Secure Non-Linear Network Code Over a One-Hop Relay Network

Reduction Theorem for Secrecy over Linear Network Code for Active Attacks

“…This result can be regarded as a kind of reduction theorem because the secrecy analysis with contamination can be reduced to that without contamination. Indeed, when the linearity is not imposed, there is a counterexample of this reduction theorem [21].…”

“…Hence, to guarantee the secrecy under an arbitrary active attack, it is sufficient to show secrecy under the passive attack. However, there is an example of non-linear network such that this kind of reduction does not hold [21]. In fact, even when the network does not have synchronization so that the information transmission on an edges starts before the end of the information transmission on the previous edge, the above reduction theorem hold under the uniqueness condition.…”

“…whose realization in a concrete network will be discussed in Sections 3.2 and 3.3. Notice that the relations (20) and (21) with H E = 0 (only the relation ( 20)) were treated as the starting point of the paper [20] (the papers [22][23][24]). In this case, regarding n transmissions of one channel as n different edges, we consider the directed graph composed of nm 5 edges.…”

“…In other words, if the network graph has a cycle, there is a possibility that a good time ordering on the network flow that causes correlation with the delayed information. However, there is no relation between the relations (20) and (21) and the acyclic condition for the network graph, and the relations (20) and (21) directly depend on the time ordering on the network flow. That is, the acyclic condition for the network graph is not equivalent to the existence of the effect of delayed information.…”

“…This reduction theorem holds even under this kind of attack. In fact, there is an example having a non-linear node operation such that Eve can increase her performance to extract information from eavesdropping an edge outgoing an intermediate node by adding artificial information to an edge incoming the intermediate node [21] (The paper [21] also discusses linear code; it discusses only code a on one-hop relay network. Our results can be applied to general networks.)…”