We prove the equivalence, up to a small polynomial approximation
factor *(n/ log(n))*^{0.5},
of the lattice problems **uSVP**
(unique Shortest Vector Problem),
**BDD** (Bounded Distance Decoding) and
**GapSVP**
(the decision version of the Shortest Vector Problem).
This
resolves a long-standing open problem about the relationship
between **uSVP** and the more standard
**GapSVP**, as well the
**BDD** problem commonly used in coding theory.
The main cryptographic application of our work is the proof that the
Ajtai-Dwork [STOC 1997]
and the Regev [J. ACM 51(6):899-942] cryptosystems,
which were previously only known to be based on the hardness of
**uSVP**, can be equivalently based on the hardness of
worst-case **GapSVP**_{O(n2.5)} and
**GapSVP**_{O(n2)},
respectively.
Also, in the case of **uSVP** and
**BDD**, our connection is very tight,
establishing the equivalence
(within a small constant approximation factor)
between the two most central problems used in lattice based
public key cryptography and coding theory.