# Example 1

$$Setup(1^{\lambda},N)$$: 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 $$vk=57$$.