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 { $1,...,\mid\mathbb{F}\mid-\mid\mathbb{H}\mid$} ={ $1,...,176$ } such as $b=2$. Also, $m=2n_g=6$, $|w|=n_g-n_r=2$, $|\mathbb{K}|=m=6$. Hence:

$d=$ { $d_{AHP}(N,i,j)$ } $_{i=0,1,...,k_AHP, j=1,2,..,s_AHP(i)}^{}$ = { 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 $vk=57$.