Example 1

Assume the following sample code: mul R1, R1, 5 => Gate 1, p=181 add R1, R1, 11 => Gate 2, p=181 mul R1, R1, 26 => Gate 3, p=181

The constraints are as follows considering $p=181$:

$R_1^{(2)}-5R_1^{(1)}=0$ $R_1^{(3)}-R_1^{(2)}-11=0$ $R_1^{(4)}-26R_1^{(3)}=0$

To keep the example simple and understandable, we continue with $n_g=3$ and $n_i=1$.

The Prover calculates square matrices $A$, $B$ and $C$ of order $n_g+n_i+1=3+1+1=5$ based on above construction:

$$A=\begin{bmatrix} 0&0&0&0&0\\ 0&0&0&0&0\\ 0&1&0&0&0\\ 1&0&0&0&0\\ 0&0&0&1&0 \end{bmatrix}$$

$$B=\begin{bmatrix} 0&0&0&0&0\\ 0&0&0&0&0\\ 5&0&0&0&0\\ 11&0&1&0&0\\ 26&0&0&0&0 \end{bmatrix}$$

$$C=\begin{bmatrix} 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1 \end{bmatrix}$$

At we see, the matrices $A$ and $B$ are $2-SLT$ and matrix $C$ is $2-Diag$. Also $Az\hspace{1mm}o\hspace{1mm}Bz=Cz$.

We consider multiplicative subgroup $\mathbb{H}$ of order $n=n_g+n_i+1=5$ with generator $\omega=2^{36}\equiv 59 (\textrm{mod}\hspace{1mm}181)$. Note that if $g$ is a generator of field $\mathbb{F}$ of order $p$, then, $g^{\frac{p-1}{n}}$ is a generator of a multiplicative subgroup of it of order $n$. Therefore, $\mathbb{H}=$ { $1,\omega,\omega^2,\omega^3,\omega^4$ } = { $1,59,42,125,135$ }.

Also, consider multiplicative subgroup $\mathbb{K}$ of order $m=2n_g=6$ where $t=n_i+1=2$ with generator $\gamma=g^{\frac{p-1}{m}}=2^{30}\equiv49(\textrm{mod}\hspace{1mm}181)$. Therefore, $\mathbb{K}=$ { $1,\gamma,\gamma^2,...,\gamma^5$ } = { $1,49,48,180,132,133$ } .

PFR Commitment

$Commit(ck=(2,66,83,91,96,24,2,66,83),m_f=(A,B,C),s\in R)$:

1- The Prover Selects $s=(s_1,...,s_{s_{AHP}(0)})$ of random space $R$.

2- The Prover calculates $\overrightarrow{O}=Enc(m_f=(A,B,C))$ as encoded index as following:.

First, calculates the polynomial $row_{PFR_A}(x)$. Since matrix $A$ has three non-zero entries that the first of them is in the second row, so $c_0=2$ and $row_{PFR_A}(\gamma^0)=\omega^2$ , note that rows numbered of zero, the second is in the third row, so $c_1=3$ and $row_{PFR_A}(\gamma^1)=\omega^3$ and the third is in the fourth row, so $c_2=4$ and $row_{PFR_A}(\gamma^2)=\omega^4$. Note that to construct all three polynomials, we read the entries in the matrix in row or column order. Here they are read in row order.

