Upper bounds for causal adversaries

Bikash Dey, Mike Langberg, Sid Jaggi, and I submitted an extended version of our (now accepted) ISIT paper on new upper bounds for binary channels with causal adversaries to the IT Transactions. The model is pretty straightforward : Alice (the encoder) transmits a message to Bob (the receiver) encoded over n uses of a binary input, binary output channel. The channel is controlled by Calvin (an adversary) who sequentially looks at each bit and can decide whether or not to flip it, up to pn total bit flips. That is, Calvin is causal : the decision to flip bit i is based on the knowledge of bits 1, 2, \ldots, i. What we show is a new upper bound on the capacity of this channel. Let \alpha(p,\bar{p}) = 1-4(p-\bar{p}). Then

C \le \min_{\bar{p} \in [0,p]} \left[ \alpha(p,\bar{p})\left(1-H\left(\frac{\bar{p}}{\alpha(p,\bar{p})}\right)\right) \right]

This is what it looks like:

New Upper Bound for Causal Adversaries

Plot of the new upper bound


So clearly causal adversaries are worse than i.i.d. noise (the 1 - H(p) bound).

To show such a bound we have to propose a new attack for the adversary. We call our attack “babble and push.” It operates in two phases. The first phase is of length \ell channel uses and the second of length n - \ell. Let \mathbf{x}(m) be the codeword for message m.

  1. (Babble) Calvin chooses a random subset \bar{p} n indices uniformly from all (\bar{p} n)-subsets of \{1, 2, \ldots, \ell\} and flips bit i for i \in \Gamma.
  2. (Push) Calvin finds all possible codewords which are consistent with what Bob has received in the first phase:

    B_{\mathbf{y}^{\ell}} = \{ u : d_H(\mathbf{y}^{\ell}, \mathbf{x}^{\ell}(u))=\bar{p}n \},

    and selects an element \hat{u} \in B_{\mathbf{y}_1} uniformly at random. For the second phase, Calvin selectively pushes the received codeword towards \mathbf{x}(\hat{u}) — if the transmitted codeword and the selected codeword match, he does nothing, and if they do not match he flips the bit with probability 1/2.

Analyzing this scheme amounts to showing that Calvin can render the channel “symmetric.” This is a common condition in arbitrarily varying channels (AVCs), a topic near and dear to my heart. Basically Bob can’t tell the difference between the real codeword and the transmitted codeword, because under Calvin’s attack, the chance that Alice chose u and Calvin chose \hat{u} is the same as the chance Alice chose \hat{u} and Calvin chose {u}. To establish this symmetry condition requires some technical excursions which are less fun to blog about, but were fun to figure out.

It’s relatively clear that this approach would extend to more general AVCs, which we could work on for the future. What is neat to me is that this shows how much value Calvin can derive by knowing the current input bit — by forcing additional uncertainty to Bob during the babble phase, Calvin can buy some time to more efficiently use his bit flipping budget in the second phase.

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