Efficient AGCD-Based Homomorphic Encryption for Matrix and Vector Arithmetic

We propose a leveled homomorphic encryption scheme based on the Approximate Greatest Common Divisor (AGCD) problem that operates natively on vectors and matrices. To overcome the limitation of large ciphertext expansion that is typical in AGCD-based schemes, we randomize the ciphertexts with a hidden matrix, which allows us to choose smaller parameters. To be able to efficiently evaluate circuits with large multiplicative depth, we use a decomposition technique à la GSW. The running times and ciphertext sizes are practical: for instance, for 100 bits of security, we can perform a sequence of 128 homomorphic products between 128-dimensional vectors and 128×128 matrices in less than one second. We show how to use our scheme to homomorphically evaluate nondeterministic finite automata and also a Naïve Bayes Classifier.