  A secure multi-party batch matrix multiplication problem (SMBMM) is
considered, where the goal is to allow a master to efficiently compute the
pairwise products of two batches of massive matrices, by distributing the
computation across S servers. Any X colluding servers gain no information about
the input, and the master gains no additional information about the input
beyond the product. A solution called Generalized Cross Subspace Alignment
codes with Noise Alignment (GCSA-NA) is proposed in this work, based on
cross-subspace alignment codes. The state of art solution to SMBMM is a coding
scheme called polynomial sharing (PS) that was proposed by Nodehi and
Maddah-Ali. GCSA-NA outperforms PS codes in several key aspects - more
efficient and secure inter-server communication, lower latency, flexible
inter-server network topology, efficient batch processing, and tolerance to
stragglers. The idea of noise alignment can also be combined with N-source
Cross Subspace Alignment (N-CSA) codes and fast matrix multiplication
algorithms like Strassen's construction. Moreover, noise alignment can be
applied to symmetric secure private information retrieval to achieve the
asymptotic capacity.
