## 1 Introduction  
In cryptography, data privacy and security during training and inference are key [1]. Decentralized learning systems like Split Learning need new ways to protect against adversarial attacks, especially in collaborative learning environments where multiple parties interact without sharing raw data [2]. This work addresses these concerns by combining Homomorphic Encryption and Zero-Knowledge Proofs, two powerful cryptographic techniques for privacy-preserving computations in distributed learning [3, 4, 5]. This is especially useful in Split Learning where clients need to collaborate with servers without exposing their data. Recent advances in HE like CKKS and BFV have shown to be effective in keeping privacy while allowing complex arithmetic operations on encrypted data [6, 7]. These schemes have been used in privacy-preserving machine learning, especially in applications like healthcare and finance where data are sensitive [8, 9]. Zero-knowledge proofs are a way to verify computations without revealing the underlying data [10, 11, 12]. This is crucial in decentralized learning, where it is necessary to ensure that computations performed by clients and servers are accurate and free from tampering. This paper utilizes Homomorphic Encryption (HE) and Zero-Knowledge Proofs (ZKP) to protect data privacy and ensure integrity in decentralized learning environments. This research proposes a new method of combining ZKP with Split Learning to ensure the learning process is not tampered with by adversarial attacks like backdoor manipulations [13, 14]. To address the security and privacy challenges in Split Learning, this study formulates the following hypotheses, grounded in mathematical and cryptographic principles, to rigorously evaluate the effectiveness of Homomorphic Encryption and Zero-Knowledge Proofs in safeguarding model integrity and data confidentiality:  
**Hypothesis 1**. : _Utilizing Homomorphic Encryption (HE) in Split Learning ensures mathematical integrity by enabling secure arithmetic operations on encrypted data, preserving data confidentiality throughout the training process without decryption, and maintaining model accuracy._  
**Hypothesis 2**. : _Integrating Zero-Knowledge Proofs (ZKP) into Split Learning provides a cryptographic framework for mathematically verifying the correctness of client computations, allowing for the detection and mitigation of unauthorized parameter modifications (e.g., backdoor attacks) without disclosing underlying data or affecting model performance._  
ZKP has already proven to be quite a powerful tool in blockchain technology and secure communication protocols, so it would be a natural progression to apply it to machine learning [15; 16; 17; 18]. The novel SafeSplit framework, presented in this paper, is grounded in these cryptographic principles as it provides a mechanism tailored to protect against client-side backdoor attacks in the context of Split Learning. SafeSplit conducts both a static and dynamic analysis of model updates, so if a client tries to inject some malicious modifications, these changes are immediately noticed and counteracted [19]. This two-tiered defense, which takes advantage of rollback mechanisms to previous authenticated states, is a great stride towards securing decentralized machine learning systems [20]. The application of Homomorphic Encryption and Zero-Knowledge Proofs in this research is a breakthrough work in the research area of privacy-preserving machine learning. These are very promising approaches to the problems of privacy, model integrity, and adversarial defense in a distributed learning environment [21; 22]. With the constant development of cryptographic techniques, the combination of these with machine learning will prove indispensable to the maintenance of the security and trustworthiness of such systems against the ever-growing intelligent threats.