Miklos Ajtai

We give a random class of n dimensional lattices so that, if

there is a probabilistic polynomial time algorithm which finds a short

vector in a random lattice with a probability of at least 1/2

then there is also a probabilistic polynomial time algorithm which

solves the following three ...
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Miklos Ajtai

We show that the shortest vector problem in lattices

with L_2 norm is NP-hard for randomized reductions. Moreover we

also show that there is a positive absolute constant c, so that to

find a vector which is longer than the shortest non-zero vector by no

more than a factor of ...
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Jin-Yi Cai, Ajay Nerurkar

Recently Ajtai showed that

to approximate the shortest lattice vector in the $l_2$-norm within a

factor $(1+2^{-\mbox{\tiny dim}^k})$, for a sufficiently large

constant $k$, is NP-hard under randomized reductions.

We improve this result to show that

to approximate a shortest lattice vector within a

factor $(1+ \mbox{dim}^{-\epsilon})$, for any

$\epsilon>0$, ...
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Phong Nguyen, Jacques Stern

Recently, Ajtai discovered a fascinating connection

between the worst-case complexity and the average-case

complexity of some well-known lattice problems.

Later, Ajtai and Dwork proposed a cryptosystem inspired

by Ajtai's work, provably secure if a particular lattice

problem is difficult. We show that there is a converse

to the ...
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Boris Hemkemeier, Frank Vallentin

A lattice in euclidean space which is an orthogonal sum of

nontrivial sublattices is called decomposable. We present an algorithm

to construct a lattice's decomposition into indecomposable sublattices.

Similar methods are used to prove a covering theorem for generating

systems of lattices and to speed up variations of the LLL ...
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Daniele Micciancio

Lattices have received considerable attention as a potential source of computational hardness to be used in cryptography, after a breakthrough result of Ajtai (STOC 1996) connecting the average-case and worst-case complexity of various lattice problems. The purpose of this paper is twofold. On the expository side, we present a rigorous ... more >>>

Mårten Trolin

We give a method for approximating any $n$-dimensional

lattice with a lattice $\Lambda$ whose factor group

$\mathbb{Z}^n / \Lambda$ has $n-1$ cycles of equal length

with arbitrary precision. We also show that a direct

consequence of this is that the Shortest Vector Problem and the Closest

Vector Problem cannot ...
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Elmar Böhler

A clone is a set of functions that is closed under generalized substitution.

The set FP of functions being computable deterministically in polynomial

time is such a clone. It is well-known that the set of subclones of every

clone forms a lattice. We study the lattice below FP, which ...
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Vadim Lyubashevsky, Daniele Micciancio

The generalized knapsack problem is the following: given $m$ random

elements $a_1,\ldots,a_m\in R$ for some ring $R$, and a target $t\in

R$, find elements $z_1,\ldots,z_m\in D$ such that $\sum{a_iz_i}=t$

where $D$ is some given subset of $R$. In (Micciancio, FOCS 2002),

it was proved that for appropriate choices of $R$ ...
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Adam Klivans, Alexander A. Sherstov

We give the first representation-independent hardness results for

PAC learning intersections of halfspaces, a central concept class

in computational learning theory. Our hardness results are derived

from two public-key cryptosystems due to Regev, which are based on the

worst-case hardness of well-studied lattice problems. Specifically, we

prove that a polynomial-time ...
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Chris Peikert, Alon Rosen

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 ... more >>>

Chris Peikert

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 ... more >>>

Zvika Brakerski, Craig Gentry, Vinod Vaikuntanathan

We present a radically new approach to fully homomorphic encryption (FHE) that dramatically improves performance and bases security on weaker assumptions. A central conceptual contribution in our work is a new way of constructing leveled fully homomorphic encryption schemes (capable of evaluating arbitrary polynomial-size circuits), {\em without Gentry's bootstrapping procedure}.

... more >>>Greg Kuperberg, Shachar Lovett, Ron Peled

We show the existence of rigid combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, $t$-designs, and $t$-wise permutations. In all cases, the sizes of the objects are optimal up to polynomial overhead. ... more >>>

Elena Grigorescu, Chris Peikert

The question of list decoding error-correcting codes over finite fields (under the Hamming metric) has been widely studied in recent years. Motivated by the similar discrete structure of linear codes and point lattices in $R^{N}$, and their many shared applications across complexity theory, cryptography, and coding theory, we initiate the ... more >>>

Karthekeyan Chandrasekaran, Mahdi Cheraghchi, Venkata Gandikota, Elena Grigorescu

Motivated by the structural analogies between point lattices and linear error-correcting codes, and by the mature theory on locally testable codes, we initiate a systematic study of local testing for membership in lattices. Testing membership in lattices is also motivated in practice, by applications to integer programming, error detection in ... more >>>

Huck Bennett

Computational problems on point lattices play a central role in many areas of computer science including integer programming, coding theory, cryptanalysis, and especially the design of secure cryptosystems. In this survey, we present known results and open questions related to the complexity of the most important of these problems, the ... more >>>