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A Jacobi Method for Lattice Basis Reduction
"... Abstract—Lattice reduction aided decoding has been successfully used in wireless communications. In this paper, we propose a Jacobi method for lattice basis reduction. Jacobi method is attractive, because it is inherently parallel. Thus high performance can be achieved by exploiting multiprocessor a ..."
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Abstract—Lattice reduction aided decoding has been successfully used in wireless communications. In this paper, we propose a Jacobi method for lattice basis reduction. Jacobi method is attractive, because it is inherently parallel. Thus high performance can be achieved by exploiting multiprocessor and/or multicore architectures. We also present our experimental results on the convergence of our method and the comparison with the LLL algorithm, a lattice basis reduction method widely used in wireless communication applications. I.
HKZ and Minkowski Reduction Algorithms for LatticeReductionAided MIMO Detection
"... Abstract—Recently, lattice reduction has been widely used for signal detection in multiinput multioutput (MIMO) communications. In this paper, we present three novel lattice reduction algorithms. First, using a unimodular transformation, a significant improvement on an existing HermiteKorkineZolot ..."
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Abstract—Recently, lattice reduction has been widely used for signal detection in multiinput multioutput (MIMO) communications. In this paper, we present three novel lattice reduction algorithms. First, using a unimodular transformation, a significant improvement on an existing HermiteKorkineZolotareffreduction algorithm is proposed. Then, we present two practical algorithms for constructing Minkowskireduced bases. To assess the output quality, we compare the orthogonality defect of the reduced bases produced by LLL algorithm and our new algorithms, and find that in practice Minkowskireduced basis vectors are the closest to being orthogonal. An errorrate analysis of suboptimal decoding algorithms aided by different reduction notions is also presented. To this aim, the proximity factor is employed as a measurement. We improve some existing results and derive upper bounds for the proximity factors of Minkowskireductionaided decoding (MRAD) to show that MRAD can achieve the same diversity order with infinite lattice decoding (ILD). Index Terms—HKZ, lattice reduction, LLL, MIMO detection, Minkowski, proximity factors.
A Parallel LLL Algorithm
"... The LLL algorithm is a wellknow and widely used lattice basis reduction algorithm. In many applications, its speed is of essential. However, it is very difficult to parallelize the original LLL algorithm. We present a multithreading LLL algorithm based on a recent improved version: an LLL algorith ..."
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The LLL algorithm is a wellknow and widely used lattice basis reduction algorithm. In many applications, its speed is of essential. However, it is very difficult to parallelize the original LLL algorithm. We present a multithreading LLL algorithm based on a recent improved version: an LLL algorithm with delayed size reduction. 1
Practical HKZ and Minkowski Lattice Reduction Algorithms
, 2011
"... Recently, lattice reduction has been widely used for signal detection in multiinput multioutput (MIMO) communications. In this paper, we present three novel lattice reduction algorithms. First, using a unimodular transformation, a significant improvement on an existing HermiteKorkineZolotareffred ..."
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Cited by 1 (1 self)
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Recently, lattice reduction has been widely used for signal detection in multiinput multioutput (MIMO) communications. In this paper, we present three novel lattice reduction algorithms. First, using a unimodular transformation, a significant improvement on an existing HermiteKorkineZolotareffreduction algorithm is proposed. Then, we present two practical algorithms for constructing Minkowskireduced (Mreduced) bases. To assess the output quality, we compare the orthogonality defect of the reduced bases produced by LLL algorithm and our new algorithms, and find that in practice Mreduced basis vectors are the closest to being orthogonal. An errorrate analysis of suboptimal decoding algorithms aided by different reduction notions is also presented. To this aim, the proximity factor is employed as a measurement. We improve some existing results and derive upper bounds for the proximity factors of Minkowskireductionaided decoding (MRAD) to show that MRAD can achieve the same diversity order with infinite lattice decoding (ILD).
