# Verifiable Threshold Multi-Party Fully Homomorphic Encryption from Share Resharing
## 1 Introduction  
In the rapidly advancing information age, cloud computing has become an integral platform for consolidating vast computing resources. Many organizations and individuals choose to upload large volumes of data to cloud servers for processing and storage. However, while cloud computing provides extensive computational resources, it also faces numerous security challenges. When users transfer their data to cloud servers, they relinquish control over it, and both cloud service providers and malicious actors may exploit the cloud to steal users' sensitive information. Fully homomorphic encryption (FHE) [1] addresses this privacy challenge by enabling additions and multiplications to be performed directly on encrypted data, without requiring decryption. In other words, FHE can perform a series of operations on ciphertexts that directly correspond to additions and multiplications on the respective plaintexts. However, traditional (single-key) FHE [2, 3, 4, 5] is limited to performing homomorphic operations on ciphertexts encrypted with a single key, and therefore does not support multi-party collaborative computation scenarios. To address this problem, several generations of schemes have been proposed that allow joint computation on encrypted data from multiple users without the need for decryption. These schemes support flexible function selection and participant configuration,while ensuring that decryption of computation results can only be performed jointly. These schemes are commonly referred to as multi-party fully homomorphic encryption (MFHE) schemes [6; 7]. Over the years, various multi-key and threshold variants of MFHE schemes have been proposed [8; 9; 10; 11; 12; 13; 14], which serve as alternatives to traditional multi-party computation (MPC) protocols in cloud-based systems. These are particularly useful in collaborative data scenarios, such as distributed training of machine learning models [15] and biomedical analysis [16; 17]. For users, compact schemes are advantageous, as the number of interactions is not affected by the function being executed or the number of parties involved, and the computation remains constant regardless of the circuit depth of the function, thereby avoiding significant computational and communication costs. Although using passively secure threshold multi-party fully homomorphic encryption (TMFHE) schemes is effective and practical in most scenarios, these schemes are extensions of traditional FHE schemes to multiple users, and can, at most, operate among honest-but-curious parties. Therefore, there is a possibility of privacy leakage when facing malicious adversaries. In TMFHE schemes, when a set of parties corrupted by a malicious adversary or a dishonest majority outputs incorrect partial decryption results, it can cause decryption failures or even leakage of private information, thereby compromising the correctness and security of the schemes. To ensure security against malicious adversaries in TMFHE schemes, a natural and feasible approach is to introduce commitment and non-interactive zero-knowledge (NIZK) proof systems based on the Fiat-Shamir transform [18]. However, constructing an actively secure TMFHE scheme presents several difficult challenges. In one respect, the workflow of TMFHE schemes involves uncertainty in the parties and their numbers, and there are constraints such as the inability to ensure consistency of secrets between different protocols. This means that the verification mechanism based on commitment and NIZK must be specifically designed according to the characteristics of the TMFHE scheme, without affecting the execution of the primary encryption scheme. In another respect, the variable ciphertext structure dimensions and complex computation processes of the current FHE schemes result in additional computations when NIZKs are introduced into the multi-party execution of prescribed protocols.