworks for handling PD-related data has become an essential research direction in intelligent healthcare systems.

Fig. 1.

Illustration of Parkinson’s Disease.

The mathematical modeling of cryptographic systems has attracted sustained attention due to its role in analyzing stability, convergence, and resistance properties of secure communication algorithms. In particular, hybrid encryption models where algebraic transformations are combined with public-key frameworks offer a fertile ground for establishing provable guarantees on confidentiality and robustness. Among such hybrids, the integration of RSA cryptosystems with matrix-based secret sharing presents a compelling balance between theoretical tractability and computational efficiency.

Recent advances have explored combining cryptography with data-concealment and transform-domain operations to enhance security and imperceptibility. Shamsi et al.⁵ developed a visually meaningful cipher data concealment method using lifting wavelet transforms and an e- hyperchaotic map. Similarly, Shamsi and Singh⁶ proposed hiding encrypted images within audio data using the Ikeda map for improved robustness. In another work, Shamsi and Singh⁷ integrated cryptography, steganography, and QR-code encoding to strengthen covert communication. These hybrid frameworks demonstrate the growing trend toward multi-domain cryptographic modeling for secure multimedia transmission.

With the rapid growth of telemedicine and Internet-of-Medical-Things (IoMT) devices, patient-specific biomedical data must be transmitted under strict confidentiality constraints. Traditional symmetric encryption schemes (e.g., AES, DES) provide efficiency but lack a mathematically transparent security foundation when extended to block-structured medical signals. Conversely, RSA ensures theoretical security but becomes computationally expensive for high-dimensional data streams. This motivates the development of mathematically grounded hybrid models that retain analytical clarity while improving performance.

Recent studies have explored matrix-driven encryption approaches in which the Hilbert matrix acts as a secret transformation kernel because of its invertibility and algebraic regularity. However, Hilbert matrices are known to be ill-conditioned, making direct numerical implementation prone to round-off errors and decryption instability. Furthermore, earlier RSA–Hilbert frameworks were validated only on small numerical examples, lacking both large-scale experimentation and quantitative analysis of security metrics such as entropy, avalanche behavior, and computational complexity. To overcome these limitations, this study introduces a novel cryptographic framework that unifies RSA encryption with block-based secret sharing governed by a Hilbert-matrix mathematical model. The Hilbert structure is employed not merely as a numerical tool but as a mathematically tunable component capable of generating block-adaptive secret keys. The framework is further analyzed through a stability-oriented mathematical formulation that quantifies the sensitivity of decryption accuracy to matrix conditioning and key perturbations.

The principal contributions of this work are summarized as follows:

• Formulation of a hybrid RSA–Hilbert block secret-sharing algorithm under a rigorous mathematical model that quantifies key-space diffusion and decryption stability.

• Derivation of analytical conditions ensuring numerical robustness of Hilbert-based encryption using rational or high-precision arithmetic.

• Theoretical and empirical evaluation of the framework through entropy, avalanche, and complexity metrics, establishing formal resistance to chosen-plaintext and replay attacks.

• Validation of the proposed model on Parkinson’s disease datasets (EEG and spiral drawing) to demonstrate its applicability to biomedical signal encryption.

The remainder of the paper is organized as follows. The section “Related work” reviews related mathematical and cryptographic works. The section “Proposed methodology and mathematical modeling” presents the proposed hybrid model and detailed mathematical derivations. The section “Security analysis” develops the formal security analysis and proof sketch. The section “Experimental results and discussion” reports the experimental evaluation and discusses performance results. The section “Conclusion” concludes the study with remarks on future theoretical extensions.

Related work

The design of secure cryptographic mechanisms grounded in mathematical modeling has evolved significantly in recent years. Early studies on public-key systems, particularly the RSA algorithm, established the theoretical foundation of number-theoretic encryption by exploiting the computational difficulty of prime factorization ⁸. However, classical RSA encryption faces limitations when applied to large-scale biomedical data, motivating research into hybrid frameworks that combine algebraic or matrix-based transformations to improve performance ^9,10. Matrix-based cryptography, especially those involving structured matrices such as Hilbert, Vandermonde, or Toeplitz forms, has attracted growing interest due to its deterministic invertibility and compact representation ¹¹. The Hilbert matrix, in particular, exhibits a unique analytic form defined by , making it suitable for key embedding and secret sharing. Several works have explored Hilbert-based cryptosystems integrated with RSA for data hiding and key scheduling ¹². Nevertheless, these approaches often neglect the matrix’s numerical instability, which leads to decryption errors under floating-point arithmetic and limits scalability for higher matrix orders ¹³.

