25,318 research outputs found
Search-to-Decision Reductions for Lattice Problems with Approximation Factors (Slightly) Greater Than One
We show the first dimension-preserving search-to-decision reductions for
approximate SVP and CVP. In particular, for any ,
we obtain an efficient dimension-preserving reduction from -SVP to -GapSVP and an efficient dimension-preserving reduction
from -CVP to -GapCVP. These results generalize the known
equivalences of the search and decision versions of these problems in the exact
case when . For SVP, we actually obtain something slightly stronger
than a search-to-decision reduction---we reduce -SVP to
-unique SVP, a potentially easier problem than -GapSVP.Comment: Updated to acknowledge additional prior wor
New Shortest Lattice Vector Problems of Polynomial Complexity
The Shortest Lattice Vector (SLV) problem is in general hard to solve, except
for special cases (such as root lattices and lattices for which an obtuse
superbase is known). In this paper, we present a new class of SLV problems that
can be solved efficiently. Specifically, if for an -dimensional lattice, a
Gram matrix is known that can be written as the difference of a diagonal matrix
and a positive semidefinite matrix of rank (for some constant ), we show
that the SLV problem can be reduced to a -dimensional optimization problem
with countably many candidate points. Moreover, we show that the number of
candidate points is bounded by a polynomial function of the ratio of the
smallest diagonal element and the smallest eigenvalue of the Gram matrix.
Hence, as long as this ratio is upper bounded by a polynomial function of ,
the corresponding SLV problem can be solved in polynomial complexity. Our
investigations are motivated by the emergence of such lattices in the field of
Network Information Theory. Further applications may exist in other areas.Comment: 13 page
Solving the Closest Vector Problem in Time--- The Discrete Gaussian Strikes Again!
We give a -time and space randomized algorithm for solving the
exact Closest Vector Problem (CVP) on -dimensional Euclidean lattices. This
improves on the previous fastest algorithm, the deterministic
-time and -space algorithm of
Micciancio and Voulgaris.
We achieve our main result in three steps. First, we show how to modify the
sampling algorithm from [ADRS15] to solve the problem of discrete Gaussian
sampling over lattice shifts, , with very low parameters. While the
actual algorithm is a natural generalization of [ADRS15], the analysis uses
substantial new ideas. This yields a -time algorithm for
approximate CVP for any approximation factor .
Second, we show that the approximate closest vectors to a target vector can
be grouped into "lower-dimensional clusters," and we use this to obtain a
recursive reduction from exact CVP to a variant of approximate CVP that
"behaves well with these clusters." Third, we show that our discrete Gaussian
sampling algorithm can be used to solve this variant of approximate CVP.
The analysis depends crucially on some new properties of the discrete
Gaussian distribution and approximate closest vectors, which might be of
independent interest
On the Lattice Isomorphism Problem
We study the Lattice Isomorphism Problem (LIP), in which given two lattices
L_1 and L_2 the goal is to decide whether there exists an orthogonal linear
transformation mapping L_1 to L_2. Our main result is an algorithm for this
problem running in time n^{O(n)} times a polynomial in the input size, where n
is the rank of the input lattices. A crucial component is a new generalized
isolation lemma, which can isolate n linearly independent vectors in a given
subset of Z^n and might be useful elsewhere. We also prove that LIP lies in the
complexity class SZK.Comment: 23 pages, SODA 201
Solving the Shortest Vector Problem in Lattices Faster Using Quantum Search
By applying Grover's quantum search algorithm to the lattice algorithms of
Micciancio and Voulgaris, Nguyen and Vidick, Wang et al., and Pujol and
Stehl\'{e}, we obtain improved asymptotic quantum results for solving the
shortest vector problem. With quantum computers we can provably find a shortest
vector in time , improving upon the classical time
complexity of of Pujol and Stehl\'{e} and the of Micciancio and Voulgaris, while heuristically we expect to find a
shortest vector in time , improving upon the classical time
complexity of of Wang et al. These quantum complexities
will be an important guide for the selection of parameters for post-quantum
cryptosystems based on the hardness of the shortest vector problem.Comment: 19 page
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