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For the fully homomorphic encryption schemes in, this paper presents attacks to solve an equivalent secret key and directly recover plaintext from ciphertext for lattice dimensions n=2048 with lattice reduction algorithm. Given the average-case behavior of LLL in it is true, then their schemes are also not secure for n=8192.The authors first presented the concept of homomorphic encryption, which had been widely used in cryptography. It became insecure by 2009 when Gentry constructed the first fully homomorphic encryptions based on ideal lattice.
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