Coded Distributed (Batch) Matrix Multiplication over Galois Ring via RMFE

Coded Distributed Matrix Multiplication (CDMM) is a distributed matrix multiplication (DMM) for large-scale matrices through a coding scheme such that any R worker nodes among all N worker nodes can recover the final product, where N corresponds to the length of the code and R ≤ N is called the recovery threshold. The state-of-art CDMM schemes, such as EP codes for Single DMM and GCSA codes for batch DMM, are defined over a Galois field GF(q) of size q ≥ N. These are inefficient for small Galois fields such as GF(2) and the integer residue ring Z_p^ℓ due to the lack of invertible elements for interpolation. DMM over Z_p^ℓ (such as Z₂⁶⁴ ) is well-motivated in practice due to their direct compatibility with hardware. In this work, we construct efficient CDMM over the Galois ring GR(p^ℓ, d) which is an extension ring over Z_p^ℓ of degree d, particularly, GR(p, d) = GF(p^d) is the Galois field and GR(_p^ℓ, 1) = Z_p^ℓ. We first give a general CDMM framework for the batch of n matrix multiplications via the famous RMFE (Cascudo et al. Crypto’18). Compared with GCSA, our construction has a smaller recovery threshold by a factor of 1/n. Next, we optimize EP codes via batch preprocessing of the input matrices. We give two types of Single CDMM, which can achieve almost the same performance as EP codes over a Galois field with size q ≥ N. Finally, we present the experimental analysis of our CDMM on Galois rings.