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Homomorphic encryption systems encode plaintext represented as rational numbers based on modular products of the rational numbers and a power of an integer basis with respect to a modulus defined by the integer basis. Decrypted ciphertexts are decoded based on modular products of the decrypted ciphertexts and an integer power of the integer basis. Typically, the integer power is one-half the number of available digits if the integer basis is odd; if the integer basis is even, the integer power is one-half the number of available digits plus one.