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The traditional hierarchical identity fully homomorphic encryption scheme can only satisfy the homomorphic operation under a single identity, and the dimension of the lattice will increase synchronously with the increase of the layering depth. Aiming at the problem of high algorithm complexity due to the high-precision real number iterative operation used in the trapdoor derivation process of the existing scheme, this paper proposes an improved multi-identity fully homomorphic encryption scheme (mIBFHE): first, we use the features of new MP12 trapdoor to optimize the Gaussian sampling of the original scheme; secondly, the encryption and decryption structure of the original scheme is modified on the basis of the dual Regev system, so that the improved hierarchical identity encryption (HIBE) scheme meets the conversion format of the GSW compiler; finally, the transformed HIBFHE scheme is further transformed into a multi-identity fully homomorphic encryption scheme through the Mask system that meets the conditions. The hardness of our scheme can be reduced to the LWE problem, and IND-ID-CPA security is achieved under the random oracle model. After comparative analysis, the size of the master private key of the mIBFHE scheme in this paper is significantly reduced compared to the size of the same type of IBE scheme, which is 1/4 of the original scheme size. Our scheme is more efficient than traditional IBFHE scheme that it has smaller size than similar schemes in both the public key and the ciphertext when satisfying the attributes of hierarchical, homomorphism, and multiple identities and at the same time.
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