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BGV-style homomorphic encryption schemes over polynomial rings rely for their security on rings of very large dimension. This large dimension is needed because of the large modulus-to-noise ratio in the key-switching matrices that are used for the top few levels of the evaluated circuit. However, larger noise (and hence smaller modulus-to-noise ratio) is used in lower levels of the circuit, so from a security standpoint it is permissible to switch to lower-dimension rings. Switching to a smaller ring, when possible, can help speeding up the homomorphic operations for the lower levels of the circuit. However, implementing such ring-switching is nontrivial, since these schemes rely on the ring algebraic structure for their homomorphic properties.
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