Algorithm 4 Decoding at legitimate Receiver 1

Require:${{\rm Y}}_{(1)}^{(V)}$, ${\Phi }_{(1),1:L}^{(V)}$, ${\kappa }_{\Theta }^{(V)}$ and ${\kappa }_{\Gamma }^{(V)}$, and ${\tilde{Y}}_{(1),1:L}^n$.

1: ${\widehat{A}}_1^n\leftarrow \left({{\rm Y}}_{(1)}^{(V)},{\Phi }_{(1),1}^{(V)},{\tilde{Y}}_{(1),1}^n\right)$  2: ${\widehat{\Lambda }}_{2:L}^{(V)}\leftarrow {\widehat{A}}_1\left[{\mathcal{R}}_{\Lambda }^{(n)}\right]$  3:for$i=1$ to $L-1$ do  4:    ${\widehat{\Psi }}_i^{(V)}\leftarrow {\widehat{A}}_i[{\mathcal{C}}_2^{(n)}]$  5:    ${\widehat{\Gamma }}_i^{(V)}\leftarrow {\widehat{A}}_i[{\mathcal{C}}_{1,2}^{(n)}]$  6:    ${\widehat{\overline{\Theta }}}_{i+1}^{(V)}\leftarrow \left({\widehat{A}}_i[{\mathcal{R}}_1^{(n)}],{\widehat{A}}_i[{\mathcal{R}}_{1,2}^{\prime (n)}]\oplus {\widehat{\Psi }}_{2,i-1}^{(V)}\right)$  7:    ${\widehat{\Theta }}_{i+1}^{(V)}\leftarrow {\widehat{\overline{\Theta }}}_{i+1}^{(V)}\oplus {\kappa }_{\Theta }^{(V)}$  8:    ${\widehat{\overline{\Gamma }}}_{i+1}^{(V)}\leftarrow \left({\widehat{A}}_i[{\mathcal{R}}_{1,2}^{(n)}]\oplus {\widehat{\Gamma }}_{1,i-1}^{(V)},{\widehat{A}}_i[{\mathcal{R}}_1^{\prime (n)}]\right)$  9:    ${\widehat{\Gamma }}_{i+1}^{(V)}\leftarrow {\widehat{\overline{\Gamma }}}_{i+1}^{(V)}\oplus {\kappa }_{\Gamma }^{(V)}$ 10:    ${\widehat{\Pi }}_i^{(V)}\leftarrow {\widehat{A}}_i[{\mathcal{I}}^{(n)}\cap {\mathcal{G}}_2^{(n)}]$ 11:    ${\widehat{{\rm Y}}}_{(1),i+1}^{\prime (V)}\leftarrow \left({\widehat{\Psi }}_{1,i}^{(V)},{\widehat{\Gamma }}_{2,i}^{(V)},{\widehat{\Theta }}_{i+1}^{(V)},{\widehat{\Gamma }}_{i+1}^{(V)},{\widehat{\Pi }}_i^{(V)},{\widehat{\Lambda }}_i^{(V)}\right)$ 12:    ${\widehat{A}}_{i+1}^n\leftarrow \left({\widehat{{\rm Y}}}_{(1),i+1}^{\prime (V)},{\Phi }_{(1),i+1}^{(V)},{\tilde{Y}}_{(1),i+1}^n\right)$ 13: end for