Asymptotic Behavior of the ZZW Embedding Construction

Abstract

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

The square root law does not require a linear key

“…One of the leading source coding schemes is given by the construction in [18], dubbed ZZW after its authors, which converts one matrix embedding scheme into another. The construction is (in a sense described in [6]) optimal when it is applied to the trivial embedding of one bit per location. It involves, like Algorithm 2, breaking the cover image into groups and using a Hamming code within each group; it also involves a wet paper code so that the changes in each block can signal further information (see [18] for the details).…”

“…This would be a curious result, because the ZZW construction is more efficient in terms of bits conveyed per change. Perhaps this can be reconciled by the comment in [6], where it is noted that both Hamming and ZZW families "parallel" the rate distortion bound, when embedding efficiency against embedding rate is charted in a particular way. We might conjecture that all schemes which have the same property are equally secure as regards their asymptotic KL divergence ESFI.…”

Minimizing Additive Distortion in Steganography Using Syndrome-Trellis Codes

“…Current state-of-the-art methods use codes based on Low Density Generator Matrices (LDGMs) [8] in combination with the ZZW construction [15]. The embedding efficiency of these codes stays rather close to the bound for arbitrarily small relative payloads [33].…”

Data Hiding in Images using Majority Vote Parity Check

“…J. Fridrich noted that the embedding construction for ±1embedding also proposed in [5] approaches the bound on embedding efficiency of ternary codes with the same limit, because the ternary bound increases by 1 compared to the binary bound. The embedding effectiveness of ±1ZZW code families is also larger by 1 [6].…”