According to the values ​​defined for $\gamma$ and $\omega$, we have $row_{PFR_A}(1)=42$, $row_{PFR_A}(43)=125$ and $row_{PFR_A}(39)=135$. Therefore based on Lagrange polynomials, have $row_{PFR_A}(x)=\sum_{i=1}^{3}y_iL_i(x)$, so $row_{PFR_A}(x)=42L_1(x)+125L_2(x)+135L_3(x)$, where $L_1(x)=\frac{(x-43)(x-39)}{(1-43)(1-39)}=\frac{(x-43)(x-39)}{(-42)(-38)}$. Now, since $-42\equiv139\hspace{1mm}(\textrm{mod}\hspace{1mm}181)$, $-38\equiv143\hspace{1mm}(\textrm{mod}\hspace{1mm}181)$, $-43\equiv138\hspace{1mm}(\textrm{mod}\hspace{1mm}181)$ and $-39\equiv142\hspace{1mm}(\textrm{mod}\hspace{1mm}181)$ , therefore $L_1(x)=\frac{(x+138)(x+142)}{(139)(143)}$ and since $(139)(143)\equiv 148\hspace{1mm}(\textrm{mod}\hspace{1mm}181)$ and $148^{-1}\equiv 170\hspace{1mm}(\textrm{mod}\hspace{1mm}181)$ , have $L_1(x)=170(x+138)(x+142)=170x^2+178x+15$ . We get in a similar way that $L_2(x)=167x^2+17x+178$ and $L_3(x)=25x^2+167x+170$ . Therefore $row_{PFR_A}(x)=77x^2+109x+37$ .

Now, calculates $col_{PFR_A}(x)$. Since matrix $A$ has three non-zero entries that the first of them is in the first column, so $r_0=1$ and $col_{PFR_A}(\gamma^0)=\omega^1$ , note that columns numbered of zero, the second is in the zero column, so $r_1=0$ and $col_{PFR_A}(\gamma^1)=\omega^0$ and the third is in the third column, so $r_2=3$ and $col_{PFR_A}(\gamma^2)=\omega^3$. Therefore $col_{PFR_A}(1)=59$, $col_{PFR_A}(43)=1$ and $col_{PFR_A}(39)=125$. So $col_{PFR_A}(x)=59L_1(x)+L_2(x)+125L_3(x)=109x^2+81x+50$.

Now, calculates $val_{PFR_A}(x)$. Since matrix $A$ has three non-zero entries such that the value of all of them is one, therefore $v_0=v_1=v_2=1$. So, $val_{PFR_A}(x)=L_1(x)+L_2(x)+L_3(x)=1$.

Therefore, the matrix $A$ is encoded by $row_{PFR_A}(x)=77x^2+109x+37$, $col_{PFR_A}(x)=109x^2+81x+50$ and $val_{PFR_A}(x)=1$.

Now, encodes the matrix $B$. First, calculates the polynomial $row_{PFR_B}(x)$. Since matrix $B$ has four non-zero entries such that the first of them is in the second row, so $c_0=2$ and $row_{PFR_B}(\gamma^0)=\omega^2$ . The second and the third are in the third row, so $c_1=c_2=3$ and $row_{PFR_B}(\gamma^1)=row_{PFR_B}(\gamma^2)=\omega^3$ and the fourth is in the fourth row, so $c_3=4$ and $row_{PFR_B}(\gamma^3)=\omega^4$. Therefore, $row_{PFR_B}(1)=42$, $row_{PFR_B}(43)=125$, $row_{PFR_B}(39)=125$ and $row_{PFR_B}(48)=135$. So $row_{PFR_B}(x)=42L_1(x)+125L_2(x)+125L_3(x)+135L_4(x)$ where $L_1(x)=\frac{(x-43)(x-39)(x-48)}{(1-43)(1-39)(1-48)}=\frac{(x+138)(x+142)(x+133)}{(139)(143)(134)}=\frac{(x+138)(x+142)(x+133)}{103}$. Since $103^{-1}\equiv58\hspace{1mm}(\textrm{mod}\hspace{1mm}181)$, so $L_1(x)=58(x+138)(x+142)(x+133)=58x^3+62x^2+116x+127$. We get in a similar way that $L_2(x)=39x^3+7x^2+19x+116$, $L_3(x)=138x^3+155x^2+7x+62$ and $L_4(x)=127x^3+138x^2+39x+58$. Therefore $row_{PFR_B}(x)=76x^3+35x^2+174x+119$.