In recent years, hybrid cryptographic methods combining RSA, AES, and secret-sharing schemes have been proposed to improve confidentiality in healthcare communication ^14,15. For instance, combinations of AES with Shamir’s Secret Sharing or ECC-based encryption have demonstrated improved key management but remain computationally intensive for low-power medical IoT environments. In contrast, lightweight Hilbert or algebraic matrix-based systems provide theoretical elegance but lack practical performance validation ¹⁶. From the perspective of biomedical security, encryption models have also been evaluated for specific disease datasets such as Parkinson’s disease, where medical signals (e.g., EEG and handwriting data) contain sensitive personal information ¹⁷. The need for mathematically verifiable encryption methods in such contexts has been widely recognized, yet most implementations remain heuristic rather than analytically grounded. To situate this research within the current literature, Table 1 summarizes key features of selected hybrid cryptographic frameworks. Notably, works such as ^18–21 demonstrate ongoing interest in integrating mathematical formulations with practical IoT encryption, but none explicitly combine RSA with Hilbert matrix-based block secret sharing under a rigorous stability analysis framework. This gap forms the core motivation of the present study. Recent studies have continued to enhance RSA-based encryption mechanisms for improving data security and computational efficiency in real-world systems. Ghrib et al.²² developed an optimized RSA framework that strengthens key management and parameter selection to enhance protection against cryptanalytic attacks while reducing execution latency across application-level deployments. Their approach demonstrates practical improvements in encryption throughput but does not explore hybrid or matrix-assisted schemes. Similarly, Qiang²³ investigated resource allocation and performance optimization strategies for RSA encryption in distributed network environments, proposing adaptive computation scheduling to minimize latency and optimize CPU and memory usage. Although both works contribute to improving RSA efficiency, they remain confined to conventional modular arithmetic operations. In contrast, the proposed RSA–Hilbert block secret-sharing framework introduces a mathematical transformation layer based on Hilbert matrices that enhances both security diffusion and numerical stability, supported by a bounded-error reconstruction theorem and validated through experiments on Parkinson’s disease data.

Table 1.

Summary of hybrid cryptographic frameworks.

References	Methodology	Application domain	Remarks
16	AES + ECC hybrid model	IoT data transmission	Improved key reuse efficiency
17	RSA + Shamir secret sharing	Cloud security	Limited scalability for large blocks
18	AES + chaotic map encryption	Medical image protection	High diffusion but lacks mathematical proof
19	RSA + Hilbert matrix encryption	Biomedical signal privacy	No stability or conditioning analysis

The proposed study extends the mathematical foundation of hybrid cryptography by introducing a Hilbert-based block secret-sharing layer within the RSA structure. Unlike earlier schemes that treated Hilbert matrices as numerical tools, this work constructs an explicit mathematical model that links encryption stability to matrix conditioning, enabling theoretical guarantees of robustness and key sensitivity.

Proposed methodology and mathematical modeling

This section presents the mathematical formulation of the proposed cryptographic framework that integrates RSA encryption with block-based secret sharing using Hilbert matrix transformations. The model is designed to ensure secure and numerically stable transmission of sensitive medical data, such as Parkinson’s EEG or spiral drawing signals, within telemedicine and IoMT systems.

Overview of the framework

The hybrid framework consists of three major phases:

1. Key generation: Public and private keys are generated using the RSA algorithm. Additionally, a Hilbert matrix of adjustable order is constructed to serve as a secondary key transformation layer.

2. Block-based encryption: The medical data (image or signal) is divided into equal-sized blocks. Each block is encrypted using an RSA exponentiation process and subsequently transformed using the Hilbert matrix to generate a block-adaptive secret.

3. Decryption and reconstruction: The receiver performs inverse Hilbert transformation followed by RSA decryption to reconstruct the original data with minimal numerical loss as shown in Fig. 2.

Fig. 2.

Overview of proposed methodology.

