Abstract-We analyze asymptotic behavior of the embedding construction for steganography proposed by Zhang, Zhang, and Wang (ZZW) at 10th Information Hiding by deriving a closedform expression for the limit between embedding efficiency of the ZZW construction and the theoretical upper bound as a function of relative payload. This result confirms the experimental observation made in the original publication.
I. MOTIVATION AND BACKGROUNDSteganography deals with secret communication by hiding messages in innocuous-looking objects, such as digital images, by slightly modifying the colors of their pixels. The goal is to make the stego images, which carry secret messages, statistically indistinguishable from the original unmodified (cover) images [2]. Statistical detectability of most steganographic schemes increases with embedding distortion (see, e.g., [10]). This is why most stegosystems limit the amplitude of embedding changes to the smallest possible value. In this case, the distortion is often measured with the number of embedding changes. The average number of bits embedded per one embedding change is called embedding efficiency and it constitutes an important numerical characteristic of a steganographic scheme.By mapping the individual pixels of the cover to elements of a finite field, for example, by associating a bit (or q-ary symbol from finite field F q ) with each pixel value 1 , one can formulate the problem of maximizing embedding efficiency within the framework of coding theory [3], [8]. In particular, it is known that a q-ary linear code C with length n, dimension k, parity check matrix H, and covering radius R can be used to communicate n − k q-ary symbols (or (n − k) lg q bits) in a cover consisting of n elements by making at most R changes in the following manner. Let x ∈ F n q be the vector of symbols assigned to n elements of the cover. Then, n − k message symbols, F n−k q , can be embedded in x by modifying the symbols of pixels to y = x + e(m − Hx), where e(s) is a coset leader of the coset corresponding to syndrome s. The recipient can read the message from the stego object as its syndrome, m = Hy, because Hy = Hx + m − Hx. The