Algorithm 1: The General Unbounded FHE Scheme without Noises

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

$•$ 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

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

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

The public key for encryption is $\left\{A_1\mathrm{,}{A}_2\mathrm{,}\cdots \mathrm{,}{A}_m\right\}\subset {\mathbb{Z}}^n$ , and each $A_i$ is given by (2.5).

$•$ 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 is given by

$c=f\left(u\right)=\overline{u_1}\otimes A_1+\overline{u_2}\otimes A_2+\cdots +\overline{u_m}\otimes A_m,$ (2.6)

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 (2.3), we have

$f^{-1}\left(c\right)=\left(u_1,u_2,\cdots ,u_m\right)=u.$