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Revision #1 to TR06-148 | 15th February 2007 00:00
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#### Limits on the Hardness of Lattice Problems in $ell_p$ Norms Revision of: TR06-148

**Abstract:**

We show that several recent ``positive'' results for lattice

problems in the $\ell_2$ norm also hold in $\ell_p$ norms, for any

$p > 2$. In particular, for lattices of dimension $n$:

* Approximating the shortest and closest vector in the $\ell_p$

norm to within $\tilde{O}(\sqrt{n})$ factors is contained in

$\coNP$.

* Approximating the length of the shortest vector in the

$\ell_p$ norm to within $\tilde{O}(n)$ factors reduces to the

average-case problems studied in related works (Ajtai, STOC

1996; Micciancio and Regev, FOCS 2004; Regev, STOC 2005).

These results improve upon the current understanding of $\ell_p$

norms by up to a $\sqrt{n}$ factor. Taken together, they can be

viewed as a partial converse to recent reductions from the $\ell_2$

norm to $\ell_p$ norms (Regev and Rosen, STOC 2006).

One of our main technical contributions is a very general analysis

of Gaussian distributions over lattices, which may be of independent

interest. Our proofs employ analytical techniques of Banaszczyk

which, to our knowledge, have yet to be exploited in computer

science.

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TR06-148 | 4th December 2006 00:00
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#### Limits on the Hardness of Lattice Problems in $\ell_p$ Norms

**Abstract:**
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

$\ell_p$ norm to within $\tilde{O}(\sqrt{n})$ factors is

contained in $\coNP$.

* Approximating the length of the shortest vector in the

$\ell_p$ norm to within $\tilde{O}(n)$ factors reduces to the

same average-case problems that have been studied in related

works (Ajtai, STOC 1996; Micciancio and Regev, FOCS 2004; Regev,

STOC 2005).

Each of these results improves upon the current understanding of

$\ell_p$ norms by up to a $\sqrt{n}$ factor. Taken together, they

can be seen as a partial converse to recent reductions from

lattice problems in the $\ell_2$ norm to corresponding problems in

$\ell_p$ norms (Regev and Rosen, STOC 2006).

One of our main technical contributions is a general analysis of

\emph{sums} of independent Gaussian distributions over lattices,

which may be of independent interest. Our proofs employ

analytical techniques of Banaszczyk which, to our knowledge, have

yet to be exploited in computer science.