Example 2

Assume the following sample code: $add\hspace{2mm} R_1, R_1, 5$ => Gate 1 $mul\hspace{2mm} R_2, R_2, R_{10}$ => Gate 2 $add\hspace{2mm} R_2,R_2,10$ => Gate 3 $mul\hspace{2mm} R_1,R_1,R_{19}$ => Gate 4

The constraints are as follows considering $p= 1678321$:

$R_1^{(2)}=R_1^{(1)}+5$ $R_2^{(2)}=R_2^{(1)}\times R_{10}^{(1)}$ $R_2^{(3)}=R_2^{(2)}+10$ $R_1^{(3)}=R_1^{(2)}\times R_{19}^{(1)}$

We continue with $n_g=4$ and $n_i=32$.

The Prover calculates square matrices $A$, $B$ and $C$ of order $n_g+n_i+1=4+32+1=37$ based on above construction:

B=\begin{bmatrix} 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&...&0&0&0&0\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&...&0&0&0&0\ .&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&...&.&.&.&.\ .&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&...&.&.&.&.\ .&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&...&.&.&.&.\ .&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&...&.&.&.&.\ 5&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&...&0&0&0&0\ 0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&..&0&0&0&0\ 10&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&...&0&1&0&0\ 0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&...&0&0&0&0 \end{bmatrix}

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