This layered architecture ensures both mathematical security (through RSA’s modular arithmetic) and structural diffusion (through Hilbert-based transformation), producing high key sensitivity and resistance against differential attacks.

RSA encryption model

The RSA algorithm operates on the number-theoretic principle of modular exponentiation, defined as follows:

3.1

3.2

where M denotes the plaintext block, C the ciphertext, e and d are the public and private exponents, and is the modulus computed from two large primes p and q. The security of RSA is ensured by the infeasibility of integer factorization of n. However, for biomedical datasets containing thousands of image or EEG samples, direct RSA computation on large data vectors is computationally expensive. Therefore, a block-partitioning strategy is introduced to parallelize encryption and integrate a second mathematical layer using the Hilbert matrix.

Hilbert matrix transformation

Let be an Hilbert matrix defined as:

3.3

The Hilbert matrix possesses closed-form inverses and deterministic eigenvalue properties that make it attractive for algebraic key embedding. However, it is ill-conditioned, implying that numerical errors can amplify during inversion. To overcome this, the proposed model employs:

• High-precision arithmetic: Rational number computation or 128-bit floating precision via mpmath.

• Normalization: Each block is scaled to [0, 1] before transformation.

• Adaptive block order: The order n of is varied dynamically according to the key size.

The encryption transformation is then expressed as:

3.4

where denotes the plaintext block and is the corresponding encrypted matrix.

Decryption is performed by:

3.5

The condition number is used to measure numerical stability:

3.6

and the decryption error is quantified by:

3.7

A smaller indicates better numerical stability and reliable reconstruction.

Block-based secret sharing mechanism

To strengthen resistance against block replacement or replay attacks, a secret-sharing process is introduced after Hilbert transformation. Each ciphertext block is divided into k shares using modular splitting:

3.8

where is a random nonce derived from the hash of the corresponding block index and key.

At the receiver’s side, reconstruction is achieved through:

3.9

This mechanism ensures that the ciphertext can only be recovered when all k shares are present, fulfilling the (k, k) threshold property and enhancing confidentiality against partial exposure.

Algorithmic representation

Algorithm 1.

Proposed RSA–Hilbert Block Secret Sharing Encryption

Mathematical stability modeling

To ensure decryption reliability, the following stability criterion is defined.

Definition 3.1

(Stability Criterion) The RSA–Hilbert encryption model is said to be numerically stable if there exists a constant such that

3.10

Empirically, guarantees accurate recovery.

Theorem 3.1

(Bounded Error Reconstruction) Let denote the Hilbert matrix of order n, and let be its condition number in a consistent norm. During the decryption process of the RSA–Hilbert block secret sharing framework, each block is reconstructed via

where represents the cumulative floating–point perturbation. If computations are performed with machine precision and , then the forward reconstruction error satisfies

3.11

where C is a moderate constant depending on the numerical stability of the inversion routine. Consequently, when remains moderate and high–precision arithmetic is employed,

Proof

The result follows from classical forward–error bounds in numerical linear algebra. Let the computed inverse satisfy , with . Then, using the perturbed decryption equation , we have

Hence , as claimed.

Remark 3.1

For normalized data (), the reconstruction error reduces to . This indicates that decryption accuracy depends jointly on the matrix conditioning and arithmetic precision. Since increases rapidly with n, selecting a bounded matrix order and employing extended–precision arithmetic ensures numerical stability in the proposed RSA–Hilbert framework.

Computational complexity

The computational complexity of the model is given by:

3.12

where arises from Hilbert matrix operations and corresponds to modular exponentiation in RSA. In practice, parallelization over B blocks reduces real execution time, making the model suitable for distributed medical IoT systems.

Security analysis

This section provides the theoretical analysis of the proposed RSA–Hilbert block secret-sharing framework. Security is evaluated through entropy analysis, key-space estimation, and resistance to classical and modern attack models. We also present a proof sketch showing that decryption without both RSA private key and Hilbert transformation parameters is computationally infeasible.

Entropy and key-space analysis

Let K denote the complete secret key consisting of the RSA modulus n, the private exponent d, and the Hilbert matrix order and elements . The total key-space size can be estimated as

4.1

where represents the effective RSA key length (typically 2048 or 4096 bits) and is the Hilbert-matrix bit representation. Because each contains rational entries, the Hilbert component contributes approximately

4.2

where p denotes the precision level (e.g., 64 or 128 bits). Hence, for and , the hybrid key space exceeds , which is computationally intractable to brute-force.

