Algorithm 2 Function form_AG

Require:i, $S_i$, ${\overline{\Theta }}_{i+1}^{(V)}$, ${\overline{\Gamma }}_{i+1}^{(V)}$, ${\Psi }_{i-1}^{(V)}$, ${\Gamma }_{i-1}^{(V)}$, ${\Pi }_{i-1}^{(V)}$, ${\Lambda }_{i-1}^{(V)}$

1:Define ${\mathcal{R}}_1^{(n)}$, ${\mathcal{R}}_1^{\prime (n)}$, ${\mathcal{R}}_2^{(n)}$, ${\mathcal{R}}_2^{\prime (n)}$, ${\mathcal{R}}_{1,2}^{(n)}$, ${\mathcal{R}}_{1,2}^{\prime (n)}$, ${\mathcal{I}}^{(n)}$, ${\mathcal{R}}_{\mathrm{S}}^{(n)}$, ${\mathcal{R}}_{\Lambda }^{(n)}$ (depending on the case)  2: if $i=1$ then ${\tilde{A}}_1[{\mathcal{I}}^{(n)}\cup {\mathcal{G}}_1^{(n)}\cup {\mathcal{G}}_{1,2}^{(n)}]\leftarrow S_1$  3: if $i\in [2,L-1]$ then ${\tilde{A}}_i[{\mathcal{I}}^{(n)}]\leftarrow S_i$  4: if $i=L$ then ${\tilde{A}}_L[{\mathcal{I}}^{(n)}\cup {\mathcal{G}}_2^{(n)}]\leftarrow S_L$  5:${\Psi }_{1,i-1}^{(V)}$, ${\Psi }_{2,i-1}^{(V)}\leftarrow {\Psi }_{i-1}^{(V)}$ (depending on the case)  6:${\Gamma }_{1,i-1}^{(V)}$, ${\Gamma }_{2,i-1}^{(V)}\leftarrow {\Gamma }_{i-1}^{(V)}$ (depending on the case)  7:${\overline{\Theta }}_{1,i+1}^{(V)}$, ${\overline{\Theta }}_{2,i+1}^{(V)}\leftarrow {\overline{\Theta }}_{i+1}^{(V)}$ (depending on the case)  8:${\overline{\Gamma }}_{1,i+1}^{(V)}$, ${\overline{\Gamma }}_{2,i+1}^{(V)}\leftarrow {\overline{\Gamma }}_{i+1}^{(V)}$ (depending on the case)  9: ${\tilde{A}}_i[{\mathcal{R}}_{1,2}^{(n)}]\leftarrow {\Gamma }_{1,i-1}^{(V)}\oplus {\overline{\Gamma }}_{1,i+1}^{(V)}$ 10: ${\tilde{A}}_i[{\mathcal{R}}_{1,2}^{\prime (n)}]\leftarrow {\Psi }_{2,i-1}^{(V)}\oplus {\overline{\Theta }}_{2,i+1}^{(V)}$ 11: if $i\in [1,L-1]$ then 12:    ${\tilde{A}}_i[{\mathcal{R}}_1^{(n)}]\leftarrow {\overline{\Theta }}_{1,i+1}^{(V)}$ 13:    ${\tilde{A}}_i[{\mathcal{R}}_1^{\prime (n)}]\leftarrow {\overline{\Gamma }}_{2,i+1}^{(V)}$ 14: end if 15: if $i\in [2,L]$ then 16:    ${\tilde{A}}_i[{\mathcal{R}}_2^{(n)}]\leftarrow {\Psi }_{1,i-1}^{(V)}$ 17:    ${\tilde{A}}_i[{\mathcal{R}}_2^{\prime (n)}]\leftarrow {\Gamma }_{2,i-1}^{(V)}$ 18:    ${\tilde{A}}_i[{\mathcal{R}}_{\mathrm{S}}^{(n)}]\leftarrow {\Pi }_{i-1}^{(V)}$ 19:    ${\tilde{A}}_i[{\mathcal{R}}_{\Lambda }^{(n)}]\leftarrow {\Lambda }_{i-1}^{(V)}$ 20: end if 21: ${\Pi }_i^{(V)}\leftarrow {\tilde{A}}_i[{\mathcal{I}}^{(n)}\cap {\mathcal{G}}_2^{(n)}]$ 22: ${\Lambda }_i^{(V)}\leftarrow {\tilde{A}}_i[{\mathcal{R}}_{\Lambda }^{(n)}]$ 23:return the sequences ${\tilde{A}}_i\left[{\mathcal{G}}^{(n)}\right]$, ${\Pi }_i^{(V)}$ and ${\Lambda }_i^{(V)}$