Algorithm 1 Setup

Input: $\Re -SIS$, security parameter n Output: System parameter $pp$;   1:Sets $q=q\left(n\right)\ge 3$ as a prime number;   2:Sets $m\ge 3nlogq$ as a positive integer;   3:KGC randomly selects an approximate random distribution matrix $mpk=A\in Z_q^{n\times m}$;   4:Generates a basis $T\in Z_q^{m\times m}$ from ${\mathsf{\Lambda }}^{\perp }\left(A\right)$, $\left|\right|\tilde{T}\left|\right|\le L$;   5:Sets $H_1,H_2,H_3,H_5:{\{0,1\}}^{*}\to Z_q^{n\times m}$ as the hash functions;   6:Sets $H_4:{\{0,1\}}^{*}\to Z_q^m$ as the other one hash function;   7:Serves the master public key $mpk=A$ and master secret key $msk=T$;   8:Opens $pp=\{n,q,m,A,H_1,H_2,H_3,H_4,H_5\}$.