The square root law of steganographic capacity for Markov covers

It is a well-established result that steganographic capacity of perfectly secure stegosystems grows linearly with the number of cover elements-secure steganography has a positive rate. In practice, however, neither the Warden nor the Steganographer has perfect knowledge of the cover source and thus it is unlikely that perfectly secure stegosystems for complex covers, such as digital media, will ever be constructed. This justifies study of secure capacity of imperfect stegosystems. Recent theoretical results from batch steganography, supported by experiments with blind steganalyzers, point to an emerging paradigm: whether steganography is performed in a large batch of cover objects or a single large object, there is a wide range of practical situations in which secure capacity rate is vanishing. In particular, the absolute size of secure payload appears to only grow with the square root of the cover size. In this paper, we study the square root law of steganographic capacity and give a formal proof of this law for imperfect stegosystems, assuming that the cover source is a stationary Markov chain and the embedding changes are mutually independent.