In some aspects, a method for generating encoded plaintext data in a plaintext vector space includes obtaining a plurality of vectors of plaintext elements, where each plaintext element is an element of a first finite field. The method further includes encoding the plurality of vectors of plaintext elements to a vector of field elements, where each vector of plaintext elements is encoded to a respective field element of the vector of field elements, each of the field elements is an element of a second finite field, and the second finite field is a finite extension field of the first finite field. The method additionally includes encoding the vector of field elements into an element of the plaintext vector space to produce the encoded plaintext data for homomorphic encryption and computation.