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Electronic Colloquium on Computational Complexity

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Reports tagged with lattices:
TR96-007 | 29th January 1996
Miklos Ajtai

Generating Hard Instances of Lattice Problems

Comments: 1

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 ... more >>>

TR97-047 | 20th October 1997
Miklos Ajtai

The Shortest Vector Problem in L_2 is NP-hard for Randomized Reductions.

Revisions: 1

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 ... more >>>

TR97-059 | 22nd December 1997
Jin-Yi Cai, Ajay Nerurkar

Approximating the SVP to within a factor $\left(1 + \frac{1}{\mathrm{dim}^\epsilon}\right)$ is NP-hard under randomized reductions

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
$\epsilon>0$, ... more >>>

TR98-010 | 22nd January 1998
Phong Nguyen, Jacques Stern

A Converse to the Ajtai-Dwork Security Proof and its Cryptographic Implications

Revisions: 1

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 ... more >>>

TR98-052 | 5th August 1998
Boris Hemkemeier, Frank Vallentin

On the decomposition of lattices

Revisions: 1

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 ... more >>>

TR03-066 | 2nd September 2003
Daniele Micciancio

Almost perfect lattices, the covering radius problem, and applications to Ajtai's connection factor

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 >>>

TR04-113 | 19th November 2004
Mårten Trolin

Lattices with Many Cycles Are Dense

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 ... more >>>

TR05-028 | 12th February 2005
Elmar Böhler

On the Lattice of Clones Below the Polynomial Time Functions

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 ... more >>>

TR05-142 | 1st December 2005
Vadim Lyubashevsky, Daniele Micciancio

Generalized Compact Knapsacks are Collision Resistant

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 $R$ ... more >>>

TR06-057 | 19th April 2006
Adam Klivans, Alexander A. Sherstov

Cryptographic Hardness Results for Learning Intersections of Halfspaces

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 ... more >>>

TR06-147 | 27th November 2006
Chris Peikert, Alon Rosen

Lattices that Admit Logarithmic Worst-Case to Average-Case Connection Factors

Revisions: 1

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 >>>

TR06-148 | 4th December 2006
Chris Peikert

Limits on the Hardness of Lattice Problems in $\ell_p$ Norms

Revisions: 1

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

* Approximating the shortest and closest vector in ... more >>>

TR11-111 | 10th August 2011
Zvika Brakerski, Craig Gentry, Vinod Vaikuntanathan

Fully Homomorphic Encryption without Bootstrapping

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 >>>

TR11-144 | 2nd November 2011
Greg Kuperberg, Shachar Lovett, Ron Peled

Probabilistic existence of rigid combinatorial structures

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 >>>

TR11-165 | 8th December 2011
Elena Grigorescu, Chris Peikert

List Decoding Barnes-Wall Lattices

Revisions: 2

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 >>>

TR16-125 | 31st July 2016
Karthekeyan Chandrasekaran, Mahdi Cheraghchi, Venkata Gandikota, Elena Grigorescu

Local Testing for Membership in Lattices

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 >>>

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