Algorithm 6: Generating Algorithm for Secret Key

· Randomly select m different prime numbers $q_1,q_2,\cdots \mathrm{,}{q}_m$ such that $q_i\ne p$ . The principal ideal lattices

$I_{q_i}=\langle {\alpha }_{q_i}\rangle ,1\le i\le m$

are the secret key, and the corresponding one dimensional modulus $t_{q_i}$ given by

$t_{q_i}=t\left(I_{q_i}\right)=q_i^n-q_i^{n-1}+\cdots -q_i+1.$ (5.1)