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2, we show that the first term on the RHS of (85) can be made arbitrarily small. 4, we perform similar analysis for the second term. 2. For any > 0, there exists some 1 > 0 and some integer n1 > 0 such that for all n > n1  1 − 1 − Cn arccos β ∗ (R1,type + δ) − 2nR1,type −2nR0,type 1 < ,  Cn (π) where type is either add or gen. Proof. There exists some 1 > 0 such that 1 log 2 1 1 − β ∗ (R1,type + δ) − This follows since the LHS of (87) equals R1,type + δ when and (71) the rate R1,type satisfies 0 < β ∗ (R 1,type > R1,type .

The covertext is thus not ergodic, but it is stationary with if Z = 1. Assume that σu,0 u,1 2 + σ 2 )/2. Even though the covertext is non-ergodic, it is still true that no rate E Uk2 = (σu,0 u,1 2 ) can be achieved, as above C ∗ (D1 , D2 , E Ui2 ) is achievable. In fact, no rate above C ∗ (D1 , D2 , σu,0 2 ) demonstrates. This type an attacker of the form (101) designed for the parameters (D1 , D2 , σu,0 of argument naturally extends to any covertext with a finite number of ergodic modes, and in fact, with the proper modifications, to more general covertexts too.

A )n , which has n−1 E particular, the maximizing A in (8)), the watermarking channel (PX|U X 2 = A, guarantees a mutual information that is strictly larger than the value of the game reported in [8, 9]. s. 3 (additive attack). 2 only briefly, because, as we will show, these rates are achievable even in the public versions. 4. 1 with some preliminary results that we will use throughout the section. 1 Preliminaries Spherical caps We now state some asymptotic properties of the surface area of a spherical cap on a unit n-sphere.

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