Now, calculates $col_{PFR_B}(x)$. Since matrix $B$ has four non-zero entries that the first, second and fourth of them are in zero column, so $r_0=r_1=r_3=0$ and $col_{PFR_B}(\gamma^0)=col_{PFR_B}(\gamma^1)=col_{PFR_B}(\gamma^3)=\omega^0$ and the third is in the second column, so $r_2=2$ and $col_{PFR_B}(\gamma^2)=\omega^2$ . Therefore $col_{PFR_B}(1)=1$, $col_{PFR_B}(43)=1$, $col_{PFR_B}(39)=42$ and $col_{PFR_B}(48)=1$. So $col_{PFR_B}(x)=L_1(x)+L_2(x)+42L_3(x)+L_4(x)=47x^3+20x^2+106x+9$.

Now, calculates $val_{PFR_B}(x)$. Since matrix $B$ has four non-zero entries that value of them are $5$, $11$, $1$ and $26$, respectively. Therefore $v_0=v_1=v_2=1$ $val_{PFR_B}(\gamma^0)=5$, $val_{PFR_B}(\gamma^1)=11$, $val_{PFR_B}(\gamma^2)=1$ and $val_{PFR_B}(\gamma^3)=26$. So, $val_{PFR_B}(1)=5$, $val_{PFR_B}(43)=11$, $val_{PFR_B}(39)=1$ and $val_{PFR_B}(48)=26$. Therefore, $val_{PFR_B}(x)=5L_1(x)+11L_2(x)+L_3(x)+26L_4(x)=177x^3+148x^2+42$.

Therefore, the matrix $B$ is encoded by $row_{PFR_B}(x)=76x^3+35x^2+174x+119$, $col_{PFR_B}(x)=47x^3+20x^2+106x+9$ and $val_{PFR_B}(x)=177x^3+148x^2+42$.

Now, calculates polynomial $row_{PFR_C}(x)$. In a similar way, Since matrix $C$ has three non-zero entries that are in the third, fourth and fifth rows, therefore $row_{PFR_C}(\gamma^0)=\omega^2$ , $row_{PFR_C}(\gamma^1)=\omega^3$ and $row_{PFR_C}(\gamma^2)=\omega^4$. Then $row_{PFR_C}(1)=42$ , $row_{PFR_C}(43)=125$ and $row_{PFR_C}(39)=135$. So $row_{PFR_C}(x)=42L_1(x)+125L_2(x)+135L_3(x)$, therefore $row_{PFR_C}(x)=77x^2+109x+37$.

Now, since $C$ is a diagonal matrix, polynomials $row_{PFR_C}(x)$ and $col_{PFR_C}(x)$ are equal. Also, since the matrix $C$ has three non-zero entries such that the value of all of them is one, therefore $v_0=v_1=v_2=1$. So, $val_{PFR_C}(x)=L_1(x)+L_2(x)+L_3(x)=1$.

Therefore, the matrix $C$ is encoded by $row_{PFR_C}(x)=77x^2+109x+37$, $col_{PFR_C}(x)=77x^2+109x+37$ and $val_{PFR_C}(x)=1$. Therefore, encoding of $m_f=(A,B,C)$ calculates as following: $\overrightarrow{O}_{PFR}=(37,109,77,0,0,0,0,0,0,50,81,109,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,119,174,35,76,0,0,$

3- The Prover calculates commitment by using of KZG commitment scheme as following:

$Com_{T}=\sum_{i=0}^{deg_T}a_ig\tau^i=\sum_{i=0}^{deg_T}a_ick(i)$ where $a_i$ is coefficient of $x^i$ in polynomial $T(x)$.

Therefore

$Com_{PFR}^0=\sum_{i=0}^{2}row_{PFR_{A_i}}(g\tau^i)=\sum_{i=0}^{2}row_{PFR_{A_i}}ck(i)=37ck(0)+109ck(1)+77ck(2)=37\times2+109\times 57+77\times 86\equiv\hspace{1mm}(\textrm{mod}\hspace{1mm}181)$

$Com_{PFR}^1=$

$Com_{PFR}^2=$

and similarly

$Com_{PFR}^3=$, $Com_{PFR}^4=$, $Com_{PFR}^5=$, $Com_{PFR}^6=$, $Com_{PFR}^7=$ and $Com_{PFR}^8=$.

