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How to Construct Strongly Secure Network Coding Scheme

Part of the Lecture Notes in Computer Science book series (LNSC,volume 8317)

Abstract

We say that a network coding scheme is strongly \(1\)-secure if a source node \(s\) can multicast \(n\) field elements \(\{m_1, \cdots , m_n\}\) to a set of sink nodes \(\{t_1, \cdots , t_q\}\) in such a way that any single edge leaks no information on any \(S \subset \{m_1, \cdots , m_n\}\) with \(|S|=n-1\), where \(n=\min _{t_i}\)max-flow\((s,t_i)\) is the maximum transmission capacity. We also say that a strongly \(h\)-secure network coding scheme is strongly \((h+1)\)-secure if any \(h+1\) edges leak no information on any \(S \subset \{m_1, \cdots , m_n\}\) with \(|S|=n-(h+1)\).

In this paper, we show the first explicit algorithm which can construct strongly \(k\)-secure network coding schemes. In particular, it runs in polynomial time for fixed \(k\).

Keywords

  • Network coding
  • Strongly secure
  • Construction

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Fig. 1.
Fig. 2.

Notes

  1. 1.

    See “Time Complexity” of [14, page 313].

  2. 2.

    Our strongly \((n-1)\)-secure is their strongly \(0\)-secure [5].

  3. 3.

    They instead analyzed a case such that the source node multicasts \(n'<n\) field elements [5, Sec. 6].

  4. 4.

    In the scheme of Kurihara et al. [7], \(T \ge n'+n\) if the source nodes multicasts \(n'\) messages. So \(T \ge 2n\) if the source nodes multicasts \(n\) messages.

  5. 5.

    Tang et al. [14] did not show such an algorithm.

  6. 6.

    Since the first row of \(U^*\) consists of nonzero elements, it holds that \(rank(U^*_{A, \{1\}})=1\) for any \(A \in \mathsf{Rank}_{2}\). Therefore there exists a \(\{1\}\)-zero projection of \(U^*\) from Lemma 3.

  7. 7.

    Since the 2nd row of \(U^*\) consists of nonzero elements, it holds that \(rank(U^*_{A, \{2\}})=1\) for any \(A \in \mathsf{Rank}_{2}\). Therefore there exists a \(\{2\}\)-zero projection of \(U^*\) from Lemma 3.

References

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Correspondence to Kaoru Kurosawa .

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Appendices

A Example of Secure Linear Network Coding Schemes

Fig. 3.
figure 3

\(1\)-Secure linear network coding scheme (mod3)

Fig. 4.
figure 4

Strongly \(1\)-secure linear network coding scheme (mod5)

Fig. 5.
figure 5

Strongly \(2\)-secure linear network coding scheme (mod11)

B Example of \(D\)-Zero Projection

Consider

$$ U=\left( \begin{array}{ccc} 1 &{} 1 &{} 2 \\ 0 &{} 1 &{} 1 \end{array} \right) $$

over \(\mathsf{F}_5\). Then

$$ U_{\{1,2\}} = \left( \begin{array}{cc} 1 &{} 1 \\ 0 &{} 1 \end{array} \right) , U_{\{1,3\}} = \left( \begin{array}{cc} 1 &{} 2 \\ 0 &{} 1 \end{array} \right) , U_{\{2,3\}} = \left( \begin{array}{cc} 1 &{} 2 \\ 1 &{} 1 \end{array} \right) . $$

Therefore

$$\begin{aligned} \mathsf{Rank}_2=\{\{1,2\}, \{1,3\}, \{2,3\}\} \end{aligned}$$

because

$$\begin{aligned} rank(U_{\{1,2\}}) = rank(U_{\{1,3\}})=rank(U_{\{2,3\}})=2. \end{aligned}$$

Next

$$\begin{aligned} U_{\{1,2\}, \{1\}}&= (1,1) \\ U_{\{1,3\}, \{1\}}&= (1,2) \\ U_{\{2,3\}, \{1\}}&= (1,2) \end{aligned}$$

Therefore from Lemma 3, there exists a \(\{1\}\)-zero projection of \(W\) because

$$\begin{aligned} rank(U_{\{1,2\}, \{1\}})= rank(U_{\{1,3\}, \{1\}})=rank(U_{\{2,3\}, \{1\}})=1 \end{aligned}$$

Let

$$\begin{aligned} \mathbf{b}_{\{1,2\}}&= -(1,0)^T+(1,1)^T=(0, 1)^T \\ \mathbf{b}_{\{1,3\}}&= -2\cdot (1,0)^T+(2,1)^T=(0, 1)^T \\ \mathbf{b}_{\{2,3\}}&= -2\cdot (1,1)^T+(2,1)^T=(0, -1)^T. \end{aligned}$$

