Related papers: LLL Algorithm for Lattice Basis Reduction
This paper presents the first combinatorial polynomial-time algorithm for minimizing submodular set functions, answering an open question posed in 1981 by Grotschel, Lovasz, and Schrijver. The algorithm employs a scaling scheme that uses a…
The often elusive Poincar\'e recurrence can be witnessed in a completely separable system. For such systems, the problem of recurrence reduces to the classic mathematical problem of simultaneous Diophantine approximation of multiple…
Recently, lattice-reduction-aided detectors have been proposed for multiple-input multiple-output (MIMO) systems to give performance with full diversity like maximum likelihood receiver, and yet with complexity similar to linear receivers.…
Lattices defined as modules over algebraic rings or orders have garnered interest recently, particularly in the fields of cryptography and coding theory. Whilst there exist many attempts to generalise the conditions for LLL reduction to…
Lattice reduction algorithms, such as the LLL algorithm, have been proposed as preprocessing tools in order to enhance the performance of suboptimal receivers in MIMO communications. In this paper we introduce a new kind of lattice…
Lenstra-Lenstra-Lovasz (LLL) algorithm, which is one of the lattice reduction (LR) techniques, has been extensively used to obtain better basis of the channel matrix. In this paper, we jointly apply Seysen's lattice reduction algorithm…
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…
The Lenstra-Lenstra-Lovasz (LLL) reduction has wide applications in digital communications. It can greatly improve the speed of the sphere decoding (SD) algorithms for solving an integer least squares (ILS) problem and the performance of…
Lattices are a popular field of study in mathematical research, but also in more practical areas like cryptology or multiple-input/multiple-output (MIMO) transmission. In mathematical theory, most often lattices over real numbers are…
Given a convex function $f$ on $\mathbb{R}^n$ with an integer minimizer, we show how to find an exact minimizer of $f$ using $O(n^2 \log n)$ calls to a separation oracle and $O(n^4 \log n)$ time. The previous best polynomial time algorithm…
We present an algorithm for list decoding codewords of algebraic number field codes in polynomial time. This is the first explicit procedure for decoding number field codes whose construction were previously described by Lenstra and…
We study the integer minimization of a quasiconvex polynomial with quasiconvex polynomial constraints. We propose a new algorithm that is an improvement upon the best known algorithm due to Heinz (Journal of Complexity, 2005). This…
Given an integer mxn matrix A satisfying certain regularity assumptions, a well-known integer programming problem asks to find an integer point in the associated knapsack polytope P(A, b)={x: A x= b, x>=0} or determine that no such point…
DeepLLL algorithm (Schnorr, 1994) is a famous variant of LLL lattice basis reduction algorithm, and PotLLL algorithm (Fontein et al., 2014) and $S^2$LLL algorithm (Yasuda and Yamaguchi, 2019) are recent polynomial-time variants of DeepLLL…
For large ranks, there is no good algorithm that decides whether a given lattice has an orthonormal basis. But when the lattice is given with enough symmetry, we can construct a provably deterministic polynomial-time algorithm to accomplish…
In this paper, we present a deterministic algorithm for the closest vector problem for all l_p-norms, 1 < p < \infty, and all polyhedral norms, especially for the l_1-norm and the l_{\infty}-norm. We achieve our results by introducing a new…
Lagarias and Odlyzko (J.~ACM~1985) proposed a polynomial time algorithm for solving ``\emph{almost all}'' instances of the Subset Sum problem with $n$ integers of size $\Omega(\Gamma_{\text{LO}})$, where $\log_2(\Gamma_{\text{LO}}) > n^2…
The Hermite-Korkine-Zolotarev reduction plays a central role in strong lattice reduction algorithms. By building upon a technique introduced by Ajtai, we show the existence of Hermite-Korkine-Zolotarev reduced bases that are arguably least…
In this paper, we show that for each lattice basis, there exists an equivalent basis which we describe as ``strongly reduced''. We show that bases reduced in this manner exhibit rather ``short'' basis vectors, that is, the length of the…
It has been shown that the parallel Lattice Linear Predicate (LLP) algorithm solves many combinatorial optimization problems such as the shortest path problem, the stable marriage problem and the market clearing price problem. In this…