The Arbitrarily Varying Wiretap Channel—Secret Randomness, Stability, and Super-Activation

Abstract

We define the common randomness-assisted capacity of an arbitrarily varying wiretap channel (AVWC) when the eavesdropper is kept ignorant about the common randomness. We prove a multi-letter capacity formula for this model. We prove that, if enough common randomness is used, the capacity formula can be given a single-shot form again. We then consider the opposite extremal case, where no common randomness is available, and derive the capacity. It is known that the capacity of the system can be discontinuous under these circumstances. We prove here that it is still stable in the sense that it is continuous around its positivity points. We further prove that discontinuities can only arise if the legal link is symmetrizable and characterize the points where it is positive. These results shed new light on the design principles of communication systems with embedded security features. At last, we investigate the effect of super-activation of the message transmission capacity of AVWCs under the average error criterion. We give a complete characterization of those AVWCs that may be super-activated. The effect is thereby also related to the (conjectured) super-activation of the common randomness assisted capacity of AVWCs with an eavesdropper that gets to know the common randomness. Super-activation is based on the idea of wasting a few bits of non-secret messages in order to enable provably secret transmission of a large bulk of data, a concept that may prove to be of further importance in the design of communication systems. In this paper, we provide further insight into this phenomenon by providing a class of codes that is capacity achieving and does not convey any information to the eavesdropper.

Super-Activation as a Unique Feature of Secure Communication in Malicious Environments

“…There has been some work done in order to understand the secrecy capacity of the AVWC [15][16][17][18][19][20][21][22] which is briefly reviewed in the following. If CR is available, the transmitter and the legitimate receiver can coordinate their choice of encoder and decoder.…”

“…Such channel conditions appear for example in fast fading environments but also, more importantly, in scenarios in which malevolent adversaries actively influence or jam the legitimate transmission. This is the AVWC [15][16][17][18][19][20][21][22] which is depicted in Figure 1. Figure 1.…”

“…If CR is not available to the transmitter and legitimate receiver, unassisted codes must be used and the corresponding unassisted secrecy capacity has been completely characterized in terms of its CR-assisted secrecy capacity [16,21].…”

“…On the other hand, reducing the state set to have only one element, i.e., |S| = 1, the AVWC becomes a compound wiretap channel [34][35][36] (as the state remains constant for the entire duration of transmission) and the corresponding secrecy capacity becomes continuous [22]. Subsequently, the discontinuous behavior was then completely characterized in [21]. …”

“…Super-activation can occur for secure communication over AVCs and there exist orthogonal "useless" AVWCs, i.e., having zero secrecy capacity, whose overall secrecy capacity is strictly positive [17]. This phenomenon of super-activation and its resulting secrecy capacity have then been completely characterized in [21].…”