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We construct the first fully homomorphic encryption (FHE) scheme that encrypts matrices and supports homomorphic matrix addition and multiplication. This is a natural extension of packed FHE and thus supports more complicated homomorphic operations. We optimize the bootstrapping procedure of Alperin-Sheriff and Peikert (CRYPTO 2014) by applying our scheme. Our optimization decreases the lattice approximation factor fromÕ(n 3) toÕ(n 2.5). By taking a lattice dimension as a larger polynomial in a security parameter, we can also obtain the same approximation factor as the best known one of standard latticebased public-key encryption without successive dimension-modulus reduction, which was essential for achieving the best factor in prior works on bootstrapping of standard lattice-based FHE.

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