We present a probabilistic public key cryptosystem which is
secure unless the following worst-case lattice problem can be solved in
polynomial time:
"Find the shortest nonzero vector in an n dimensional lattice
L where the shortest vector v is unique in the sense that any other
vector whose length ... more >>>

We show that computing the approximate length of the shortest vector
in a lattice within a factor c is NP-hard for randomized reductions
for any constant c<sqrt(2). We also give a deterministic reduction
based on a number theoretic conjecture.

This paper shows finding the closest vector in a lattice
to be NP-hard to approximate to within any factor up to
$2^{(\log{n})^{1-\epsilon}}$ where
$\epsilon = (\log\log{n})^{-\alpha}$
and $\alpha$ is any positive constant $<{1\over 2}$.

Computing the Hermite Normal Form
of an $n\times n$ matrix using the best current algorithms typically
requires $O(n^3\log M)$ space, where $M$ is a bound on the length of
the columns of the input matrix.
Although polynomial in the input size (which is ... more >>>

A measure $\mu_{n}$ on $n$-dimensional lattices with
determinant $1$ was introduced about fifty years ago to prove the
existence of lattices which contain points from certain sets. $\mu_{n}$
is the unique probability measure on lattices with determinant $1$ which
is invariant under linear transformations with determinant $1$, where a
linear ... more >>>

We describe a public-key cryptosystem with worst-case/average case
equivalence. The cryptosystem has an amortized plaintext to
ciphertext expansion of $O(n)$, relies on the hardness of the
$\tilde O(n^2)$-unique shortest vector problem for lattices, and
requires a public key of size at most $O(n^4)$ bits. The new
cryptosystem generalizes a conceptually ... more >>>