The generalized knapsack function is defined as $f_{\a}(\x) = \sum_i
a_i \cdot x_i$, where $\a = (a_1, \ldots, a_m)$ consists of $m$
elements from some ring $R$, and $\x = (x_1, \ldots, x_m)$ consists
of $m$ coefficients from a specified subset $S \subseteq R$.
Micciancio (FOCS ... more >>>

We demonstrate an \emph{average-case} problem which is as hard as
finding $\gamma(n)$-approximate shortest vectors in certain
$n$-dimensional lattices in the \emph{worst case}, where $\gamma(n)
= O(\sqrt{\log n})$. The previously best known factor for any class
of lattices was $\gamma(n) = \tilde{O}(n)$.

To obtain our results, we ... more >>>

We show that for any $p \geq 2$, lattice problems in the $\ell_p$
norm are subject to all the same limits on hardness as are known
for the $\ell_2$ norm. In particular, for lattices of dimension
$n$:

* Approximating the shortest and closest vector in the
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