The Shannon entropy of the key distribution is computed as

4.3

where denotes the probability of selecting the key element. Empirically, indicates maximal randomness when all key bits are equiprobable, implying that the proposed system attains near-uniform entropy.

Resistance to attack models

The model’s defense mechanisms against several classical and modern cryptographic attacks are summarized in Table 2. Two layers of security operate simultaneously: (i) RSA modular arithmetic for number-theoretic hardness and (ii) Hilbert block transformation for structural diffusion.

Table 2.

Attack-resistance evaluation of the proposed RSA–Hilbert model.

Attack Type	Vulnerability in Classical RSA	Resistance in Proposed Framework
Chosen-Plaintext (CPA)	Possible through multiple plaintext–ciphertext pairs	Each block encrypted with dynamic Hilbert key; random nonce prevents repetition.
Chosen-Ciphertext (CCA)	Decryption oracle may reveal partial plaintext	Secret-sharing layer prevents oracle reconstruction; ciphertext must be recombined before decryption.
Replay Attack	Identical ciphertext retransmission undetected	Session-based random in Eq. (3.8) ensures uniqueness of every encryption.
Side-Channel / Timing Attack	Correlation between timing and key bits	Random padding and uniform modular exponentiation time reduce leakage.
Brute-Force Attack	Complexity limited to RSA key size	Combined RSA and Hilbert key space () makes exhaustive search infeasible.
Statistical / Differential Attack	Ciphertext correlations may leak patterns	Hilbert diffusion creates high avalanche effect; adjacent pixel correlation .

Proof of security

Theorem 4.1

(Indistinguishability under Adaptive Chosen-Plaintext Attack) Let be a probabilistic polynomial-time adversary attempting to distinguish between two ciphertexts and generated by the proposed RSA–Hilbert model. Then the advantage of is negligible in , i.e.,

4.4

Proof

Each ciphertext block is computed as . Recovering from requires both (i) inversion of RSA exponentiation and (ii) inversion of . Without access to the private key d, solving reduces to the RSA inversion problem, which is provably as hard as integer factorization. Even if RSA were compromised, the attacker must still reconstruct with precision ; any error in propagates exponentially in n, making recovery statistically negligible. Therefore, ’s success probability is bounded by , establishing indistinguishability under adaptive chosen-plaintext attack.

Avalanche and correlation analysis

The avalanche criterion measures sensitivity of ciphertext to small changes in plaintext. For a one-bit change in , denote the number of flipped bits in ciphertext as f. The avalanche ratio is computed as

4.5

where L is the ciphertext length. Experimental results show between and , satisfying the ideal 50% avalanche property.

Pixel-wise correlation between adjacent elements before and after encryption is defined as

4.6

where x and y denote adjacent pixel intensities in plaintext and ciphertext images. For the proposed scheme, , indicating strong decorrelation and high diffusion capability.

Summary of security strength

The hybrid RSA–Hilbert framework achieves:

• Extremely large key-space () and near-uniform entropy.

• Resistance to CPA, CCA, replay, and side-channel attacks.

• High avalanche ratio and minimal plaintext–ciphertext correlation.

• Dual-layer mathematical hardness–integer factorization and ill-conditioned matrix inversion.

These theoretical guarantees, supported by experimental validation in the section “Experimental results and discussion”, demonstrate that the proposed model satisfies the desired criteria of confidentiality, unpredictability, and numerical robustness required for secure biomedical data transmission.

Experimental results and discussion

This section presents the experimental validation of the proposed RSA–Hilbert block secret-sharing framework. Experiments were conducted to evaluate numerical stability, encryption efficiency, and security robustness compared with existing hybrid encryption models.

Dataset description

Two publicly available Parkinson’s disease datasets were used for performance evaluation:

• EEG Dataset: Obtained from the PhysioNet repository, containing multi-channel EEG signals recorded from Parkinson’s and healthy subjects. Each segment was sampled at 512 Hz, converted into numerical matrices, and normalized to [0, 1] before encryption.

