When talking about FHE, refresh process is a little different from bootstrapping process. Bootstrapping always means that a scheme homomorphic decrypting its process, while refresh imply that use another scheme, always in large scale, to perform its decryption process. In EUROCRYPT'2015, Ducas and Micciancio proposed a FHE which can perform refresh process in less than a second, called DM14, while the scheme only support bite plaintext space, which is cumbersome for many applications. Extending DM14 to a large plaintext space becomes an open problem. In order to solve it, we improved the msbExtract process to endure a large base, by mapping the element to position. As a result, we constructed an efficient FHE with large plaintext space and quickly refresh process. We implemented our scheme in computer, and made a comparison between our performance and DM14. The result is that the running time is almost same, when extend the plaintext space from 2 to 8.
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