Reduction Theorem for Secrecy over Linear Network Code for Active Attacks

Abstract

We discuss the effect of sequential error injection on information leakage under a network code. We formulate a network code for the single transmission setting and the multiple transmission setting. Under this formulation, we show that the eavesdropper cannot increase the power of eavesdropping by sequential error injection when the operations in the network are linear operations. We demonstrated the usefulness of this reduction theorem by applying a concrete example of network.

Advantage of the Key Relay Protocol Over Secure Network Coding

“…This lemma is a variant of Theorem 1, Ref. [16], which was previously obtained for the secure network coding (SNC). As the proof is essentially the same as in Ref.…”

“…As the proof is essentially the same as in Ref. [16], we here only give a sketch: Suppose for example that the active adversary modifies a local key r e ′ to r e ′ + ∆r, which is to be input to a node v. With v being linear, v's subsequent outputs all change linearly in ∆r; for example, a public message p e , which v outputs, changes to p e + f (∆r) with f being a linear function. Since those linear response to tampering, such as f (∆r), are all predictable, we can conclude that the adversary gains nothing by tampering with communication.…”

Advantage of the key relay protocol over secure network coding

“…This lemma is a variant of Theorem 1, Ref. [11], which was previously obtained for the secure network coding (SNC). As the proof is essentially the same as in Ref.…”

“…As the proof is essentially the same as in Ref. [11], we here only give a sketch: Suppose for example that the active adversary modifies a local key r e to r e + ∆r, which is to be input to a node v. With v being linear, v's subsequent outputs all change linearly in ∆r; for example, a public message p e , which v outputs, changes to p e + f (∆r) with f being a linear function. Since those linear response to tampering, such as f (∆r), are all predictable, we can conclude that the adversary gains nothing by tampering with communication.…”

“…[12]). Relations (11) and (12) claim that when two bits a 1 , a 2 are fixed, three bits e 1 , e 4 , e 6 are in one-to-one correspondence with each other, and are uniformly distributed individually. Hence e i can be expressed as…”