4- The Prover sends $Com_{PFR}=$ to the Verifier.

AHP Commitment

$Commit(ck,m_f=(A,B,C),s\in R)$:

1 - The Prover Selects $s=(s_1,...,s_{s_{AHP}(0)})$ of random space $R$. 2- The Prover calculates $\overrightarrow{O}_{AHP}=Enc(m_f=(A,B,C))$ as encoded index as following:

The polynomial $row_{AHP_A}:\mathbb{K}=$ { $1,49,48,180,132,133$ } $\to\mathbb{H}=$ { $1,59,42,125,135$ } with $row_{AHP_A}(k=\gamma^i)=\omega^{r_i}$ for $1\leq i\leq ||A||$ , and otherwise $row_{AHP_A}(k)$ returns an arbitrary element in $\mathbb{H}$. So $row_{AHP_A}(k)$ on $\mathbb{K}$ is a polynomial so that $row_{AHP_A}(1)=42$ , $row_{AHP_A}(49)=125$, $row_{AHP_A}(48)=135$ and otherwise $row_{AHP_A}(k)$ returns arbitrary elements of $\mathbb{H}$, for example $row_{AHP_A}(180)=125$, $AHProw_A(132)=135$, $AHProw_A(133)=1$ . Therefore, $row_{AHP_A}(x)=\sum_{i=1}^{6}y_iL_i(x)=162x^5+62x^4+161x^3+169x^2+88x+124$.

Also, $col_{AHP_A}:\mathbb{K}=$ { $1,49,48,180,132,133$ } $\to\mathbb{H}=$ { $1,59,42,125,135$ } with $col_{AHP_A}(k=\gamma^i)=\omega^{c_i}$ for $1\leq i\leq ||A||$ and otherwise $col_{AHP_A}(k)$ returns an arbitrary element in $\mathbb{H}$. So $col_{AHP_A}(k)$ on $\mathbb{K}$ is a polynomial so that $col_{AHP_A}(1)=59$ , $col_{AHP_A}(49)=1$, $col_{AHP_A}(48)=125$ and otherwise $col_{AHP_A}(k)$ returns arbitrary elements of $\mathbb{H}$, for example $col_{AHP_A}(180)=125$, $col_{AHP_A}(132)=135$ and $col_{AHP_A}(133)=1$. Therefore, $col_{AHP_A}(x)=\sum_{i=1}^{6}y_iL_i(x)=128x^5+150x^4+32x^3+109x^2+169x+14$

and , $val_{AHP_A}:\mathbb{K}=$ { $1,49,48,180,132,133$ } $\to\mathbb{H}=$ { $1,59,42,125,135$ } with $val_{AHP_A}(k=\gamma^i)=\frac{v_i}{u_{\mathbb{H}}(row_{AHP_A}(k),row_{AHP_A}(k))u_{\mathbb{H}}(col_{AHP_A}(k),col_{AHP_A}(k))}$ for $1\leq i\leq ||A||$ where $v_i$ is value of $i^{th}$ nonzero entry and otherwise $val_{AHP_A}(k)$ returns zero. Note that based on definition of $u_{\mathbb{H}}(x,y)$, for each $x \in \mathbb{H}$, $u_{\mathbb{H}}(x,x)=|\mathbb{H}|x^{|\mathbb{H}|-1}=5x^4$. So $val_{AHP_A}(1)=\frac{1}{(5row_{AHP_A}^4(1))(5col_{AHP_A}^4(1))}=\frac{1}{5\times 42^4\times5\times59^4}=\frac{1}{145}\equiv 5\hspace{1mm}(\textrm{mod}\hspace{1mm}181)$, $val_{AHP_A}(49)=\frac{1}{(5row_{AHP_A}^4(49))(5col_{AHP_A}^4(49))}=\frac{1}{5\times 125^4\times5\times1^4}=\frac{1}{145}\equiv 5\hspace{1mm}(\textrm{mod}\hspace{1mm}181)$ , $val_{AHP_A}(48)=\frac{1}{(5row_{AHP_A}^4(48))(5col_{AHP_A}^4(48))}=\frac{1}{5\times 135^4\times5\times 125^4}=\frac{1}{48}\equiv 132\hspace{1mm}(\textrm{mod}\hspace{1mm}181)$ and $val_{AHP_A}(k)=0$ for $k\in \mathbb{K}-$ { $1,49,48$ }. Therefore $val_{AHP_A}(x)=\sum_{i=1}^{6}y_iL_i(x)=72x^5+79x^4+22x^3+111x^2+180x+84$.

