  We present novel homomorphic encryption schemes for integer arithmetic,
intended for use in secure single-party computation in the cloud. These schemes
are capable of securely computing only low degree polynomials homomorphically,
but this appears sufficient for most practical applications. In this setting,
our schemes lead to practical key and ciphertext sizes. We present a sequence
of generalisations of our basic schemes, with increasing levels of security,
but decreasing practicality. We have evaluated the first four of these
algorithms by computing a low-degree inner product. The timings of these
computations are extremely favourable. Finally, we use our ideas to derive a
fully homomorphic system, which appears impractical, but can homomorphically
evaluate arbitrary Boolean circuits.
