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**Additional info for Digital Watermarking, Fingerprinting and Compression An Information-Theoretic Perspective**

**Example text**

Error Events: Without loss of generality, we assume W = 1. , there exists no q ∈ {1, . . , 2nR1 } such that (I n , Y˜ n (1, q)) ∈ TI,Yˆ . • E2 : There exists a Y˜ n (1, q) = Yˆ n (1) such that (I n , Yˆ n (1)) ∈ TI,Yˆ , but (I n , Yˆ n (1), Z n ) ∈ TI,Yˆ ,Z . • E3 : (I n , Yˆ n (1), Z n ) ∈ TI,Yˆ ,Z but there also exists a k > 1 such that (I n , Yˆ n (k), Z n ) ∈ TI,Yˆ ,Z . The probability of error is then ˆ = 1} = Pr(E1 ) + Pr(E2 ) + Pr(E3 ) Pr{W 31 Behavior of P r(E1 ): From standard rate-distortion theorems [33], we know that if R1 = RQ − RW > I(I; Yˆ ) (the mutual information of the bivariate pI,Yˆ deﬁned above), then Pr(E1 ) → 0 as n → ∞.

However, this does not immediately guarantee the existence of a single deterministic code which is simultaneously good for all pZ|Yˆ ∈ MA (pI , pYˆ n |I n , DA ). In principle, for each pZ|Yˆ , a diﬀerent deterministic code could achieve small probability of error. 46 Existence of a deterministic code that achieves Rdsc D,DA can indeed be established using the same technique as in proving the existence of a deterministic code for compound channels [37]. This technique consists of: (a) ﬁrst approximating MA (pI , (p∗Yˆ |I )n , DA ) by a discrete set that contains suﬃciently many channel distributions; (b) showing existence of a good code for a single (not compound) channel whose distribution function is an “average” of the distribution functions of the discrete set; and (c) proving that the code obtained in (b) can be used for the entire compound channel, with the original set of channel distributions.

2nRW }. The technique is similar to the private version of regular QIM [16], in that 2nRW quantizers, each one indexed by a diﬀerent watermark, are employed. d. sequences Y˜ n , is generated, such that each dimension is distributed according to some pmf pYˆ . , RQ = RW + R1 The wth subset, consisting of sequences Y˜ n (w, 1), . . , Y˜ n (w, 2nR1 ), becomes the codebook for the wth watermark. 1) is satisﬁed. The output of the embedder (encoder) is denoted by Yˆ n (w) = Y˜ n (w, q). If none of the codewords in the wth codebook is jointly typical with I n , then the embedder outputs Yˆ n (w) = 0.