Information-Theoretic Limits for Steganography in Multimedia

Steganography in multimedia aims to embed secret data into an innocent multimedia cover object. The embedding introduces some distortion to the cover object and produces a corresponding stego-object. The embedding distortion is measured by a cost function that determines the probability of detection of the existence of secret embedded data. An accurate definition of the cost function and its relation to the maximum embedding rate is the keystone for the proper evaluation of a steganographic system. Additionally, the statistical distribution of multimedia sources follows the Gibbs distribution which is a complex statistical model that prohibits a thorough mathematical analysis. Previous multimedia steganographic approaches either assume a relaxed statistical distribution of multimedia sources or presume a proposition on the maximum embedding rate then try to prove the correctness of the proposition. Alternatively, this paper introduces an analytical procedure for calculating the maximum embedding rate within multimedia cover objects through a constrained optimization problem that governs the relationship between the maximum embedding rate and the probability of detection by any steganographic detector. In the optimization problem, we use the KL-divergence between the statistical distributions for the cover and the stego-objects to be our cost function as it upper limits the performance of the optimal steganographic detector. To solve the optimization problem, we establish an equivalence between the Gibbs and the correlated-multivariate-quantized-Gaussian distributions for mathematical thorough analysis. The solution to our optimization problem provides an analytical form for the maximum embedding rate in terms of the WrightOmega function. Moreover, we prove that the achieved maximum embedding rate comes in agreement with the well-known square root law (SRL) of steganography. We also establish the relationship between the achieved maximum embedding rate and the experimental results obtained from several embedding and detection steganographic techniques.