Upper Bounds on the Capacity of Binary Channels With Causal Adversaries

In this paper, we consider the communication of information in the presence of a causal adversarial jammer. In the setting under study, a sender wishes to communicate a message to a receiver by transmitting a codeword <b>x</b>=(<i>x</i>₁, ..., <i>x</i>_n) bit-by-bit over a communication channel. The sender and the receiver do not share common randomness. The adversarial jammer can view the transmitted bits <i>xi</i> one at a time and can change up to a <i>p</i>-fraction of them. However, the decisions of the jammer must be made in a causal manner. Namely, for each bit <i>x</i>_i, the jammer's decision on whether to corrupt it or not must depend only on <i>x</i>_j for <i>j</i> ≤ <i>i</i>. This is in contrast to the “classical” adversarial jamming situations in which the jammer has no knowledge of <b>x</b>, or knows <b>x</b> completely. In this study, we present upper bounds (that hold under both the average and maximal probability of error criteria) on the capacity which hold for both deterministic and stochastic encoding schemes.