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sm/sm9/doc/验证sm9私钥.md

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+# 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 $de$:
+$$
+e(P_{pube} + [h_1]\cdot P_1, de) \overset{?}{=}  e(P_{pube}, P_2).
+$$
+
+