Efficient Modular Arithmetic in Adapted Modular Number System Using Lagrange Representation
ACISP '08 Proceedings of the 13th Australasian conference on Information Security and Privacy
Tail behavior of sphere-decoding complexity in random lattices
ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 2
Bounds for solid angles of lattices of rank three
Journal of Combinatorial Theory Series A
Performance and complexity analysis of infinity-norm sphere-decoding
IEEE Transactions on Information Theory
On bounded distance decoding for general lattices
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
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Upper and lower bounds are derived for the decoding complexity of a general lattice L. The bounds are in terms of the dimension n and the coding gain γ of L, and are obtained based on a decoding algorithm which is an improved version of Kannan's (1983) method. The latter is currently the fastest known method for the decoding of a general lattice. For the decoding of a point x, the proposed algorithm recursively searches inside an, n-dimensional rectangular parallelepiped (cube), centered at x, with its edges along the Gram-Schmidt vectors of a proper basis of L. We call algorithms of this type recursive cube search (RCS) algorithms. It is shown that Kannan's algorithm also belongs to this category. The complexity of RCS algorithms is measured in terms of the number of lattice points that need to be examined before a decision is made. To tighten the upper bound on the complexity, we select a lattice basis which is reduced in the sense of Korkin-Zolotarev (1873). It is shown that for any selected basis, the decoding complexity (using RCS algorithms) of any sequence of lattices with possible application in communications (γ⩾1) grows at least exponentially with n and γ. It is observed that the densest lattices, and almost all of the lattices used in communications, e.g., Barnes-Wall lattices and the Leech lattice, have equal successive minima (ESM). For the decoding complexity of ESM lattices, a tighter upper bound and a stronger lower bound result are derived