Symmetrizability for Myopic AVCs

Amitalok J. Budkuley, Bikash Kumar Dey, Sidharth Jaggi, Michael Langberg, Anand D. Sarwate, Carol Wang

Abstract: Myopic arbitrarily varying channels (AVCs) are point-to-point communication models in which a channel state is controlled by a malicious adversary (a jammer) who receives side-information about the transmitted codeword via a side-channel (wiretapping) and wishes to maximize the probability of error. Compared to standard "oblivious" AVCs, myopic AVCs can potentially use the side information to launch a more effective attack, lowering the capacity of the channel. In this paper, we define a novel property, myopic symmetrizability, and prove it is a sufficient condition for the capacity of any myopic AVC to be zero. We also study the sufficiently myopic setting, in which, roughly speaking, the jammer's side information reveals less information on the codeword transmitted than eventually available at the receiver. In this scenario we show that myopic symmetrizability is also a necessary condition for the capacity to equal zero, by providing a novel code construction using non-i.i.d. codebooks. A key technical lemma, interesting in its own right, is an argument showing that for any positive-rate code (whether for myopic AVCs or not) one can identify a corresponding distribution P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X,X'</sub> that is a convex combination of product distributions, and such that a constant fraction of pairs of codewords have an empirical distribution approximately equaling P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X,X'</sub> .