In recent years, distributed machine learning has garnered significant attention. However, privacy continues to be an unresolved issue within this field. Multi-key homomorphic encryption over torus (MKTFHE) is one of the promising candidates for addressing this concern. Nevertheless, there may be security risks in the decryption of MKTFHE. Moreover, to our best known, the latest works about MKTFHE only support Boolean operation and linear operation which cannot directly compute the non-linear function like Sigmoid. Therefore, it is still hard to perform common machine learning such as logistic regression and neural networks in high performance. In this paper, we first discover a possible attack on the existing distributed decryption protocol for MKTFHE and subsequently introduce secret sharing to propose a securer one. Next, we design a new MKTFHE-friendly activation function via \emph{homogenizer} and \emph{compare quads}. Finally, we utilize them to implement logistic regression and neural network training in MKTFHE. Comparing the efficiency and accuracy between using Taylor polynomials of Sigmoid and our proposed function as an activation function, the experiments show that the efficiency of our function is 10 times higher than using 7-order Taylor polynomials straightly and the accuracy of the training model is similar to using a high-order polynomial as an activation function scheme.

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