## 2 Materials and Methods  
Before diving into the specifics of the methodologies employed in this study, it is important to outline the core objectives that drive this research. This work focuses on enhancing the security and privacy of Split Learning through the integration of advanced cryptographic techniques, including Homomorphic Encryption and Zero-Knowledge Proofs. The SafeSplit framework is proposed as a defense mechanism against client-side backdoor attacks, which are a significant threat in distributed learning systems. The methods detailed in this section describe the cryptographic tools, datasets, model architectures, and experimental protocols used to evaluate the effectiveness of the proposed solutions. All materials, data, and protocols have been made accessible to ensure replicability and transparency. The cryptographic techniques used in the Split Learning framework ensure privacy, security, and integrity of the data. We primarily focus on Homomorphic Encryption and Zero-Knowledge Proofs to protect the confidentiality of client data and verify the correctness of computations. We employed the CKKS (Cheon-Kim-Kim-Song) and BFV encryption schemes, both of which support arithmetic operations on encrypted data. These HE schemes enable the server to process the encrypted data without decrypting it, ensuring that the raw data remains secure. Specifically, the client \(C_{i}\) encrypts its dataset \(X_{i}\) using its \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Feature** & **Federated Learning** & **Split Learning** & **Layered Learning** \\ \hline Data Sharing & Model updates only & Encrypted & \\  & & intermediate activations & Layer-wise model sharing \\ \hline Privacy & Medium & High (encrypted & \\  & & computations) & Medium \\ \hline Security Risks & Backdoor attacks & \
\begin{tabular}{c} Backdoor attacks, \\ adversarial modifications \\ \end{tabular} & Model leakage \\ \hline Computational Load & Heavy on clients &
\begin{tabular}{c} Shared between clients \\ and server \\ \end{tabular} & Moderate \\ \hline Application in HavenSL & Not directly applied & Primary focus & Limited \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison between FL, SL, and LL. public key \(pk_{i}\) and sends the encrypted data to the server, which performs computations without decrypting:  
\[X_{i}^{enc}=HE.Enc(X_{i},pk_{i})\]  
The notation \(X=HE.Enc(Xi,\ pki)\) represents the encryption of the dataset using a homomorphic encryption scheme (e.g., CKKS, BFV) and the public key. The operation ensures that data confidentiality is preserved while enabling secure computation. This formulation follows the standard homomorphic encryption approach as described in [5]. Equation describes the transformation of encrypted data as it propagates through the split learning model. This ensures that computations are performed while preserving confidentiality, which is a core property of HE-based training systems. The server performs operations on the encrypted data and returns the results to the client for decryption:  
\[y_{S}^{enc}=HE.Eval(Z_{i}^{enc};\theta_{S})\]  
The client then decrypts the result using its private key. Equation represents the application of Zero-Knowledge Proofs (ZKP) to validate that model updates remain unaltered. This prevents adversarial modifications from being injected into the encrypted training process. We implemented ZKP to verify the integrity of computations during training without revealing any sensitive information. The client provides a ZKP that proves the correctness of its computations up to the point where its part of the neural network is trained. This ensures that no backdoor or malicious modifications are introduced:  
\[\pi_{i}=ZKP.Prove(f(X_{i};\theta_{i}),pk_{s})\]  
The server verifies the proof without needing to access the raw data:  
\[ZKP.Verify(f(X_{i};\theta_{i}),\pi_{i},\ pk_{s})\ =\ \text{True}\]  
We developed and implemented the SafeSplit framework to defend against client-side backdoor attacks in Split Learning.