• Spiral Drawing Dataset: Collected from the Kaggle Parkinson’s Drawing dataset, consisting of scanned spiral images (size ). Images were converted to grayscale and divided into non-overlapping blocks prior to encryption.

Both datasets are open-access and used in accordance with their licensing policies, ensuring no human data collection or ethical concerns.

Experimental setup

All simulations were implemented in Python 3.10 using NumPy, mpmath, and OpenCV libraries. Experiments were performed on a workstation equipped with:

• Intel Core i7 CPU @ 3.2 GHz,

• 32 GB RAM,

• Ubuntu 22.04 LTS, and

• 128-bit precision arithmetic.

The model parameters were selected as:

For performance comparison, three baseline models were included:

1. AES–Shamir hybrid model,

2. ECC–RSA model, and

3. Classical RSA encryption.

Performance metrics

The following quantitative measures were used:

• Encryption Time (ET) and Decryption Time (DT): measured in milliseconds for each block.

• Peak Signal-to-Noise Ratio (PSNR):

  5.1

  where is the mean squared error between the original and decrypted image.

• Structural Similarity Index (SSIM): evaluates perceptual quality between input and reconstructed images.

• CPU Usage (%): indicates computational overhead.

• Decryption Error (): as defined in Eq. (3.7).

Quantitative results

Table 3 presents the comparative results of the proposed and baseline models.

Table 3.

Performance comparison of encryption models on Parkinson’s datasets.

Method	ET (ms)	DT (ms)	PSNR (dB)	SSIM	CPU (%)
AES + Shamir	12.3	11.8	45.6	0.94	35
ECC–RSA	14.1	13.9	46.1	0.95	41
Classical RSA	18.7	17.2	42.9	0.91	39
Proposed RSA–Hilbert	10.5	10.2	47.9	0.96	33

The results demonstrate that the proposed model achieves the lowest encryption/decryption time while maintaining superior image fidelity (PSNR and SSIM). The improvement is attributed to adaptive block-based processing and optimized matrix inversion using rational arithmetic.

Numerical stability evaluation

To evaluate the stability of Hilbert matrix operations, the decryption error and condition number were measured for varying Hilbert orders as shown in Table 4.

Table 4.

Numerical stability evaluation of Hilbert transformation.

Hilbert Order (n)	Condition Number	Error	Stability Indicator ()
3			Stable
5			Stable
8			Stable
10			Marginally Stable

Even though grows rapidly with n, the controlled arithmetic precision (128-bit) ensures that for , confirming theoretical stability as predicted by Theorem 1.

Scalability and efficiency analysis

Figure 3 illustrates the trend of encryption time with respect to block size and Hilbert order. The proposed approach scales linearly with data volume due to independent block operations, which can be parallelized for distributed environments.

Fig. 3.

Scalability analysis of the RSA–Hilbert block secret-sharing encryption. Encryption time increases linearly with block size, confirming the parallelizable design. Higher Hilbert orders show moderately higher computation cost due to matrix inversion complexity.

Discussion

Theorem 3.1 formalizes the stability of the decryption stage under finite-precision arithmetic. It adapts the classical forward-error bound from numerical linear algebra to the context of Hilbert-based encryption, showing that the reconstruction error is linearly proportional to both the condition number of the Hilbert matrix and the machine precision.

While the inequality is well known for generic linear systems^20,21, its application to a cryptographic RSA–Hilbert pipeline is new. This analysis quantifies the trade-off between numerical stability and matrix order, ensuring that for moderate Hilbert orders () and high-precision arithmetic, the reconstruction error remains negligible (). The experimental outcomes validate the theoretical claims presented in the sections “Proposed methodology and mathematical modeling” and “Security analysis”. Key observations are summarized below:

• The hybrid RSA–Hilbert framework achieved a PSNR improvement of approximately 2.3 dB and reduced computation time by nearly 25% compared to AES–Shamir.

• The average entropy of encrypted data reached 7.998 bits/pixel, demonstrating excellent randomness and diffusion properties.

• The numerical stability analysis confirmed that high-precision computation mitigates Hilbert conditioning issues, ensuring .

• Experiments with EEG matrices confirmed zero decryption failure across all test samples, validating the theoretical bound .

These findings affirm that the proposed cryptographic model is both mathematically sound and computationally efficient for secure transmission of Parkinson’s disease data.

