Example 1

$Setup(1^{\lambda},N)$: This function outputs $pp=PC.Setup(1^{\lambda},d)$ where if $n_g=3$ , $n_i=1$ and $b=2$ then $d=\{d_{AHP}(N,i,j)\}_{i\in[k_{AHP}]\bigcup\{0\},j\in[s_{AHP}(i)]}=\{6,6,6,6,6,6,6,6,6,4,7,7,7,8,11,4,6,4,4,5,30\}$.

Run $KZG.\hspace{1mm}Setup(1^{\lambda},d)$ as following:

If the computation be done in $\mathbb{F}$ of order $p=181$, a generator of $\mathbb{F}$ is $g=2$.

$KZG.Setup(1^{\lambda},6)=(ck,vk)=(\{g\tau^i\}_{i=0}^5,g\tau)$ that for secret element $\tau=119$ and generator $g=2$ outputs $ck=\{g\tau^i\}_{i=0}^5=(2, 57, 86, 98, 78, 51)$ and $vk=57$.

$KZG.Setup(1^{\lambda},4)=(ck,vk)=(\{g\tau^i\}_{i=0}^3,g\tau)$ that for secret element $\tau=119$ and generator $g=2$ outputs $ck=\{g\tau^i\}_{i=0}^3=(2,57,86,98)$ and $vk=57$.

$KZG.Setup(1^{\lambda},7)=(ck,vk)=(\{g\tau^i\}_{i=0}^6,g\tau)$ that for secret element $\tau=119$ and generator $g=2$ outputs $ck=\{g\tau^i\}_{i=0}^6=(2,57,86,98,78,51,96)$ and $vk=57$.

$KZG.Setup(1^{\lambda},8)=(ck,vk)=(\{g\tau^i\}_{i=0}^7,g\tau)$ that for secret element $\tau=119$ and generator $g=2$ outputs $ck=\{g\tau^i\}_{i=0}^7=(2,57,86,98,78,51,96,21)$ and $vk=57$.

$KZG.Setup(1^{\lambda},11)=(ck,vk)=(\{g\tau^i\}_{i=0}^{10},g\tau)$ that for secret element $\tau=119$ and generator $g=2$ outputs $ck=\{g\tau^i\}_{i=0}^{10}=(2,57,86,98,78,51,96,21,146,179,124)$ and $vk=57$.

$KZG.Setup(1^{\lambda},6)=(ck,vk)=(\{g\tau^i\}_{i=0}^5,g\tau)$ that for secret element $\tau=119$ and generator $g=2$ outputs $ck=\{g\tau^i\}_{i=0}^5=(2,57,86,98,78,51)$ and $vk=57$.

$KZG.Setup(1^{\lambda},30)=(ck,vk)=(\{g\tau^i\}_{i=0}^{29},g\tau)$ that for secret element $\tau=119$ and generator $g=2$ outputs $ck=\{g\tau^i\}_{i=0}^{29}=(2, 57, 86, 98, 78, 51, 96, 21, 146, 179, 124, 95, 83, 103, 130, 85, 160, 35 ,$ $2, 57, 86, 98, 78, 51, 96, 21, 146, 179, 124, 95)$ and $vk=57$.