  Fueled by its successful commercialization, the recommender system (RS) has
gained widespread attention. However, as the training data fed into the RS
models are often highly sensitive, it ultimately leads to severe privacy
concerns, especially when data are shared among different platforms. In this
paper, we follow the tune of existing works to investigate the problem of
secure sparse matrix multiplication for cross-platform RSs. Two fundamental
while critical issues are addressed: preserving the training data privacy and
breaking the data silo problem. Specifically, we propose two concrete
constructions with significantly boosted efficiency. They are designed for the
sparse location insensitive case and location sensitive case, respectively.
State-of-the-art cryptography building blocks including homomorphic encryption
(HE) and private information retrieval (PIR) are fused into our protocols with
non-trivial optimizations. As a result, our schemes can enjoy the HE
acceleration technique without privacy trade-offs. We give formal security
proofs for the proposed schemes and conduct extensive experiments on both real
and large-scale simulated datasets. Compared with state-of-the-art works, our
two schemes compress the running time roughly by 10* and 2.8*. They also attain
up to 15* and 2.3* communication reduction without accuracy loss.
