  The distributed source coding problem is considered when the sensors, or
encoders, are under Byzantine attack; that is, an unknown group of sensors have
been reprogrammed by a malicious intruder to undermine the reconstruction at
the fusion center. Three different forms of the problem are considered. The
first is a variable-rate setup, in which the decoder adaptively chooses the
rates at which the sensors transmit. An explicit characterization of the
variable-rate achievable sum rates is given for any number of sensors and any
groups of traitors. The converse is proved constructively by letting the
traitors simulate a fake distribution and report the generated values as the
true ones. This fake distribution is chosen so that the decoder cannot
determine which sensors are traitors while maximizing the required rate to
decode every value. Achievability is proved using a scheme in which the decoder
receives small packets of information from a sensor until its message can be
decoded, before moving on to the next sensor. The sensors use randomization to
choose from a set of coding functions, which makes it probabilistically
impossible for the traitors to cause the decoder to make an error. Two forms of
the fixed-rate problem are considered, one with deterministic coding and one
with randomized coding. The achievable rate regions are given for both these
problems, and it is shown that lower rates can be achieved with randomized
coding.
