Asymptotic Analysis Sample Clauses

Asymptotic Analysis. The offline cost of the protocol is dominated by setup and encryption phases. In the setup, sorting a list of N of vertices offline requires O(N log N) offline comparisons. During the encryption phase, each party encrypts the other party’s N labels and performs N XOR operations to retrieve its own encrypted labels. The garbled circuit performs the following computations for the percolate-and-match algorithm. Merg- ing two sorted lists of size N requires O(N log N) oblivious comparisons using a Batcher merge. For each iteration of the loop, it computes C (the number of components) min-circuits over m-sized lists to perform the percolation phase. We can find the minimum of a list with m elements using m comparisons; therefore, in total the min circuits require N = Cm comparisons per iteration. To perform matching in each iteration, we require O(N) pairwise comparisons. In addition, each Waksman network requires O(N log N) oblivi- ous swaps, and two permutation networks are computed per iteration. Therefore, each iteration of the loop procedure REVEALORDER(Lp1 , Xx0 , xx0 , xx0 , xx0 , xx0 ) // Merge, Permute, and Reveal the permuted index of each vertex to the vertex’s owner L ← BATCHERMERGE(Lp1 , Lp2 , CmpImg ) L ← XXXXXXX(X, xx0 , xx0 ) ← ← op1 []2N , op2 []2N ∈ for all j [2N] do if L [ j].party = p1 then ← op1 [ j] L [ j].img ← ⊕ op2 [ j] L [ j].img σp1 , j else ← ⊕ op1 [ j] L [ j].img σp2 , j ← op2 [ j] L [ j].img end if end for Output op1 to p1 and op2 to p2 end procedure procedure PERC&MATCH( X, xx0 , xx0 , xx0 , xx0 ) // Ip1 , Ip2 , outp1 , outp2 must be hard-coded // L is passed to this subcircuit already permuted ∈ − for all i [m 1] do ← L PERCOLATE(L, Ip1 ) ← L PERCOLATE(L, Ip2 ) ← L XXXXXXXXXXXXX(X, xx0 , xx0 ) ← L MATCH(L) ← L XXXXXXX(X, xx0 , xx0 ) end for op1 ←MASKFIXEDOUTPUTS(L, ρp2 , outp2 ) op2 ←MASKFIXEDOUTPUTS(L, ρp1 , outp1 ) Output op1 to p1 and op2 to p2 end procedure function XXXXXXX(X, x0, x0) return Waksman(Waksman(L, s1), s2) end function function XXXXXXXXXXXXX(X, x0, x0) return Waksman−1(Waksman−1(L, s2), s1) end function function PERCOLATE(L, I ) // I provides the indices in L that compose each component C in some party’s graph. ∈ for all idx(C) I do ← ψ []m ∈ for all j [m] do ← ψ[ j] L [idx(C)[ j]].lbl end for minlbl ← min j∈[m] ψ[ j] ∈ for all j [m] do ← L [idx(C)[ j]].lbl minlbl end for end for return L end function function MATCH(L) ∈ − for all i [2N 1] do ← lbl∗ min(L [i].lbl, L [i + 1].lbl) if L [i].img = L [i + 1].img then ...