GCSA Codes with Noise Alignment for Secure Coded Multi-Party Batch Matrix Multiplication
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A secure multi-party batch matrix multiplication problem (SMBMM) is considered, where the goal is to allow a master node to efficiently compute the pairwise products of two batches of massive matrices that originate at external source nodes, by distributing the computation across S honest but curious servers. Any group of up to X colluding servers should gain no information about the input matrices, and the master should gain no additional information about the input matrices beyond the product. A solution called Generalized Cross Subspace Alignment codes with Noise Alignment (GCSA-NA in short) is proposed in this work, based on cross-subspace alignment codes. These codes originated in secure private information retrieval, and have recently been applied to distributed batch computation problems where they generalize and improve upon the state of art schemes such as Entangled Polynomial Codes and Lagrange Coded Computing. The prior 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 (which can entirely take place beforehand, i.e., even before the input matrices are determined), flexible inter-server network topology, efficient batch processing, and tolerance to stragglers. The idea of noise-alignment can also be applied to construct schemes for N sources based on N-CSA codes, and to construct schemes for symmetric secure private information retrieval to achieve the asymptotic capacity.
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