Algorithm 1 Generic encoding scheme

Require: Private and confidential messages $W_{1:L}$ and $S_{1:L}$; randomization sequences $R_{1:L}$; random sequence ${\Lambda }_0^{(X)}$; and secret keys ${\kappa }_{\Theta }^{(V)}$, ${\kappa }_{\Gamma }^{(V)}$, ${\kappa }_{{{\rm Y}\Phi }_{(1)}}^{(V)}$ and ${\kappa }_{{{\rm Y}\Phi }_{(2)}}^{(V)}$.

1:${\Psi }_0^{(V)}$, ${\Gamma }_0^{(V)}$, ${\Pi }_0^{(V)}$, ${\Lambda }_0^{(V)}$, ${\overline{\Theta }}_{L+1}^{(V)}$, ${\overline{\Gamma }}_{L+1}^{(V)}\leftarrow ⌀$  2: ${\tilde{A}}_1[{\mathcal{C}}^{(n)}]\leftarrow W_1$  3: ${\Psi }_1^{(V)},{\Gamma }_1^{(V)}\leftarrow {\tilde{A}}_1[{\mathcal{C}}^{(n)}]$  4:for$i=1$ to L do  5:    if $i\ne L$ then  6:        ${\tilde{A}}_{i+1}[{\mathcal{C}}^{(n)}]\leftarrow W_{i+1}$  7:        ${\Psi }_{i+1}^{(V)},{\Gamma }_{i+1}^{(V)},{\overline{\Theta }}_{i+1}^{(V)},{\overline{\Gamma }}_{i+1}^{(V)}\leftarrow \left({\tilde{A}}_{i+1}[{\mathcal{C}}^{(n)}],{\kappa }_{\Theta }^{(V)},{\kappa }_{\Gamma }^{(V)}\right)$  8:    end if  9:    ${\tilde{A}}_i\left[{\mathcal{G}}^{(n)}\right]$, ${\Pi }_i^{(V)}$, ${\Lambda }_i^{(V)}\leftarrow$form_AG$\left(i,S_i,{\overline{\Theta }}_{i+1}^{(V)},{\overline{\Gamma }}_{i+1}^{(V)},{\Psi }_{i-1}^{(V)},{\Gamma }_{i-1}^{(V)},{\Pi }_{i-1}^{(V)},{\Lambda }_{i-1}^{(V)}\right)$ 10:    if $i=1$then ${{\rm Y}}_{(1)}^{(V)}\leftarrow {\tilde{A}}_1\left[{\mathcal{H}}_V^{(n)}\cap {\left({\mathcal{L}}_{V|Y_{(1)}}^{(n)}\right)}^{\mathrm{C}}\right]$ 11:    if $i=L$then ${{\rm Y}}_{(2)}^{(V)}\leftarrow {\tilde{A}}_L\left[{\mathcal{H}}_V^{(n)}\cap {\left({\mathcal{L}}_{V|Y_{(2)}}^{(n)}\right)}^{\mathrm{C}}\right]$ 12:    for $j\in {\left({\mathcal{H}}_V^{(n)}\right)}^{\mathrm{C}}$ do 13:        if $j\in {\left({\mathcal{H}}_V^{(n)}\right)}^{\mathrm{C}}\setminus {\mathcal{L}}_V^{(n)}$ then 14:           ${\tilde{A}}_i(j)\leftarrow p_{A(j)|A^{1:j-1}}\left({\tilde{A}}_i(j)|{\tilde{A}}_i^{1:j-1}\right)$ 15:        else if $j\in {\mathcal{L}}_V^{(n)}$ then 16:           ${\tilde{A}}_i(j)\leftarrow {arg\; max}_{a\in \mathcal{V}}{p}_{A(j)|A^{1:j-1}}\left({\tilde{a}}_i(j)|{\tilde{A}}_i^{1:j-1}\right)$ 17:        end if 18:    end for 19:    ${\Phi }_{(1),i}^{(V)}\leftarrow {\tilde{A}}_i\left[{\left({\mathcal{H}}_V^{(n)}\right)}^{\mathrm{C}}\cap {\left({\mathcal{L}}_{V|Y_{(1)}}^{(n)}\right)}^{\mathrm{C}}\right]$ 20:    ${\Phi }_{(2),i}^{(V)}\leftarrow {\tilde{A}}_i\left[{\left({\mathcal{H}}_V^{(n)}\right)}^{\mathrm{C}}\cap {\left({\mathcal{L}}_{V|Y_{(2)}}^{(n)}\right)}^{\mathrm{C}}\right]$ 21:    ${\tilde{X}}_i^n,{\Lambda }_i^{(X)}\leftarrow \mathtt{pb\_ch\_pref}\left({\tilde{A}}_i^n{G}_n,R_i,{\Lambda }_{i-1}^{(X)}\right)$ 22: end for 23:Send $\left({\Phi }_{(k),i}^{(V)},{{\rm Y}}_{(k)}^{(V)}\right)\oplus {\kappa }_{{{\rm Y}\Phi }_{(k)}}^{(V)}$ to Receiver $k\in [1,2]$ 24: return ${\tilde{X}}_{1:L}^n$