Now, $\hat{row_{AHP_A}}$, $\hat{col_{AHP_A}}$ and $\hat{val_{AHP_A}}$ are extensions of $row_{AHP_A}$, $col_{AHP_A}$ and $val_{AHP_A}$ so that are agree on $\mathbb{K}$.

Similarly, $row_{AHP_B}:\mathbb{K}=$ { $1,49,48,180,132,133$ } $\to\mathbb{H}=$ { $1,59,42,125,135$ } so that $row_{AHP_B}(1)=42$, $row_{AHP_B}(49)=125$, $row_{AHP_B}(48)=125$, $row_{AHP_B}(180)=135$ and for the rest values of $\mathbb{K}$, $row_{AHP_B}(k)$ returns arbitrary elements of $\mathbb{H}$, for example $row_{AHP_B}(132)=135$ and $row_{AHP_B}(133)=1$. Therefore $row_{AHP_B}(x)=\sum_{i=1}^{6}y_iL_i(x)=20x^5+85x^4+37x^3+151x^2+168x+124$.

Also, $col_{AHP_B}:\mathbb{K}=$ { $1,49,48,180,132,133$ } $\to\mathbb{H}=$ { $1,59,42,125,135$ } so that $col_{AHP_B}(1)=1$ , $col_{AHP_B}(49)=1$, $col_{AHP_B}(48)=42$, $col_{AHP_B}(180)=1$ and for the rest values of $\mathbb{K}$ , $col_{AHP_B}(k)$ returns arbitrary elements of $\mathbb{H}$, for example $col_{AHP_B}(132)=135$ and $col_{AHP_B}(133)=1$. Therefore $col_{AHP_B}(x)=\sum_{i=1}^{6}y_iL_i(x)=18x^5+164x^4+180x^3+18x^2+164x$

and , $val_{AHP_B}:\mathbb{K}=$ { $1,49,48,180,132,133$ } $\to\mathbb{H}=$ { $1,59,42,125,135$ } so that $val_{AHP_B}(1)=\frac{5}{(5row_{AHP_B}^4(1))(5col_{AHP_B}^4(1))}=\frac{5}{5\times 42^4\times5\times1^4}=\frac{5}{48}\equiv 117\hspace{1mm}(\textrm{mod}\hspace{1mm}181)$, $val_{AHP_B}(49)=\frac{11}{(5row_{AHP_B}^4(49))(5col_{AHP_B}^4(49))}=\frac{11}{5\times 125^4\times5\times1^4}=\frac{11}{145}\equiv 55\hspace{1mm}(\textrm{mod}\hspace{1mm}181)$ , $val_{AHP_B}(48)=\frac{1}{(5row_{AHP_B}^4(48))(5col_{AHP_B}^4(48))}=\frac{1}{5\times 125^4\times5\times 42^4}=\frac{1}{25}\equiv 29\hspace{1mm}(\textrm{mod}\hspace{1mm}181)$ , $val_{AHP_B}(180)=\frac{26}{(5row_{AHP_B}^4(180))(5col_{AHP_B}^4(180))}=\frac{26}{5\times 135^4\times5\times 1^4}=\frac{26}{27}\equiv 68\hspace{1mm}(\textrm{mod}\hspace{1mm}181)$ and $val_{AHP_B}(k)=0$ for $k\in \mathbb{K}-\{1,49,48,180\}$. Therefore $val_{AHP_B}(x)=\sum_{i=1}^{6}y_iL_i(x)=86x^5+53x^4+34x^3+55x^2+176x+75$.

