## 5 Conclusions  
This research introduces a mathematical approach to improve the security and integrity of Split Learning by using Homomorphic Encryption and Zero-Knowledge Proofs. HE allows calculations to be performed on encrypted data, ensuring that data privacy is maintained without needing to decrypt it. This mathematical handling of encrypted data, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Experiment** & **Training Time (s)** & **Communication Overhead (\%)** & **Cryptographic Overhead (\%)** \\ \hline Standard Split Learning & 340 & 15 & 0 \\ \hline HavenSL (HE + ZKP) & 390 & 18 & 7 \\ \hline HavenSL (HE + ZKP + DCT) & 420 & 20 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Experiment results. using the CKKS and BFV schemes, which both support arithmetic operations on encrypted data, is a major step forward for security in decentralized systems. ZKP provides a mathematical way to verify the correctness of computations without exposing the underlying data. By using these proofs, we ensure that each part of the neural network trained by clients is verified and free from unauthorized changes. The mathematical formula for verifying computation accuracy guarantees that the model segment trained by the client up to a certain point is free from malicious modifications. This study validates the effectiveness of Homomorphic Encryption and Zero-Knowledge Proofs in strengthening the security and integrity of Split Learning. Our results confirm Hypothesis 1, as Homomorphic Encryption enabled secure arithmetic operations on encrypted data, preserving data confidentiality without the need for decryption and maintaining high model accuracy. This mathematical handling of encrypted data through HE was effective across various datasets, demonstrating both feasibility and accuracy in real-world applications. Hypothesis 2 was supported by the successful implementation of Zero-Knowledge Proofs in the SafeSplit framework. The integration of ZKP provided a robust cryptographic mechanism for mathematically verifying the correctness of client computations without disclosing sensitive data, enabling the detection and correction of unauthorized changes in model parameters, such as backdoor attacks. This mechanism did not impact model performance, confirming the utility of ZKP for securing collaborative learning environments. These findings affirm the hypotheses, underscoring the potential of HE and ZKP as mathematically grounded approaches to enhancing data privacy and model integrity in decentralized learning frameworks. The SafeSplit framework uses Discrete Cosine Transform (DCT) analysis to track changes in the frequency domain of the model's parameters. This helps identify unusual changes that could suggest a backdoor attack. By using a rotational distance metric to measure the angle between model updates, this framework provides real-time detection of suspicious activities, adding an extra layer of security. The integration of mathematical methods like Homomorphic Encryption and Zero-Knowledge Proofs in this Split Learning framework is a significant step forward in protecting data and maintaining model integrity. The combination of static and dynamic analysis in this dual-layer defense system effectively detects backdoor attacks and restores the model to a previously verified state. This mathematical work aims to increase the trustworthiness of decentralized learning by expanding the use of cryptography to protect data privacy and model integrity in collaborative AI situations. Conceptualization, A.K. and E.M.; methodology, A.K. ; software, A.K. ; validation, A.K. and E.M.; formal analysis, A.K. ; investigation, E.M.; resources, A.K. ; data curation, E.M., writing--review and editing, A.K. ; visualization, A.K. ; supervision, E.M.; project administration, E.M.; funding acquisition, A.K. and E.M. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The original contributions presented in this study are included in the article.