Example 1

In this section, we provide an example for the Setup phase settings.

$Setup(1^{\lambda},N)$: This function outputs $pp=PC.Setup(1^{\lambda},d)$.

We consider an input $x$ with size of 1. Hence, $n_i=1$. Considering a program which requires three gates for its arithmatization, we have $n_g=3$. In this example, the maximum number of registers which are changed during the execution is $n_r=1$. If the computation is done in $\mathbb{F}$ of order $p=181$, $|\mathbb{F}|=181$. Also,$|\mathbb{H}|=n=n_g+n_i+1=5$. Also, $b$ is a random number in {1,...,$|\mathbb{F}|-|\mathbb{H}|$}=$\{1,...,176\}$ such as $b=2$. Also, $m=2n_g=6$, $|w|=n_g-n_r=2$, $|\mathbb{K}|=m=6$. Hence:

$d=\{d_{AHP}(N,i,j)\}_{i\in[k_{AHP}]\bigcup\{0\},j\in[s_{AHP}(i)]}=\{6,6,6,6,6,6,6,6,6,4,7,7,7,8,11,4,6,4,4,5,30\}$

Now, we run $KZG.\hspace{1mm}Setup(1^{\lambda},d)$, considering a generator of $\mathbb{F}$, $g=2$, for each element in $d$:

$KZG.Setup(1^{\lambda},6)=(ck,vk)=(\{g\tau^i\}_{i=0}^5,g\tau)$ 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 $vk=57$.