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Several public-key encryption schemes used to solve the problem of ciphertext data processing on the fly are discussed. A new targeted fully homomorphic encryption scheme based on the discrete logarithm problem is presented. Public-key encryption cryptosystems are classified to examine homomorphic encryption. Without employing techniques proposed by Gentry such as somewhat homomorphic and bootstrapping techniques, or relinearization technique proposed by Brakerski et al., a new method called "Double Decryption Algorithm" is employed in our cryptography to satisfy a fully or targeted fully homomorphic property. Inspired by ElGamal and BGN cryptography, we obtain the desired fully homomorphic property by selecting a new group and adding an extra component to the ciphertext. Proof of semantic security is also demonstrated.

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