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.

翻译：近年来，分布式机器学习受到广泛关注，然而隐私问题仍是该领域尚未解决的难题。基于环面的多密钥同态加密（MKTFHE）是解决该问题的有前景方案之一，但MKTFHE的解密过程可能存在安全风险。此外，据我们所知，现有关于MKTFHE的最新工作仅支持布尔运算和线性运算，无法直接计算Sigmoid等非线性函数，因此仍难以高效执行逻辑回归、神经网络等常见机器学习任务。本文首先发现现有MKTFHE分布式解密协议存在潜在攻击，并引入秘密共享技术提出更安全的协议。随后，我们通过同质化器（homogenizer）和比较四元组（compare quads）设计了一种新的MKTFHE友好型激活函数。最后，利用上述成果在MKTFHE中实现了逻辑回归和神经网络训练。通过对比使用Sigmoid的泰勒多项式与本文所提函数作为激活函数的效率及精度，实验表明本函数效率较直接使用7阶泰勒多项式提升10倍，且训练模型精度与使用高阶多项式激活函数的方案相当。