$begin{align}

& {{sigma }_{1}}=e(X,{{K}_{B}})/left( prodlimits_{iin {{I}_{B}}}{{{(e({{U}_{i}},{{L}_{B}})e({{D}_{i}},{{K}_{B,{{
ho }_{A}}(i)}}))}^{{{w}_{B}}(i)}}}
ight) \

& ext{ }=e({{g}^{{{s}_{A}}}},{{g}^{alpha }}{{g}^{a{{t}_{B}}}})/left( prodlimits_{iin {{I}_{B}}}{{{(e({{g}^{a{{lambda }_{i}}}}h_{{{
ho }_{A}}(i)}^{-{{r}_{i}}},{{g}^{{{t}_{B}}}})e({{g}^{{{r}_{i}}}},h_{{{
ho }_{A}}(i)}^{{{t}_{B}}}))}^{{{w}_{B}}(i)}}}
ight) \

& ext{ }=e({{g}^{{{s}_{A}}}},{{g}^{alpha }})e({{g}^{{{s}_{A}}}},{{g}^{a{{t}_{B}}}})/left( prodlimits_{iin {{I}_{B}}}{e{{({{g}^{a{{lambda }_{i}}}},{{g}^{{{t}_{B}}}})}^{{{w}_{B}}(i)}}e{{(h_{{{
ho }_{A}}(i)}^{-{{r}_{i}}},{{g}^{{{t}_{B}}}})}^{{{w}_{B}}(i)}}e{{({{g}^{{{r}_{i}}}},h_{{{
ho }_{A}}(i)}^{{{t}_{B}}})}^{{{w}_{B}}(i)}}}
ight) \

& ext{ }=e({{g}^{{{s}_{A}}}},{{g}^{alpha }})e({{g}^{{{s}_{A}}}},{{g}^{a{{t}_{B}}}})/e{{({{g}^{a}},{{g}^{{{t}_{B}}}})}^{sumlimits_{iin {{I}_{B}}}{{{lambda }_{i}}{{w}_{B}}(i)}}} \

& ext{ }=e({{g}^{{{s}_{A}}}},{{g}^{alpha }})e({{g}^{{{s}_{A}}}},{{g}^{a{{t}_{B}}}})/e{{({{g}^{a}},{{g}^{{{t}_{B}}}})}^{{{s}_{A}}}} \

& ext{ }={{(g_{T}^{alpha })}^{{{s}_{A}}}}, \

& {{sigma }_{2}}={{(g_{T}^{alpha })}^{{{s}_{B}}}}, \

& {{sigma }_{3}}={{X}^{{{s}_{B}}}}={{g}^{{{s}_{A}}{{s}_{B}}}}. \

end{align}$