Algorithm 8: A Practical Scheme for Unbounded FHE

$•$ Secret key: Let $q_1\mathrm{,}{q}_2\mathrm{,}\cdots \mathrm{,}{q}_m$ be m different prime numbers such that $q_i\ne p$ . The m pairwise relatively prime principal ideal lattices $I_{q_1}\mathrm{,}{I}_{q_2}\mathrm{,}\cdots \mathrm{,}{I}_{q_m}$ are the secret keys for decryption.

$•$ Public Key: Let $A_i=d_i\otimes {\mathcal{D}}_i\left(1\le i\le m\right)$ , then $A_{\chi }=\left\{{A^{\prime }}_1\mathrm{,}{A^{\prime }}_2\mathrm{,}\cdots \mathrm{,}{A^{\prime }}_m\right\}$ are the public key for encryption, where ${A^{\prime }}_i=A_i\otimes E_i$ and $E_i$ is given by (3.4) according to the probabilistic distribution $\chi$ . The plaintext space is

$\mathcal{P}={\mathbb{Z}}_{t_1}\oplus {\mathbb{Z}}_{t_2}\oplus \cdots \oplus {\mathbb{Z}}_{t_m}\mathrm{,}$

where $t_i$ is the one dimensional modulus of $I_{q_i}$ given by (5.1).

$•$ Encryption: For any plaintext $u=\left(u_1,u_2,\cdots ,u_m\right)\in \underset{i=1}{\overset{m}{{\displaystyle {{\displaystyle \oplus }}_{}}}}{\mathbb{Z}}_{t_i}$ , then

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

where $\overline{u_i}$ is the embedding of $u_i$ into ${\mathbb{Z}}^n$ .

$•$ Decryption: For given ciphertext $c\in {\mathbb{Z}}^n$ , there is an unique vector $\overline{u_i}$ in the orthogonal parallelepiped $\mathcal{F}\left(I_{q_i}\right)$ , such that $c\equiv \overline{u_i}\left(\mathrm{mod}{I}_{q_i}\right)$ , $1\le i\le m$ . Thus, one has

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