On Steganographic Embedding Efficiency

Abstract

In this paper, we study embedding efficiency, which is an important attribute of steganographic schemes directly influencing their security. It is defined as the expected number of embedded random message bits per one embedding change. Constraining ourselves to embedding realized using linear covering codes (so called matrix embedding), we show that the quantity that determines embedding efficiency is not the covering radius but the average distance to code. We demonstrate that for linear codes of fixed block length and dimension, the highest embedding efficiency (the smallest average distance to code) is not necessarily achieved using codes with the smallest covering radius. Nevertheless, we prove that with increasing code length and fixed rate (i.e., fixed relative message length), the relative average distance to code and the relative covering radius coincide. Finally, we describe several specific examples of q-ary linear codes with q matched to the embedding operation and experimentally demonstrate the improvement in steganographic security when incorporating the coding methods to digital image steganography.

Product perfect codes and steganography

“…Given a code of length n, dimension k = n − t and covering radius ρ we can always compose the conventional steganographic protocol by using the coset algorithm (Algorithm 1) with an embedding rate k/n and average distortion bounded by ρ/n. However, as it is pointed out in [4], for the same embedding rate, the smallest average distortion is not always obtained with the code of smallest covering radius. The average distortion is determined by the encoding algorithm.…”

The Square Root Law in Stegosystems with Imperfect Information

“…matrix embedding [16]. Effectively, the location of the changes, as well as their content, conveys information to the recipient, allowing the payload transmitted to be (slightly) superlinear in γ.…”

Benchmarking for Steganography

“…Even though universal kernels guarantee that MMD[F, p, q] = 0 if and only if p = q, the choice of the kernel parameter γ has obviously a major influence on the finite sample estimate of MMD (14). If γ is large, the kernel is very narrow and thus k(x i , x j ) ≈ 0 (the discrete approximation to the RKHS "overfits" the data).…”

“…For example, methods that use matrix embedding [14] exhibit sharp non-linear decrease in detectability with decreasing payload due to significantly lower number of embedding changes, while other methods do not allow matrix embedding (e.g., adaptive schemes). Moreover, some steganographic algorithms are inherently limited to binary codes, such as methods based on perturbed quantization [25,15], while methods that use ±1 type of embedding can utilize more powerful ternary codes [14]. Thus, one steganographic method can be embedding significantly higher payload than some other method for the same distortion budget.…”