  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.
