  Cross-silo federated learning (FL) enables multiple clients to
collaboratively train a machine learning model without sharing training data,
but privacy in FL remains a major challenge. Techniques using homomorphic
encryption (HE) have been designed to solve this but bring their own
challenges. Many techniques using single-key HE (SKHE) require clients to fully
trust each other to prevent privacy disclosure between clients. However, fully
trusted clients are hard to ensure in practice. Other techniques using
multi-key HE (MKHE) aim to protect privacy from untrusted clients but lead to
the disclosure of training results in public channels by untrusted third
parties, e.g., the public cloud server. Besides, MKHE has higher computation
and communication complexity compared with SKHE. We present a new FL protocol
ESAFL that leverages a novel efficient and secure additively HE (ESHE) based on
the hard problem of ring learning with errors. ESAFL can ensure the security of
training data between untrusted clients and protect the training results
against untrusted third parties. In addition, theoretical analyses present that
ESAFL outperforms current techniques using MKHE in computation and
communication, and intensive experiments show that ESAFL achieves approximate
204 times-953 times and 11 times-14 times training speedup while reducing the
communication burden by 77 times-109 times and 1.25 times-2 times compared with
the state-of-the-art FL models using SKHE.
