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Secure Network Code for Adaptive and Active Attacks with No-Randomness in Intermediate Nodes

Ning Cai, Masahito Hayashi

TL;DR

It is shown that active and adaptive attacks cannot improve the performance of the eavesdropper when the code is linear, and a non-linear example is given, in which an adaptive attack improves the performance.

Abstract

In secure network coding, there is a possibility that the eavesdropper can improve her performance when she changes (contaminates) the information on the attacked edges (active attack) and chooses the attacked edges adaptively (adaptive attack). We analyze the security for network code over such types of attacks. We show that active and adaptive attacks cannot improve the performance of the eavesdropper when the code is linear. Further, we give a non-linear example, in which an adaptive attack improves the performance of the eavesdropper. We derive the capacity for the unicast case and the capacity region for the multicast case or the multiple multicast case in several examples of relay networks, beyond the minimum cut theorem, when no additional random number is allowed as scramble variables in the intermediate nodes. No prior study compared the difference of the capacity and the capacity region between the existence and the non-existence of randomness in the intermediate nodes under these network models even with non-adaptive and non-active attacks.

Secure Network Code for Adaptive and Active Attacks with No-Randomness in Intermediate Nodes

TL;DR

It is shown that active and adaptive attacks cannot improve the performance of the eavesdropper when the code is linear, and a non-linear example is given, in which an adaptive attack improves the performance.

Abstract

In secure network coding, there is a possibility that the eavesdropper can improve her performance when she changes (contaminates) the information on the attacked edges (active attack) and chooses the attacked edges adaptively (adaptive attack). We analyze the security for network code over such types of attacks. We show that active and adaptive attacks cannot improve the performance of the eavesdropper when the code is linear. Further, we give a non-linear example, in which an adaptive attack improves the performance of the eavesdropper. We derive the capacity for the unicast case and the capacity region for the multicast case or the multiple multicast case in several examples of relay networks, beyond the minimum cut theorem, when no additional random number is allowed as scramble variables in the intermediate nodes. No prior study compared the difference of the capacity and the capacity region between the existence and the non-existence of randomness in the intermediate nodes under these network models even with non-adaptive and non-active attacks.

Paper Structure

This paper contains 27 sections, 19 theorems, 77 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Adaptive and active attack for general network
  3. Formulation and reduction to non-adaptive attack
  4. Proofs of Theorem \ref{['T7B']} and Lemma \ref{['L5-10']}
  5. Network with powerful adaptive attack
  6. Asymptotic formulation
  7. Relay network
  8. Formulation and capacities
  9. Converse part
  10. Code construction to achieve capacity $C_{1,L,D}$
  11. Code construction to achieve capacity $C[(\gamma_i)_i]_{L,D}$ and $C_{2,L,D}$
  12. Scalar linearity
  13. Important lemmas
  14. homogeneous multicast relay network
  15. Formulation and capacity regions
  16. ...and 12 more sections

Key Result

Theorem 1

Assume that a network code $\Phi$ satisfies the following condition. Given an arbitrary element $\bm{s}=\{s_1,\ldots,s_\zeta\} \in \mathfrak{A}^0$, we have for any element $(z_{1}, \ldots, z_{\zeta})$. Then, any general adaptive attack $\bm{\alpha} \in \mathfrak{A}^2$ satisfies $\square$

Figures (5)

  • Figure 1: Non-linear code.
  • Figure 2: Network with equality in \ref{['E9-2B']}. Node 1 is the source node and Node 5 is the terminal node. Node 4 is a pseudo source node. Hence, $\mathop{\rm mincut}\nolimits_2=1$ and $\mathop{\rm mincut}\nolimits_1=2$. It also shows a linear code to achieve $\mathop{\rm mincut}\nolimits_1-r$ when $r=1$. The source node (Node 1) has the message $M$ and a scramble variable $L_1$. The pseudo source node (Node 4) has another scramble variable $L_2$. Even when Eve wiretaps any one edge, she cannot obtain any information for the message $M$.
  • Figure 3: Unicast relay network.
  • Figure 4: homogeneous multicast relay network
  • Figure 5: homogeneous multiple multicast relay network

Theorems & Definitions (27)

  • Theorem 1
  • Corollary 1
  • Lemma 1
  • Theorem 2
  • Proposition 1: HOKC
  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 2
  • Remark 4
  • ...and 17 more