Information-Theoretic Limits for Steganography in Multimedia

TL;DR

The paper addresses the fundamental question of how much data can be reliably embedded in multimedia covers while keeping detection probability low. It introduces a Gibbs-based modeling framework, approximated by correlated-multivariate-quantized-Gaussian distributions (CMQGD), and casts the problem as maximizing $I(P_s;P_m)$ under a detectability constraint $\mathcal{D}(P_s \parallel P_c) \le 2\epsilon^2$. The authors derive a closed-form solution with $\vec{\mu}_m=0$, $\Sigma_m = (a^*-1)\Sigma_c$ and $a^* = -W(-\frac{4\epsilon^2}{n} -1 - i\pi)$, yielding the maximum rate $I(P_s;P_m) = \frac{n}{2}\ln(-W(-\frac{4\epsilon^2}{n} -1 - i\pi))$, and show that it scales as $O(\sqrt{n})$ in agreement with the Square Root Law, with an achievability proof leveraging random coding. The results provide a detector-agnostic, information-theoretic upper bound for multimedia steganography and illuminate how clique structure influences embedding limits, offering theoretical guidance for designing near-boundary embedding strategies in practice.

Abstract

Steganography is the art and science of hiding data within innocent-looking objects (cover objects). Multimedia objects such as images and videos are an attractive type of cover objects due to their high embedding rates. There exist many techniques for performing steganography in both the literature and the practical world. Meanwhile, the definition of the steganographic capacity for multimedia and how to be calculated has not taken full attention. In this paper, for multivariate quantized-Gaussian-distributed multimedia, we study the maximum achievable embedding rate with respect to the statistical properties of cover objects against the maximum achievable performance by any steganalytic detector. Toward this goal, we evaluate the maximum allowed entropy of the hidden message source subject to the maximum probability of error of the steganalytic detector which is bounded by the KL-divergence between the statistical distributions for the cover and the stego objects. We give the exact scaling constant that governs the relationship between the entropies of the hidden message and the cover object.

Figures (6)

• Figure 1: General communication model for steganography.
• Figure 2: Maximum achievable embedding rate $I(P_s;P_m)$ compared to the number of cover elements ($n$) for $P_E = 0.1$ and $P_E = 0.2$. Note that, we use the lower bound for $a^*$ in \ref{['Multimedia_Limits_Journal:eq.result4.upper']} to compute $I(P_s;P_m)$ in \ref{['Multimedia_Limits_Journal:eq.I']}.
• Figure 3: Results from markov2 comparing steganographic methods: S-UNIWARD, MIPOD, GMRF_BASE and GMRF with steganalyzer utilizing SRM feature.
• Figure 4: Results from markov2 comparing steganographic methods: S-UNIWARD, MIPOD, GMRF_BASE and GMRF with steganalyzer utilizing maxSRMd2 feature.
• Figure 5: Results from markov2 comparing steganographic methods: MiPOD, HILL and GMRF enhanced by low-pass-filtered-cost method with steganalyzer utilizing SRM feature.

Theorems & Definitions (7)

• Lemma 1
• Theorem 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6