· For any ciphertext $c \in ℤ^{n}$ , given a full rank ideal lattice $I_{i}$ with HNF basis $B = [β_{1}, β_{2}, \dots, β_{n}]$ and the corresponding orthogonal basis $B^{*} = [β_{1}^{*}, β_{2}^{*}, \dots, β_{n}^{*}]$ , we can find only one vector $\bar{u_{i}} = c - \sum_{i = 1}^{n} k_{i} β_{i} \in F (I)$ such that $c \equiv \bar{u_{i}} (\mod I_{i})$ , where

k_{n} = [\frac{〈 c, β_{n}^{*} 〉}{〈 β_{n}^{*}, β_{n}^{*} 〉}], k_{i} = [\frac{〈 c, β_{i}^{*} 〉}{〈 β_{i}^{*}, β_{i}^{*} 〉} - \frac{〈 \sum_{j = i + 1}^{n} k_{j} β_{j}, β_{i}^{*} 〉}{〈 β_{i}^{*}, β_{i}^{*} 〉}], i = n - 1, n - 2, \dots, 1.