Algorithm 2 Access Tree Description.

Input:

$T_P$: an access tree;

Output:

$\{s,q_{lf}\left(0\right)|lf\mathrm{is}\mathrm{a}\mathrm{leaf}\mathrm{node}\mathrm{of}{T}_P\}$: $s\in {\mathbb{Z}}_p$ is a randomly-picked secret integer;

1: for all u in $T_P$ do

2:   define a polynomial $q_u\left(x\right)={\displaystyle \sum _{i=0}^{k_u-1}}{a}_{ui}{x}^i$, where the coefficients $a_{ui}$ are undetermined;

3: end for

4: Pick a random integer s

5: Set $a_{R0}=q_R\left(0\right)=s$, where R is the root node of $T_P$;

6: Set other coefficients of $q_R\left(x\right)$, i.e., $a_{Ri},i=1,2,\dots ,k_R-1$, by randomly-picked secret integers;

7: From top to bottom, set all of the coefficients of other nodes (except for the root node) that satisfy the following equation; $$q_u\left(0\right)={q_{parent}}_{\left(u\right)}\left(index\left(u\right)\right).$$

return $\{s,q_{lf}\left(0\right)|lf\mathrm{is}\mathrm{a}\mathrm{leaf}\mathrm{node}\mathrm{of}{T}_p\}$.