Perfectly Secure Steganography: Capacity, Error Exponents, and Code Constructions

Abstract

An analysis of steganographic systems subject to the following perfect undetectability condition is presented in this paper. Following embedding of the message into the covertext, the resulting stegotext is required to have exactly the same probability distribution as the covertext. Then no statistical test can reliably detect the presence of the hidden message. We refer to such steganographic schemes as perfectly secure. A few such schemes have been proposed in recent literature, but they have vanishing rate. We prove that communication performance can potentially be vastly improved; specifically, our basic setup assumes independently and identically distributed (i.i.d.) covertext, and we construct perfectly secure steganographic codes from public watermarking codes using binning methods and randomized permutations of the code. The permutation is a secret key shared between encoder and decoder. We derive (positive) capacity and random-coding exponents for perfectly-secure steganographic systems. The error exponents provide estimates of the code length required to achieve a target low error probability.In some applications, steganographic communication may be disrupted by an active warden, modelled here by a compound discrete memoryless channel. The transmitter and warden are subject to distortion constraints. We address the potential loss in communication performance due to the perfect-security requirement. This loss is the same as the loss obtained under a weaker order-1 steganographic requirement that would just require matching of first-order marginals of the covertext and stegotext distributions. Furthermore, no loss occurs if the covertext distribution is uniform and the distortion metric is cyclically symmetric; steganographic capacity is then achieved by randomized linear codes. Our framework may also be useful for developing computationally secure steganographic systems that have near-optimal communication performance.

The Square Root Law in Stegosystems with Imperfect Information

“…In this scenario it has already been shown that the embedder can communicate completely undetectably at a nonzero (linear) rate by preserving all the statistics of the covers [1]. However, no such system, for embedding in real-world cover media, has ever been constructed.…”

“…1 (1). Write X for the number of 1 bits in the cover stream, and Y for the number in the object to be classified.…”

“…But knowledge about the covers is fundamental to the problem. It is known that perfect steganography is possible, conveying an amount of payload linear in the cover size with zero risk of detection, as long as the embedder has perfect knowledge of the cover distribution [1]. This is so because the embedder can ensure that all the statistics of the cover source are preserved by their embedding function 1 .…”

“…It is known that perfect steganography is possible, conveying an amount of payload linear in the cover size with zero risk of detection, as long as the embedder has perfect knowledge of the cover distribution [1]. This is so because the embedder can ensure that all the statistics of the cover source are preserved by their embedding function 1 . On the other hand, we have a completely different situation when the embedder does not know, and fails to preserve, all the statistics of the covers: this leads to the Square Root Law [3][4][5] stating that the information conveyed can grow at most of order √ n, where n represents the size of the cover, or face eventual certain detection.…”

Estimating Steganographic Fisher Information in Real Images

“…Information theoretic quantities, particularly the Kullback-Leibler (KL) divergence between cover and stego distributions, can bound the secure embedding capacity [1,2] but, naturally, they require knowledge of these distributions and this seems infeasible for digital media. Whilst information theory has produced interesting results about the rate of capacity growth [3,4], these do not specify concrete payload sizes for particular, real-world, covers.…”