Algorithm 3 Sign

Input: $pp$, $(Pk,Sk)$, and $\mu$ Output: ID-US e   1:Randomly chooses $y\in D_{\sigma }^m$;   2:Calculates $U=H_3\left(\mu \right)\in Z_q^{n\times m}$;   3:Calculates $e_1=H_4(X+Aymodq,U)$;   4:Calculates $e_2=(y+s_{I{D}_i}{e}_1)modq$;   5:Calculates $e_3=Uxmodq$;   6:Opens ID-US $e=(e_1,e_2,e_3,\mu )$.