Algorithm 1 Anonymous Tree Generation

1:function$Create$ ($node$,$size$) 2:    if $size\ne 1$ then 3:        if $size$ is odd then 4:           Let last node of $newNode$ be $node\left[size\right]$ 5:        end if 6:        for $i=1;i<size;i+=2$ do 7:           $newNode\left[\right(i+1)/2].sk\leftarrow t\left(node\right[i].sk·node[i+1].pk)$ 8:           $newNode\left[\right(i+1)/2].pk\leftarrow newNode\left[\right(i+1)/2].sk·P$ 9:           Let $newNode\left[\right(i+1)/2]$ be the parent of $node\left[i\right]$ and $node[i+1]$ 10:        end for 11:        return $Create(newNode,size(newNode\left)\right)$ 12:    else 13:        return $node$ 14:    end if 15:end function 16:procedure$Init$($i{k}_A,IK,EK$,size n) 17:    $size\leftarrow 2n,suk\stackrel{\$}{\leftarrow }{\mathbb{Z}}_q^{*},SUK\leftarrow suk·P,c{k}_0\stackrel{\$}{\leftarrow }\mathcal{K}$ 18:    Send $I{K}_A,SUK,c{k}_0$ to other members through trust third-party 19:    for each $i\in [1,2n]$ do 20:        if $imod2=0$ or $i=A$ then 21:           $node\left[i\right].sk\stackrel{\$}{\leftarrow }{\mathbb{Z}}_q^{*}$ 22:        else 23:           $node\left[i\right].sk\leftarrow KeyExchange(i{k}_A,I{K}_i,suk,E{K}_i)$ 24:        end if 25:    end for 26:    $agt\leftarrow Create(node,size)$, $gpk\leftarrow agt$, delete all $sk$ from $gpk$ 27:    Run ${\sigma }_0\leftarrow Sig(i{k}_A,gp{k}_1)$ and broadcast $(gp{k}_1,{\sigma }_0)$ to other group members 28:    return $gpk,agt,node$ 29:end procedure