A GPU Implementation of a Jacobi Method for Lattice Basis Reduction
, 2013
"... This paper describes a parallel Jacobi method for lattice basis reduction and a GPU implementation using CUDA. Our experiments have shown that the parallel implementation is more than fifty times as fast as the serial counterpart, which is about twice as fast as the wellknown LLL lattice reduction ..."
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This paper describes a parallel Jacobi method for lattice basis reduction and a GPU implementation using CUDA. Our experiments have shown that the parallel implementation is more than fifty times as fast as the serial counterpart, which is about twice as fast as the wellknown LLL lattice reduction algorithm.
Practical algorithms for constructing HKZ and Minkowski reduced bases
, 2011
"... In this paper, three practical lattice basis reduction algorithms are presented. The first algorithm constructs a Hermite, Korkine and Zolotareff (HKZ) reduced lattice basis, in which a unimodular transformation is used for basis expansion. Our complexity analysis shows that our algorithm is signifi ..."
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In this paper, three practical lattice basis reduction algorithms are presented. The first algorithm constructs a Hermite, Korkine and Zolotareff (HKZ) reduced lattice basis, in which a unimodular transformation is used for basis expansion. Our complexity analysis shows that our algorithm is significantly more efficient than the existing HKZ reduction algorithms. The second algorithm computes a Minkowski reduced lattice basis. It is the first practical algorithm for Minkowski reduced bases for lattices of arbitrary dimensions. The third algorithm is an improvement of the second algorithm by drastically reducing the number of lattice points being searched. Since the original LLL algorithm is no longer applicable to the third algorithm, we propose a notion of quasiLLL reduction to accelerate the computation.
General Terms Algorithms and Theory
"... The famous LLL algorithm is the first polynomial time lattice reduction algorithm which is widely used in many applications. In this paper, we prove the convergence of a novel polynomial time lattice reduction algorithm, called the Jacobi method introduced by S. Qiao [23], and show that it has the s ..."
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The famous LLL algorithm is the first polynomial time lattice reduction algorithm which is widely used in many applications. In this paper, we prove the convergence of a novel polynomial time lattice reduction algorithm, called the Jacobi method introduced by S. Qiao [23], and show that it has the same complexity as the LLL algorithm. Our experimental results show that the Jacobi method outperforms the LLL algorithm in not only efficiency, but also orthogonality defect of the bases it produces.
A Polynomial Time Jacobi Method for Lattice Basis Reduction
, 2012
"... Among all lattice reduction algorithms, the LLL algorithm is the first and perhaps the most famous polynomial time algorithm, and it is widely used in many applications. In 2012, S. Qiao [24] introduced another algorithm, the Jacobi method, for lattice basis reduction. S. Qiao and Z. Tian [25] impro ..."
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Among all lattice reduction algorithms, the LLL algorithm is the first and perhaps the most famous polynomial time algorithm, and it is widely used in many applications. In 2012, S. Qiao [24] introduced another algorithm, the Jacobi method, for lattice basis reduction. S. Qiao and Z. Tian [25] improved the Jacobi method further to be polynomial time but only produces a QuasiReduced basis. In this paper, we present a polynomial time Jacobi method for lattice basis reduction (short as PolyJacobi method) that can produce a reduced basis. Our experimental results indicate that the bases produced by PolyJacobi method have almost equally good orthogonality defect as the bases produced by the Jacobi method.
A PARALLEL JACOBITYPE LATTICE BASIS REDUCTION ALGORITHM
"... Abstract. This paper describes a parallel Jacobi method for lattice basis reduction and a GPU implementation using CUDA. Our experiments have shown that the parallel implementation is more than fifty times as fast as the serial counterpart, which is twice as fast as the wellknown LLL lattice reduct ..."
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Abstract. This paper describes a parallel Jacobi method for lattice basis reduction and a GPU implementation using CUDA. Our experiments have shown that the parallel implementation is more than fifty times as fast as the serial counterpart, which is twice as fast as the wellknown LLL lattice reduction algorithm. Key words. Lattice basis reduction, Jacobi method, GPU. 1.