A Multi-hop Multi-source Algebraic Watchdog

In our previous work “An Algebraic Watchdog for Wireless Network Coding”, we proposed a new scheme in which nodes can detect malicious behaviors probabilistically, police their downstream neighbors locally using overheard messages; thus, provide a secure global “self-checking network”. As the first building block of such a system, we focused on a two-hop network, and presented a graphical model to understand the inference process by which nodes police their downstream neighbors and to compute the probabilities of misdetection and false detection. In this paper, we extend the Algebraic Watchdog to a more general network setting, and propose a protocol in which we can establish “trust” in coded systems in a distributed manner. We develop a graphical model to detect the presence of an adversarial node downstream within a general two-hop network. The structure of the graphical model (a trellis) lends itself to well-known algorithms, such as Viterbi algorithm, that can compute the probabilities of misdetection and false detection. Using this as a building block, we generalize our scheme to multi-hop networks. We show analytically that as long as the min-cut is not dominated by the Byzantine adversaries, upstream nodes can monitor downstream neighbors and allow reliable communication with certain probability. Finally, we present preliminary simulation results that support our analysis.

💡 Research Summary

The paper extends the “Algebraic Watchdog” concept, originally introduced for a two‑hop wireless network with network coding, to general multi‑hop, multi‑source scenarios. In the original setting, a node could probabilistically detect misbehaving downstream neighbors by overhearing their transmissions, using a combination of coding coefficients, hash values, and error‑correcting codes. This work builds on that foundation by formalizing a graphical model—a trellis—through which a node can infer the set of possible linear combinations that a downstream node should transmit, and then compute the likelihood that the observed packet matches any of those possibilities.

The system model assumes each node transmits a packet consisting of a header (coding coefficients, hashes of neighboring packets, and a hash of its own payload) and a payload that is a linear combination of received packets, encoded with an (n, k) error‑correcting code of minimum distance d. The header is protected strongly enough to survive the noisy broadcast channel (modeled as a binary symmetric channel, BSC). An adversary can eavesdrop, inject, or corrupt packets but does not know the exact random errors introduced by the channel.

For each overheard packet ˜x_i, the receiving node constructs a transition matrix T_i that captures the probability that a particular codeword y generated the observation, conditioned on hash consistency. The size of T_i depends on the code redundancy (1 – H(d/n)) and the hash length δ. By chaining these matrices across the m intermediate nodes, a trellis with m layers is formed; each layer represents the cumulative linear combination Σ_{j=1}^{i} α_j x_j. Edges exist only when a feasible intermediate codeword links two states, and edge weights are given by the corresponding entries of T_i.

A Viterbi‑like dynamic programming algorithm traverses the trellis, computing for each state s at layer i the total probability w(s,i) of reaching that state. The final layer yields w(s,m) for all candidate linear combinations that the downstream node v_{m+1} could legitimately transmit. To evaluate the actual downstream transmission, an inverse transition matrix T^{-1} is defined, providing the probability of overhearing the observed packet ˜x_{m+1} given each candidate. The overall likelihood p* = Σ_{s} w(s,m)·T^{-1}(s,˜x_{m+1}) is then compared against a threshold t to decide whether v_{m+1} behaved maliciously.

The authors analytically derive the expected number of “matched” codewords (those consistent with both the overheard data and the hash) as 2^{n·