Federated learning (FL) is a distributed machine learning paradigm that enables multiple clients to collaboratively train a shared model without disclosing their local data. To address privacy issues of gradient, several privacy-preserving machine-learning schemes based on multi-client functional encryption (MCFE) have been proposed. However, existing MCFE-based schemes cannot support client dropout or flexible threshold selection, which are essential for practical FL. In this paper, we design a flexible threshold multi-client functional encryption for inner product (FTMCFE-IP) scheme, where multiple clients generate ciphertexts independently without any interaction. In the encryption phase, clients are able to choose a threshold flexibly without reinitializing the system. The decryption can be performed correctly when the number of online clients satisfies the threshold. An authorized user are allowed to compute the inner product of the vectors associated with his/her functional key and the ciphertext, respectively, but cannot learning anything else. Especially, the presented scheme supports clients drop out. Furthermore, we provide the definition and security model of our FTMCFE-IP scheme,and propose a concrete construction. The security of the designed scheme is formally proven. Finally, we implement and evaluate our FTMCFE-IP scheme.

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