Byzantine Multiple Access Channels -- Part II: Communication With Adversary Identification

TL;DR

This work considers a two-user discrete memoryless multiple access channel where either user may deviate from the prescribed behaviour, and introduces the problem of determining the identity of a byzantine user (internal adversary) in a communication system.

Abstract

We introduce the problem of determining the identity of a byzantine user (internal adversary) in a communication system. We consider a two-user discrete memoryless multiple access channel where either user may deviate from the prescribed behaviour. Since small deviations may be indistinguishable from the effects of channel noise, it might be overly restrictive to attempt to detect all deviations. When neither user deviates, correct decoding is required. When one user deviates, the decoder must either output a pair of messages of which the message of the non-deviating user is correct or identify the deviating user. The users and the receiver do not share any randomness. The results include a characterization of the set of channels where communication is feasible, and an inner and outer bound on the capacity region. We also show that whenever the rate region has non-empty interior, the capacity region is same as the capacity region under randomized encoding, where each user shares independent randomness with the receiver. We also give an outer bound for this randomized coding capacity region.

Figures (11)

• Figure 1: Communication with adversary identification in a $\text{byzantine-MAC}$: Reliable decoding of both the messages is required when neither user deviates. When a user (say, user $\mathsf{B}$) deviates, the decoded message should either be correct for the honest user or the decoder should identify the deviating user (by outputting $\mathbf{b}\xspace$) with high probability.
• Figure 2: When \ref{['eq:spoof1']} holds for a $\text{byzantine-MAC}$$W$, for $(\vecx', \tilde{\vecx}, \tilde{\vecy})\in \cX^n\times\cX^n\times\cY^n$, the output distributions in the three cases above will be the same.
• Figure 3: A $\text{byzantine-MAC}$$W$ is $\mathsf{B}$-spoofable if for each $\tilde{x},\, \tilde{y},\, y', \,z$ the conditional output distributions $P(z|\tilde{x},\tilde{y},y')$ in \ref{['fig:spoof2a']}, \ref{['fig:spoof2b']} and \ref{['fig:spoof2c']} are the same.
• Figure 4: The channel output $\bm{z}$ is such that the tuples $(f_{\mathsf{A}\xspace}(m_{\mathsf{A}\xspace}), \vecy, \bm{z})$, $(\vecx, f_{\mathsf{B}\xspace}(\tilde{m}_{\mathsf{B}\xspace}), \bm{z})$ and $(f_{\mathsf{A}\xspace}(\tilde{m}_{\mathsf{A}\xspace}), \tilde{\vecy}, \bm{z})$ are consistent according to channel law for some $\vecy$, $\vecx$ and $\tilde{\vecy}$ as shown in (a), (b) and (c) respectively. Suppose $(f_{\mathsf{A}\xspace}(m_{\mathsf{A}\xspace}), {\vecy}, \bm{z}, {f_{\mathsf{A}\xspace}(\tilde{m}_{\mathsf{A}\xspace})}, {f_{\mathsf{B}\xspace}(\tilde{m}_{\mathsf{B}\xspace})})\in T^n_{X{Y}Z{\tilde{X}}{\tilde{Y}}}$ is such that $I({\tilde{X}}{\tilde{Y}};XZ|{Y})$ is small ( i.e. the Markov chain $\tilde{X}\tilde{Y}-{Y}-XZ$ approximately holds). Then, subfigure (d) is a plausible explanation where user $\mathsf{A}$ is honest with input message $m_{\mathsf{A}\xspace}\xspace$, and $\tilde{m}_{\mathsf{A}\xspace}$ and $\tilde{m}_{\mathsf{B}\xspace}$ can be explained by an attack strategy of user $\mathsf{B}$ (compare subfigures (d) and (a)).
• Figure 5: The set of overwritable $\text{byzantine-MACs}$ is a strict subset of the set of spoofable $\text{byzantine-MACs}$ which, in turn, is a strict subset of the set of symmetrizable $\text{byzantine-MACs}$.

Theorems & Definitions (52)

• Example 1: Binary erasure MAC YHKEG
• Definition 1: Adversary identifying code
• Remark 1
• Definition 2: Achievable rate pair and capacity region for communication with adversary identification
• Remark 2
• Definition 3: see Jahn81
• Theorem 1: Jahn81
• Theorem 2: Gubner
• Remark 3
• Theorem 3: AhlswedeC99