For the decryption of the fully homomorphic encryption (FHE) over the integers with the message space ZQ, Nuida and Kurosawa proposed a Q4λ-multiplicative-degree circuit to compute it at Eurocrypt 2015, where λ is the security parameter and the message size Q is a constant. Since the degree of the decryption circuit is polynomial in Q, the range of the message size Q is limited. In this work, we solve this open problem as long as Q is large enough (larger than λ). We represent the decryption

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... cuit as a arithmetic polynomial of multiplicative degree 108·λ log3λ, which is independent of the message size Q except a constraint Q>λ. Moreover, the bootstrapping process requires only O(λ·log⁡λ) number of multiplications to implement the decryption circuit, which is significantly lower than O(λ4) of Nuida and Kurosawa's work. We also show the efficiency of the FHE scheme with message space ZQ compared to the FHE scheme with binary message space. As a result, we have that the former is preferable.