Now, $\hat{row_{AHP_B}}$, $\hat{col_{AHP_B}}$ and $\hat{val_{AHP_B}}$ are extensions of $row_{AHP_B}$, $col_{AHP_B}$ and $val_{AHP_B}$ so that are agree on $\mathbb{K}$.

Similarly, $row_{AHP_C}:\mathbb{K}=$ { $1,49,48,180,132,133$ } $\to\mathbb{H}=$ { $1,59,42,125,135$ } so that $row_{AHP_C}(1)=42$, $row_{AHP_C}(49)=125$, $row_{AHP_C}(48)=135$ and for the rest values of $\mathbb{K}$, $row_{AHP_C}(k)$ returns arbitrary elements of $\mathbb{H}$, for example $row_{AHP_C}(180)=125$, $row_{AHP_C}(132)=135$ and $row_{AHP_C}(133)=1$. Therefore $row_{AHP_C}(x)=\sum_{i=1}^{6}y_iL_i(x)=162x^5+62x^4+161x^3+169x^2+88x+124$.

Also, $col_{AHP_C}:\mathbb{K}=$ { $1,49,48,180,132,133$ } $\to\mathbb{H}=$ { $1,59,42,125,135$ }so that $col_{AHP_C}(1)=42$ , $col_{AHP_C}(49)=125$, $col_{AHP_C}(48)=135$, $col_{AHP_C}(180)=125$ and for the rest values of $\mathbb{K}$ , $col_{AHP_C}(k)$ returns arbitrary elements of $\mathbb{H}$, for example $col_{AHP_C}(132)=135$ and $col_{AHP_C}(133)=1$. Therefore $col_{AHP_C}(x)=\sum_{i=1}^{6}y_iL_i(x)=162x^5+62x^4+161x^3+169x^2+88x+124$.

and , $val_{AHP_C}:\mathbb{K}=$ { $1,49,48,180,132,133$ } $\to\mathbb{H}=$ { $1,59,42,125,135$ } so that $val_{AHP_C}(1)=\frac{1}{(5row_{AHP_C}^4(1))(5col_{AHP_C}^4(1))}=\frac{1}{5\times 42^4\times5\times42^4}=\frac{1}{27}\equiv 114\hspace{1mm}(\textrm{mod}\hspace{1mm}181)$, $val_{AHP_C}(49)=\frac{1}{(5row_{AHP_C}^4(49))(5col_{AHP_C}^4(49))}=\frac{1}{5\times 125^4\times5\times125^4}=\frac{1}{117}\equiv 82\hspace{1mm}(\textrm{mod}\hspace{1mm}181)$ , $val_{AHP_C}(48)=\frac{1}{(5row_{AHP_C}^4(48))(5col_{AHP_C}^4(48))}=\frac{1}{5\times 135^4\times5\times135 ^4}=\frac{1}{145}\equiv 5\hspace{1mm}(\textrm{mod}\hspace{1mm}181)$ and $val_{AHP_C}(k)=0$ for $k\in \mathbb{K}-\{1,49,48\}$. Therefore $val_{AHP_C}(x)=\sum_{i=1}^{6}y_iL_i(x)=65x^5+61x^4+157x^3+53x^2+16x+124$.

Now, $\hat{row_{AHP_C}}$, $\hat{col_{AHP_C}}$ and $\hat{val_{AHP_C}}$ are extensions of $row_{AHP_C}$, $col_{AHP_C}$ and $val_{AHP_C}$ so that are agree on $\mathbb{K}$.

