  We study two dual settings of information processing. Let $ \mathsf{Y}
\rightarrow \mathsf{X} \rightarrow \mathsf{W} $ be a Markov chain with fixed
joint probability mass function $ \mathsf{P}_{\mathsf{X}\mathsf{Y}} $ and a
mutual information constraint on the pair $ (\mathsf{W},\mathsf{X}) $. For the
first problem, known as Information Bottleneck, we aim to maximize the mutual
information between the random variables $ \mathsf{Y} $ and $ \mathsf{W} $,
while for the second problem, termed as Privacy Funnel, our goal is to minimize
it. In particular, we analyze the scenario for which $ \mathsf{X} $ is the
input, and $ \mathsf{Y} $ is the output of modulo-additive noise channel. We
provide analytical characterization of the optimal information rates and the
achieving distributions.
