  In this work we show that the hardness of the Learning with Errors problem
with errors taken from the discrete Gaussian distribution implies the hardness
of the Learning with Errors problem with errors taken from the symmetric
Skellam distribution. Due to the sample preserving search-to-decision reduction
by Micciancio and Mol the same result applies to the decisional version of the
problem. Thus, we provide a variant of the Learning with Errors problem that is
hard based on conjecturally hard lattice problems and uses a discrete error
distribution that is similar to the continuous Gaussian distribution in that it
is closed under convolution. As an application of this result we construct a
post-quantum cryptographic protocol for differentially private data anlysis in
the distributed model. The security of this protocol is based on the hardness
of the new variant of the Decisional Learning with Errors problem. A feature of
this protocol is the use of the same noise for security and for differential
privacy resulting in an efficiency boost.
