A Multi-Bit Fully Homomorphic Encryption With Shorter Public Key From LWE

There has been a great deal of work on improving the efficiency of fully homomorphic encryption (FHE) scheme. Our approach, in this regard, is to use the idea of packed ciphertexts to construct a multi-bit FHE with a short public key on the basis of the learning with errors (LWE) problem. More specifically, our FHE scheme builds on a basic encryption scheme that chooses LWE samples from the Gaussian distribution and adds Gaussian error to it. This results in decreasing the number of LWE samples from <inline-formula> <tex-math notation="LaTeX">$2n$ </tex-math></inline-formula>log<inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">$n+1$ </tex-math></inline-formula>. We prove that our FHE scheme is pragmatically feasible and its security relies on the hardness of the LWE problem. In addition, we form a new process of key switching for multi-bit FHE based on the ideas adopted by Brakerski <italic>et al.</italic> for optimizing the process of key switching. Finally, we analyze and compare the concrete parameters between our FHE scheme and BGH13 scheme. The result shows that compared with the BGH13 scheme, our scheme has a smaller public key by a factor about log<inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>.

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