Therefore, $\overrightarrow{O}_{AHP}=(124,88,169,161,62,162,14,169,109,32,150,128,84,180,111,22,79,$ $72,124,168,151,37,85,20,0,164,18,180,164,18,75,176,55,34,53,86,124,88,169,$ $161,62,162,124,88,169,161,62,162,124,16,53,157,61,65)$.

3- The Prover calculates commitment by using of $KZG$ commitment scheme as following:

$Com_{T}=\sum_{i=0}^{deg_T}a_ig\tau^i=\sum_{i=0}^{deg_T}a_ick(i)$ where $a_i$ is coefficient of $x^i$ in polynomial $T(x)$.

4- The Prover sends

$Com_{AHP}^0=\sum_{i=0}^{5}\hat{row_{AHP_{A_i}}}\hspace{1.1mm}ck(i)=124ck(0)+88ck(1)+169ck(2)+161ck(3)+62ck(4)+162ck(5) \equiv 166\hspace{1mm}(\textrm{mod}\hspace{1mm} 181)$

and similarly

$Com_{AHP}^1=36$, $Com_{AHP}^2=108$, $Com_{AHP}^3=58$, $Com_{AHP}^4=73$, $Com_{AHP}^5=157$, $Com_{AHP}^6=166$, $Com_{AHP}^7=166$ and $Com_{AHP}^8=36$.

AHP Commitment JSON File Example 1

IoT_Manufacturer_Name = "zkIoT" IoT_Device_Name = "MultiSensor" Device_Hardware_Version = "1.0" Firmware_Version = "1.0" Device Picture= <> Lines = [200, 350, 4000-4010]

{
    "commitmentId": String,
    "deviceType": String,
    "deviceIdType": String,
    "deviceModel": String,
    "manufacturer": String,
    "softwareVersion": String,
    "class": 32-bit Integer,
    "m": 6,
    "n": 5,
    "p": 181,
    "g": 2,   
        
    // PFR Commitment   
    "row_PFR_A": 64-bit Array,
    "col_PFR_A": 64-bit Array,
    "val_PFR_A": 64-bit Array,
    "row_PFR_B": 64-bit Array,
    "col_PFR_B": 64-bit Array,
    "val_PFR_B": 64-bit Array,
    "row_PFR_C": 64-bit Array,
    "col_PFR_C": 64-bit Array,
    "val_PFR_C": 64-bit Array,
    
    "Com_PFR0": 64-bit Integer,
    "Com_PFR1": 64-bit Integer,
    "Com_PFR2": 64-bit Integer,
    "Com_PFR3": 64-bit Integer,
    "Com_PFR4": 64-bit Integer,
    "Com_PFR5": 64-bit Integer,
    "Com_PFR6": 64-bit Integer,
    "Com_PFR7": 64-bit Integer,
    "Com_PFR8": 64-bit Integer,   

    // AHP Commitment       
    "row_AHP_A":[124,88,169,161,62,162],
    "col_AHP_A":[14,169,109,32,150,128],
    "val_AHP_A":[84,180,111,22,79,72],
    "row_AHP_B":[124,168,151,37,85,20],
    "col_AHP_B":[0,164,18,180,164,18],
    "val_AHP_B":[75,176,55,34,53,86],
    "row_AHP_C":[124,88,169,161,62,162],
    "col_AHP_C":[124,88,169,161,62,162],
    "val_AHP_C":[124,16,53,157,61,65],
    
    "Com_AHP0":166,
    "Com_AHP1":36,
    "Com_AHP2":108,
    "Com_AHP3":58,
    "Com_AHP4":73,
    "Com_AHP5":157,
    "Com_AHP6":166,
    "Com_AHP7":166,
    "Com_AHP8":36,
    
    "curve": "bn128",
    "PolynomialCommitment": "KZG"
}