Protocol 1: Submission Protocol

Let di,1, di,2, …, di,N be N data items collected by si during time slot t. For simplicity, we assume dmin ≤ di,1 ≤ … ≤ di,N ≤ dmax. Let Di = {dmin, di,1, …, di,N, dmax}. Then si performs the following steps.  (1) Compute the 0-1 code of each data item.  (2) Build the encrypted constraint chain Cτ,i of Di with τ and ki,t. Assuming Cτ,i = Fi,1⋈Fi,2⋈...$\bowtie$Fi,δ, we can compute δ and Fi,j as follows. $$\delta =\{\begin{array}{c}1\tau \ge N+2\\ 1+\lceil \frac{N+2-\tau }{\tau -1}\rceil \tau <N+2\end{array}$$ (6) $$F_{i,j}=\{\begin{array}{ll}{(d_{\min }|\left|d_{i,1}\right|\left|\mathrm{...}\right||d_{i,\tau -1})}_{k_{i,t}}& j=1\\ {(d_{i,(j-1)\cdot (\tau -1)}|\left|\mathrm{...}\right||d_{i,j\cdot (\tau -1)})}_{k_{i,t}}& 1<j<\delta \\ {(d_{i,(\delta -1)\cdot (\tau -1)}|\left|\mathrm{...}\right|\left|d_n\right||d_{\max })}_{k_{i,t}}& j=\delta \end{array}$$ (7)  (3) Compute the comparator of UB(Fi,j), where 1 ≤ j ≤ δ − 1. It means computing HNE0(UB(Fi,j)) and HNE1(UB(Fi,j)). Then, add the comparator set which contains the fewest elements into Ωi. Thus we have, Ωi = {min{HNE0(UB(Fi,j)), HNE1(UB(Fi,j)) | Fi,j ∈ Cτ,i ∧ 1 ≤ j ≤ δ − 1}} (8) where min(X, Y) denotes the set containing the fewest elements.  (4) Send the following message to M, where id(si) denotes the ID of si in networks. si → M: <id(si), t, Cτ,i, Ωi>