On the Distinguishability of Distance-Bounded Permutations in Ordered Channels

Ordered channels, such as those provided by Internet protocol and transmission control protocol protocols, rely on sequence numbers to recover from packet reordering due to network dynamics. The existence of covert channels in any ordered channel is a well-known fact: Two parties can reorder the elements (packets) to be sent according to some predefined code. Schemes based on distance-bounded permutations have been proposed for steganographic communication with the aim of keeping and controling the increase of latency due to reordering. In this paper, we demonstrate that distance-bounded permutations are highly anomalous from a metric point of view. Our analysis is based on the study of the distribution of distances between normal permutations generated by the channel, and those produced when embedding hidden information. We provide results for four different distances: Kendall´s tau, Spearman´s rho, Spearman´s footrule, and Levenshtein´s distance (which is equivalent to Ulam´s distance for permutations). In all cases, it is shown how sequences with hidden information can be separated from the normal ones. As a result, very accurate and efficient distinguishers can be easily constructed. Finally, we study the detection capabilities of the associated detectors through a receiver operating characteristic analysis.