This paper is an attempt to analyze the interaction between Alice and Warden in Steganography using the Game Theory. We focus on the modern steganographic embedding paradigm based on minimizing an additive distortion function. The strategies of both players comprise of the probabilistic selection channel. The Warden is granted the knowledge of the payload and the embedding costs, and detects embedding using the likelihood ratio. In particular, the Warden is ignorant about the embedding probabilities chosen by Alice. When adopting a simple multivariate Gaussian model for the cover, the payoff function in the form of the Warden’s detection error can be numerically evaluated for a mutually independent embedding operation. We demonstrate on the example of a two-pixel cover that the Nash equilibrium is different from the traditional Alice’s strategy that minimizes the KL divergence between cover and stego objects under an omnipotent Warden. Practical implications of this case study include computing the loss per pixel of Warden’s ability to detect embedding due to her ignorance about the selection channel.
Recently, a new steganographic method was introduced that utilizes a universal distortion function called UNIWARD. The distortion between the cover and stego image is computed as a sum of relative changes of wavelet coefficients representing both images. As already pointed out in the original publication, the selection channel of the spatial version of UNIWARD (the version that hides messages in pixel values called S-UNIWARD) exhibits unusual properties – in highly textured and noisy regions the embedding probabilities form interleaved streaks of low and high embedding probability. While the authors of UNIWARD themselves hypothesized that such an artifact in the embedding probabilities may jeopardize its security, experiments with state-of-the-art rich models did not reveal any weaknesses. Using the fact that the cover embedding probabilities can be approximately estimated from the stego image, we introduce the novel concept of content-selective residuals and successfully attack S-UNIWARD. We also show that this attack, which is made possible by a faulty probabilistic selection channel, can be prevented by properly adjusting the stabilizing constant in the UNIWARD distortion function.
Today, the most reliable detectors of steganography in empirical cover sources, such as digital images coming from a known source, are built using machine-learning by representing images with joint distributions (co-occurrences) of neighboring noise residual samples computed using local pixel predictors. In this paper, we propose an alternative statistical description of residuals by binning their random projections on local neighborhoods. The size and shape of the neighborhoods allow the steganalyst to further diversify the statistical description and thus improve detection accuracy, especially for highly adaptive steganography. Other key advantages of this approach include the possibility to model long-range dependencies among pixels and making use of information that was previously underutilized in the marginals of co-occurrences. Moreover, the proposed approach is much more flexible than the previously proposed spatial rich model, allowing the steganalyst to obtain a significantly better trade off between detection accuracy and feature dimensionality. We call the new image representation the Projection Spatial Rich Model (PSRM) and demonstrate its effectiveness on HUGO and WOW – two current state-of-the-art spatial-domain embedding schemes.