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

SFPDML: Securer and Faster Privacy-Preserving Distributed Machine Learning based on MKTFHE