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 \(k\).
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Notes
- 1.
See “Time Complexity” of [14, page 313].
- 2.
Our strongly \((n-1)\)-secure is their strongly \(0\)-secure [5].
- 3.
They instead analyzed a case such that the source node multicasts \(n'<n\) field elements [5, Sec. 6].
- 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.
Tang et al. [14] did not show such an algorithm.
- 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.
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.
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Appendices
A Example of Secure Linear Network Coding Schemes
B Example of \(D\)-Zero Projection
Consider
over \(\mathsf{F}_5\). Then
Therefore
because
Next
Therefore from Lemma 3, there exists a \(\{1\}\)-zero projection of \(W\) because
Let
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
is a \(\{1\}\)-zero projection of \(U\). Finally the second row of \(W\) consists of nonzero elements. Therefore
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).
In the above figure, since all the elements of \(U^*\) are nonzeros, it is clear that
for any \(A \in \mathsf{Rank}_1\). Next from Lemma 4 and from the above figure, we have
for any \(A \in \mathsf{Rank}_2\). Similarly, we can see that
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
Proof
Let \(\#A\) denote the number of columns of a matrix \(A\). Then
If \(|D|=h\), then we have
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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