13,186 research outputs found
On the micciancio-voulgaris algorithm to solve the long-horizon direct MPC optimization problem
Conference ProceedingsIt is widely accepted that model predictive control
(MPC) with long prediction horizons yields, in general, a better
performance than with short horizons. In the context of power
electronic systems, the main advantages include improved closedloop
stability and lower current distortion per switching frequency.
A shortcoming of MPC with long prediction horizons is
the computational burden associated with solving the optimization
problem in real time, which limits the minimum possible
sampling interval. The solution to the MPC optimization problem
is a polyhedral partition of the state-space. Pre-processing of
the state-space and storing representative information thereof
offline assists in reducing the online computational burden. The
problem structure is a special case in the form of a truncated
lattice. Exploiting this characteristic enables representation of
the partitioned space to be is stored as a minimal set of Voronoi
relevant vectors describing the basic Voronoi cell of a lattice. We
evaluate the algorithm proposed by Micciancio and Voulgaris
known as the MV-algorithm to solve the closest vector problem
with pre-processing (CVPP). The performance of the algorithm is
evaluated in a simulated three-level neutral point clamped (NPC)
voltage source inverter with an RL load
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
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
On the Quantitative Hardness of CVP
For odd
integers (and ), we show that the Closest Vector Problem
in the norm (\CVP_p) over rank lattices cannot be solved in
2^{(1-\eps) n} time for any constant \eps > 0 unless the Strong Exponential
Time Hypothesis (SETH) fails. We then extend this result to "almost all" values
of , not including the even integers. This comes tantalizingly close
to settling the quantitative time complexity of the important special case of
\CVP_2 (i.e., \CVP in the Euclidean norm), for which a -time
algorithm is known. In particular, our result applies for any
that approaches as .
We also show a similar SETH-hardness result for \SVP_\infty; hardness of
approximating \CVP_p to within some constant factor under the so-called
Gap-ETH assumption; and other quantitative hardness results for \CVP_p and
\CVPP_p for any under different assumptions
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
Decoding by Embedding: Correct Decoding Radius and DMT Optimality
The closest vector problem (CVP) and shortest (nonzero) vector problem (SVP)
are the core algorithmic problems on Euclidean lattices. They are central to
the applications of lattices in many problems of communications and
cryptography. Kannan's \emph{embedding technique} is a powerful technique for
solving the approximate CVP, yet its remarkable practical performance is not
well understood. In this paper, the embedding technique is analyzed from a
\emph{bounded distance decoding} (BDD) viewpoint. We present two complementary
analyses of the embedding technique: We establish a reduction from BDD to
Hermite SVP (via unique SVP), which can be used along with any Hermite SVP
solver (including, among others, the Lenstra, Lenstra and Lov\'asz (LLL)
algorithm), and show that, in the special case of LLL, it performs at least as
well as Babai's nearest plane algorithm (LLL-aided SIC). The former analysis
helps to explain the folklore practical observation that unique SVP is easier
than standard approximate SVP. It is proven that when the LLL algorithm is
employed, the embedding technique can solve the CVP provided that the noise
norm is smaller than a decoding radius , where
is the minimum distance of the lattice, and . This
substantially improves the previously best known correct decoding bound . Focusing on the applications of BDD to decoding of
multiple-input multiple-output (MIMO) systems, we also prove that BDD of the
regularized lattice is optimal in terms of the diversity-multiplexing gain
tradeoff (DMT), and propose practical variants of embedding decoding which
require no knowledge of the minimum distance of the lattice and/or further
improve the error performance.Comment: To appear in IEEE Transactions on Information Theor
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