  We introduce the use of Fourier analysis on lattices as an integral part of a
lattice based construction. The tools we develop provide an elegant description
of certain Gaussian distributions around lattice points. Our results include
two cryptographic constructions which are based on the worst-case hardness of
the unique shortest vector problem. The main result is a new public key
cryptosystem whose security guarantee is considerably stronger than previous
results ($O(n^{1.5})$ instead of $O(n^7)$). This provides the first alternative
to Ajtai and Dwork's original 1996 cryptosystem. Our second result is a family
of collision resistant hash functions which, apart from improving the security
in terms of the unique shortest vector problem, is also the first example of an
analysis which is not based on Ajtai's iterative step. Surprisingly, both
results are derived from one theorem which presents two indistinguishable
distributions on the segment $[0,1)$. It seems that this theorem can have
further applications and as an example we mention how it can be used to solve
an open problem related to quantum computation.
