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 ... 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.

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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 $<{1\over 2}$.

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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 ... 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
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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 >>>

A new probabilistic technique for establishing the existence of certain regular combinatorial structures has been introduced by Kuperberg, Lovett, and Peled (STOC 2012). Using this technique, it can be shown that under certain conditions, a randomly chosen structure has the required properties of a $t-(n,k,?)$ combinatorial design with tiny, yet ... more >>>

This paper aims to derandomize the following problems in the smoothed analysis of Spielman and Teng. Learn Disjunctive Normal Form (DNF), invert Fourier Transforms (FT), and verify small circuits' unsatisfiability. Learning algorithms must predict a future observation from the only $m$ i.i.d. samples of a fixed but unknown joint-distribution $P(G(x),y)$ ... more >>>