Entropy Amplification Property and the Loss for Writing on Dirty Paper

Costa's celebrated ldquowriting on dirty paperrdquo (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 ldquointerferencerdquo
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 ldquobadrdquo noise for writing on dirty paper, namely, ldquodifference set noiserdquo. 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 ldquoentropy amplification propertyrdquo (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.