Abstract
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.
Highlights
Recent interest in coding for secure, private, and distributed computing combines a variety of elements such as coded distributed massive matrix multiplication, straggler tolerance, batch computing and private information retrieval [1]–[40]
Since Generalized Cross Subspace Alignment (GCSA) codes are efficient at batch processing and already encompass prior approaches to coded distributed computing, in this work we explore whether GCSA codes can be applied to the problem identified by Nodehi et al In particular, we focus on the problem of multiplication of two matrices
In GCSA-Noise Alignment (NA), each server only needs to multiply the two shares received from the sources and add noise
Summary
Recent interest in coding for secure, private, and distributed computing combines a variety of elements such as coded distributed massive matrix multiplication, straggler tolerance, batch computing and private information retrieval [1]–[40] These related ideas converged recently in Generalized Cross Subspace Alignment (GCSA) codes presented in [40]. Let us refer to the additional terms that are contained in the answers sent by the servers to the master, which may collectively reveal information about the inputs beyond the result of the computation, as interference terms To secure these interference terms, we use the idea of Noise Alignment (NA) – the workers communicate among themselves to share noise terms (unknown to the master) that are structured in the same manner as the interfering terms. It may be replaced with O(a log b) if the field F supports the Fast Fourier Transform (FFT), and with O(a log b log log(b)) if it does not
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