Algorithm 5: The General Unbounded FHE Scheme

$•$ Secret Key: One randomly selects m pairwise relatively prime principal ideal lattices $I_1\mathrm{,}{I}_2\mathrm{,}\cdots \mathrm{,}{I}_m$ in ${\mathbb{Z}}^n$ as the decryption key according to Algorithm 3.

$•$ Public Key: Let $t_i=t\left(I_i\right)$ be the one dimensional modulus of $I_i\left(1\le i\le m\right)$ . The plaintexts space is the direct sum of ring $\mathcal{P}=\underset{i=1}{\overset{m}{{\displaystyle {{\displaystyle \oplus }}_{}}}}{\mathbb{Z}}_{t_i}$ , the addition and multiplication in $\mathcal{P}$ are given by

$\left(a_1\mathrm{,}{a}_2\mathrm{,}\cdots \mathrm{,}{a}_m\right)+\left(b_1\mathrm{,}{b}_2\mathrm{,}\cdots \mathrm{,}{b}_m\right)=\left(a_1+b_1\mathrm{,}{a}_2+b_2\mathrm{,}\cdots \mathrm{,}{a}_m+b_m\right)\mathrm{,}$

$\left(a_1\mathrm{,}{a}_2\mathrm{,}\cdots \mathrm{,}{a}_m\right)\cdot \left(b_1\mathrm{,}{b}_2\mathrm{,}\cdots \mathrm{,}{b}_m\right)=\left(a_1{b}_1\mathrm{,}{a}_2{b}_2\mathrm{,}\cdots \mathrm{,}{a}_m{b}_m\right)\mathrm{.}$

Let $\chi =\left({\chi }_1\mathrm{,}{\chi }_2\mathrm{,}\cdots \mathrm{,}{\chi }_m\right)$ be a probabilistic distribution over ${\mathbb{Z}}_{t_1}\times {\mathbb{Z}}_{t_2}\times \cdots \times {\mathbb{Z}}_{t_m}$ . The public key for encryption is $A_{\chi }=\left\{{A^{\prime }}_1\mathrm{,}{A^{\prime }}_2\mathrm{,}\cdots \mathrm{,}{A^{\prime }}_m\right\}\subset {\mathbb{Z}}^n$ , and each ${A^{\prime }}_i=A_i\otimes E_i$ , where $A_i$ given by Algorithm 4 and $E_i$ given by (3.4).

$•$ Encryption: For any plaintext $u=\left(u_1,u_2,\cdots ,u_m\right)\in \mathcal{P}=\underset{i=1}{\overset{m}{{\displaystyle {{\displaystyle \oplus }}_{}}}}{\mathbb{Z}}_{t_i}$ , the encryption function $f_{\chi }$ is given by

$c=f_{\chi }\left(u\right)=\overline{u_1}\otimes {A^{\prime }}_1+\overline{u_2}\otimes {A^{\prime }}_2+\cdots +\overline{u_m}\otimes {A^{\prime }}_m,$ (3.5)

where $\overline{u_i}$ is the embedding of $u_i$ .

$•$ Decryption: For any ciphertext $c\in {\mathbb{Z}}^n$ , we use the secret key $I_1\mathrm{,}{I}_2\mathrm{,}\cdots \mathrm{,}{I}_m$ to decrypt c. Since for every i, $1\le i\le m$ , we have $c\equiv \overline{u_i}\left(\mathrm{mod}{I}_i\right)$ , and c mod $I_i$ is a unique vector in the orthogonal parallelepiped $\mathcal{F}\left(I_i\right)$ of $I_i$ , thus one has c mod $I_i=\overline{u_i}$ , and by Lemma 3.1 and (2.3), we have

$f_{\chi }^{-1}\left(c\right)=f^{-1}\left(c\right)=\left(u_1\mathrm{,}{u}_2\mathrm{,}\cdots \mathrm{,}{u}_m\right)=u\mathrm{.}$