Limitations, and future work

Limitations

While the proposed RSA–Hilbert block secret-sharing model exhibits strong theoretical security and empirical stability, several limitations must be acknowledged:

• Dataset scale: The validation employed moderate-sized EEG and image datasets. Large-scale multi-modal medical datasets (e.g., MRI or continuous sensor streams) may require further optimization of block partitioning and memory management.

• Numerical conditioning: Although high-precision arithmetic mitigates Hilbert ill-conditioning, the matrix order must remain below 10 to avoid excessive computational cost.

• Hardware dependency: The current implementation assumes sufficient CPU precision and memory. Embedded IoT devices with limited hardware resources may need lightweight approximations of the Hilbert inverse.

• Security proof generalization: The present work provides an analytical proof sketch for CPA resistance; a formal reduction-based proof under the RSA assumption could further strengthen mathematical rigor.

Future work

Future research will aim to extend the mathematical and computational aspects of this study in several directions:

1. Quantum-resistant extension: Integrate lattice-based or post-quantum key exchange mechanisms with the Hilbert secret-sharing layer to enhance long-term resilience.

2. Dynamic Hilbert modeling: Develop adaptive Hilbert kernels whose order evolves with data entropy, allowing self-adjusting stability during streaming encryption.

3. Parallel and GPU implementation: Employ CUDA and OpenCL to accelerate large-block matrix operations for real-time medical data encryption.

4. Formal verification: Apply symbolic and probabilistic model-checking tools to establish provable guarantees for confidentiality and correctness.

5. Clinical integration: Combine the encryption module with edge-based diagnostic systems for Parkinson’s disease to ensure end-to-end secure analytics.

The authors believe that these extensions will bridge the theoretical and practical domains of mathematical cryptography and contribute to the development of reliable, privacy-preserving telemedicine infrastructures.

Conclusion

This paper introduced a novel cryptographic framework and mathematical model for the secure transmission of Parkinson’s disease data using a hybrid RSA–Hilbert block secret-sharing mechanism. The proposed system unifies number-theoretic encryption with matrix-based algebraic modeling to achieve dual-layer protection: modular arithmetic ensures theoretical confidentiality, while Hilbert-matrix transformation introduces adaptive diffusion and numerical robustness. A comprehensive mathematical formulation was developed, including the derivation of stability bounds, condition-number analysis, and a proof sketch of resistance to chosen-plaintext attacks. Experimental results on publicly available Parkinson’s EEG and spiral drawing datasets confirmed the practical validity of the theoretical model. Quantitatively, the proposed RSA–Hilbert framework achieved an average encryption time reduction of approximately and a PSNR improvement of 2.3 dB compared with conventional AES–Shamir and ECC-based hybrid models. High entropy (7.998 bits/pixel) and a near-ideal avalanche ratio () demonstrated strong diffusion and unpredictability characteristics. The results further verify that the mathematical stability of the Hilbert transformation–when combined with high-precision computation–preserves decryption accuracy with negligible numerical error (). Moreover, the security analysis established the framework’s resistance to brute-force, replay, and side-channel attacks through its exponentially large key-space () and adaptive secret-sharing scheme. In conclusion, the RSA–Hilbert block secret-sharing model offers a mathematically rigorous and computationally efficient solution for secure biomedical data transmission, effectively bridging the gap between theoretical cryptography and real-world telemedicine applications. Future work will extend the proposed approach toward quantum-resistant and GPU-accelerated implementations, enabling large-scale, real-time protection of medical IoT data streams

Author contributions

Thalapathiraj Sambandham: Writing - original draft, Validation, Software. Hari Mohan Srivastava: Conceptualization, Methodology, Data curation, Writing - review & editing. Stalin Thangamani: Data curation, Conceptualization, Supervision, Methodology, Writing - review & editing. Majeed Ahmad Yousif: Software, Validation, Visualization, Writing - review & editing. Abdelhamid Mohammed Djaouti: Methodology, Validation, Funding acquisition, Writing - original draft. Pshtiwan Othman Mohammed: Project administration, Writing - original draft. The authors unanimously concur on the publication of this research work.

Funding

This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (Project No. KFU254508).

Data availability

The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

Ethics approval and consent to participate

All methods were carried out in accordance with relevant guidelines and regulations.