Entropy amplification property and the loss for writing on dirty paper

Costa's celebrated "writing on dirty paper" (WDP) shows that the power-constrained channel Y = X + S + Z, with Gaussian Z, has the same capacity as the standard AWGN channel Y = X + Z, provided that the "interference" S (no matter how strong it is) is known at the transmitter. While this ability for perfect interference cancelation is very appealing, it relies heavily on the Gaussianity of the (unknown) noise Z. We construct an example of "bad" noise for writing on dirty paper, namely, "difference set noise". If the interference S is strong, then difference-set noise limits the WDP capacity to at most 2 bits. At the same time, like in the AWGN case, the zero-interference capacity grows without bound with the input constraint. Thus almost 100% of the available capacity is lost in WDP in the presence of difference-set noise. This high capacity loss is due to the "entropy amplification property" (EAP) of noise with an aperiodic probability distribution. Using the EAP and the duality between WDP and Wyner--Ziv source coding, we also give an example of dramatic rate-loss in quantizing encrypted source. Index Terms--Difference sets, side information, writing on dirty paper (WDP), Wyner--Ziv seup.