Algoruthm 1 Recovery ${\overline{d}}_c$ captured at segment c

Input: $(a_1,a_2,···,a_m)$, ${\beta }_c$, $\phi$, $\delta$, $M_q^j$ and $N_q^j$

Output:  ${\overline{d}}_c$

begin:

set $X_1=M_q^j$, $X_2=N_q^j$;

for $k=m$ to c do

$d_k=\frac{X_1-X_{1}mod{a}_k}{a_k}$, $e_k=\frac{X_2-X_{2}mod{a}_k}{a_k}$;

$X_1=X_{1}mod{a}_k$, $X_2=X_{2}mod{a}_k$;

return ${\overline{d}}_c=\frac{d_c-{\beta }_c-\delta \phi }{e_c-{\beta }_c-\delta \phi }$.

end