Then \(\mathbf{b}_A\) is a \(\{1\}\)-zero projection of \(U_A\) for \(A=\{1,2\}, \{1,3\}\) and \(\{2,3\}\). Therefore \(W\) such that

$$ W=(\mathbf{b}_{\{1,2\}}, \mathbf{b}_{\{1,3\}}, \mathbf{b}_{\{2,3\}}) =\left( \begin{array}{ccc} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} -1 \end{array} \right) $$

is a \(\{1\}\)-zero projection of \(U\). Finally the second row of \(W\) consists of nonzero elements. Therefore

$$\begin{aligned} rank(U_{\{1,2\},\{1,2\}})=rank(U_{\{1,3\},\{1,2\}})=rank(U_{\{2,3\},\{1,2\}})=2 \end{aligned}$$

from Lemma 4.

C Proof of Theorem 3

At line 5 and line 6, we can show that there exists such a \(D\)-zero projection \(W_D\) of \(U^*\) by induction on \(i\) based on Lemma 3. (See the footnotes of Sec. 5.3.) At line 8, the \((i+1)\)th row of \(Y_i\) consists nonzero elements, and the other rows are the same as those of \(X_i\). Therefore the final \(Y_{n-1}\) looks as follows, where \(U^*=T \cdot U\). It is also easy to see that \(W_D^*\) is a \(D\)-zero projection of \(U^*\) for all \(D \subset \{1, \cdots , n\}\) such that \(|D|<k\) (Fig. 6).

Fig. 6.
figure 6

The final \(Y_{n-1}\)

In the above figure, since all the elements of \(U^*\) are nonzeros, it is clear that

$$\begin{aligned} rank(U^*_A)=rank(U^*_{A, \{1\}})= \cdots = rank(U^*_{A, \{n\}})=1 \end{aligned}$$

for any \(A \in \mathsf{Rank}_1\). Next from Lemma 4 and from the above figure, we have

$$\begin{aligned} rank(U^*_A)=rank(U^*_{A, \{1,2\}})= \cdots = rank(U^*_{A, \{n-1,n\}})=2 \end{aligned}$$

for any \(A \in \mathsf{Rank}_2\). Similarly, we can see that

$$\begin{aligned} rank(U^*_A)=rank(U^*_{A,B}) \end{aligned}$$

for any \(A \in \mathsf{Rank}_j\) and any \(B \subset \{1, \cdots , n\}\) such that \(|B|= j\) for \(j=1, \cdots , k\). Therefore \(U^*\) is strongly \(k\)-secure from Proposition 3.

Lemma 5

Let \(L_k\) be the number of columns of the final \(X\). Then

$$\begin{aligned} L_k \le |\mathcal{E}| + \sum _{i=1}^{k-1} {n-1 \atopwithdelims ()i} {|\mathcal{E}| \atopwithdelims ()i+1}. \end{aligned}$$

Proof

Let \(\#A\) denote the number of columns of a matrix \(A\). Then

$$\begin{aligned} L_k=\#U + \sum _{h=1}^{k-1} {n-1 \atopwithdelims ()h} \sum _{|D|=h} \#W_D \end{aligned}$$

If \(|D|=h\), then we have

$$\begin{aligned} \#W_D = |\mathsf{Rank}_{h+1}| \le {|\mathcal{E}| \atopwithdelims ()h+1} \end{aligned}$$

from Eq. (1), Therefore we have this lemma. \(\square \)

Therefore at line 7, we can compute each \(T_i\) if \(|\mathsf{F}| \ge L \ge L_k\) in time \(O(nL)\) from Theorem 1. To compute all \(T_i\), it takes time \(O(n^2L)\).

At line 5, it takes \(O(n|D|^2)\) time to compute each \(W_D\). To compute all \(W_D\), it takes time \(O(\sum _{i=1}^k ni^2 {n-1 \atopwithdelims ()i})\) which is bounded by \(O(n^2L)\).

Finally the time complexity of line 2 and line 9 is bounded by \(O(n^2L)\). Therefore our algorithm runs in time \(O(n^2L)\).  

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Kurosawa, K., Ohta, H., Kakuta, K. (2014). How to Construct Strongly Secure Network Coding Scheme. In: Padró, C. (eds) Information Theoretic Security. ICITS 2013. Lecture Notes in Computer Science(), vol 8317. Springer, Cham. https://doi.org/10.1007/978-3-319-04268-8_1

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