Algorithm 2 Init

1:Let $G:{\{0,1\}}^{\lambda }\to {\{0,1\}}^{2\lambda +2}$ a random generator. 2:Let ${\mathbb{G}}^{\prime }={\{0,1\}}^{\lambda }$, where${\{0,1\}}^{\lambda }$ employs bitwise addition 3:and ${\mathrm{Convert}}_{{\mathbb{G}}^{\prime }}:{\{0,1\}}^{\lambda }\to {\mathbb{G}}^{\prime }$ denote a function that transforms a $\lambda$-bit random string to a random group element in $\mathbb{G}$. ${\mathrm{Convert}}_{\mathbb{G}}\left(m\right)$: 4:Let $u\leftarrow \left|\mathbb{G}\right|$ 5:if $u=2^s$ for an integer s then 6:     Output the group element corresponding to $G\left(m\right)$, for PRG $G:{\{0,1\}}^{\lambda }\to {\{0,1\}}^s$ 7:else 8:     Let $l\leftarrow \lceil {log}_{2}u\rceil +\lambda$ 9:     Output the group element corresponding to $G\left(m\right)modu$, for a PRG $G:{\{0,1\}}^{\lambda }\to {\{0,1\}}^l$ 10:end if