  With the development of machine learning, it is difficult for a single server
to process all the data. So machine learning tasks need to be spread across
multiple servers, turning the centralized machine learning into a distributed
one. However, privacy remains an unsolved problem in distributed machine
learning. Multi-key homomorphic encryption is one of the suitable candidates to
solve the problem. However, the most recent result of the Multi-key homomorphic
encryption scheme (MKTFHE) only supports the NAND gate. Although it is Turing
complete, it requires efficient encapsulation of the NAND gate to further
support mathematical calculation. This paper designs and implements a series of
operations on positive and negative integers accurately. First, we design basic
bootstrapped gates with the same efficiency as that of the NAND gate. Second,
we construct practical $k$-bit complement mathematical operators based on our
basic binary bootstrapped gates. The constructed created can perform addition,
subtraction, multiplication, and division on both positive and negative
integers. Finally, we demonstrated the generality of the designed operators by
achieving a distributed privacy-preserving machine learning algorithm, i.e.
linear regression with two different solutions. Experiments show that the
operators we designed are practical and efficient.
