  This paper investigates the problem of Secure Multi-party Batch Matrix
Multiplication (SMBMM), where a user aims to compute the pairwise products
$\mathbf{A}\divideontimes\mathbf{B}\triangleq(\mathbf{A}^{(1)}\mathbf{B}^{(1)},\ldots,\mathbf{A}^{(M)}\mathbf{B}^{(M)})$
of two batch of massive matrices $\mathbf{A}$ and $\mathbf{B}$ that are
generated from two sources, through $N$ honest but curious servers which share
some common randomness. The matrices $\mathbf{A}$ (resp. $\mathbf{B}$) must be
kept secure from any subset of up to $X_{\mathbf{A}}$ (resp. $X_\mathbf{B}$)
servers even if they collude, and the user must not obtain any information
about $(\mathbf{A},\mathbf{B})$ beyond the products
$\mathbf{A}\divideontimes\mathbf{B}$. A novel computation strategy for single
secure matrix multiplication problem (i.e., the case $M=1$) is first proposed,
and then is generalized to the strategy for SMBMM by means of cross subspace
alignment. The SMBMM strategy focuses on the tradeoff between recovery
threshold (the number of successful computing servers that the user needs to
wait for), system cost (upload cost, the amount of common randomness, and
download cost) and system complexity (encoding, computing, and decoding
complexities). Notably, compared with the known result, the strategy for the
degraded case $X= X_{\mathbf{A}}=X_{\mathbf{B}}$ achieves better recovery
threshold, amount of common randomness, download cost and decoding complexity
when $X$ is less than some parameter threshold, while the performance with
respect to other measures remain identical.
