  We study robust testing and estimation of discrete distributions in the
strong contamination model. We consider both the "centralized setting" and the
"distributed setting with information constraints" including communication and
local privacy (LDP) constraints. Our technique relates the strength of
manipulation attacks to the earth-mover distance using Hamming distance as the
metric between messages(samples) from the users. In the centralized setting, we
provide optimal error bounds for both learning and testing. Our lower bounds
under local information constraints build on the recent lower bound methods in
distributed inference. In the communication constrained setting, we develop
novel algorithms based on random hashing and an $\ell_1/\ell_1$ isometry.
