Important Properties of Normalized Kl Divergence under the Hmc Model

Normalized Kullback-Leibler (KL) divergence between cover and stego objects was shown to be important quantity for proofs around Square Root Law. In these proofs, we need to find Taylor expansion of this function w.r.t. change-rate β around β = 0 or be able to find an upper bound for some derivatives of this function. We can expect that this can be done since normalized KL divergence should be arbitrarily smooth in β. In this report, we show that every derivative of this function is uniformly upper-bounded and Lipschitz-continuous w.r.t. β and thus Taylor expansion of any order can be done. 1. Report in a nutshell If you are reading this report just because you need its main result, you can only read this section. If cover image is modeled as Markov Chain and embedding operation is mutually independent (LSB, ±1) and done with change-rate β ∈ [0, βMAX ] (we call this HMC model), then we define normalized KL divergence between nelement cover distribution P (n) and n-element stego distribution Q (n) β as (1.1) 1 n dn(β) = 1 n DKL ( P (n)||Q β ) . Main result of this report, formally stated in Theorem 3, tells us that every derivative of dn(β)/n w.r.t. β (and function dn(β)/n itself) is uniformly bounded and Lipschitz-continuous (or simply continuous) on [0, β0]. These properties are independent of n ≥ 1. 2. Introduction Normalized KL divergence between n-element cover and stego distributions, defined by equation (1.1), was shown to be a key quantity for studying and proving Square Root Law (SRL). Although it is not hard to believe, that this function and its derivatives are well behaved functions (bounded and continuous), the proofs are somewhat lengthy and not inspirational and thus they are presented in this report, separate from other and more important results. In the rest of this section, we describe the notation used in this report. Section 3 gives a summary of the assumptions we are using and derives some of their basic consequences. Main result of this report is formulated in Section 4 and proved in Section 5. All necessary results from the theory of hidden Markov chains are presented in Appendix A. In addition to the notation developed before, we use the following symbols. We [notation]