# 判断标识，用户私钥和主公钥的一致性
Let $h_1 = H_1(ID||hid, N)$, for the user sign key $ds$, 
$$
ds = \left[ \frac{ks}{h_1+ks} \right]\cdot P_1,
$$
and $P_{pubs} = [ks]\cdot P_2$, we have
$$
\begin{array}{rcl}
& & e(ds, P_{pubs} + [h_1]\cdot P2) \\
&=& e(\left[ \frac{ks}{h_1+ks} \right]\cdot P_1, [ks+h_1]\cdot P_2)\\
&=& e(P_1, P_{2})^{ks} \\
&=& e(P_1, P_{pubs}).
\end{array}
$$


Thus, we need to compute if the following equation holds:
$$
e(ds, P_{pubs} + [h_1]\cdot P_2) \overset{?}{=}  e(P_1, P_{pubs}).
$$

The same for encryption user key $de$:
$$
e(P_{pube} + [h_1]\cdot P_1, de) \overset{?}{=}  e(P_{pube}, P_2).
$$


