On the communication cost of determining an approximate nearest lattice point

We consider the closest lattice point problem in a distributed network setting and study the communication cost and the error probability for computing an approximate nearest lattice point, using the nearest-plane algorithm, due to Babai. Two distinct communication models, centralized and interactive, are considered. The importance of proper basis selection is addressed. Assuming a reduced basis for a two-dimensional lattice, we determine the approximation error of the nearest plane algorithm. The communication cost for determining the Babai point, or equivalently, for constructing the rectangular nearest-plane partition, is calculated in the interactive setting. For the centralized model, an algorithm is presented for reducing the communication cost of the nearest plane algorithm in an arbitrary number of dimensions.

[1]  Toby Berger,et al.  Rate distortion theory : a mathematical basis for data compression , 1971 .

[2]  Michael E. Pohst,et al.  On the computation of lattice vectors of minimal length, successive minima and reduced bases with applications , 1981, SIGS.

[3]  Alexander Vardy,et al.  Closest point search in lattices , 2002, IEEE Trans. Inf. Theory.

[4]  Andrew Chi-Chih Yao,et al.  Some complexity questions related to distributive computing(Preliminary Report) , 1979, STOC.

[5]  Graham Cormode,et al.  Communication-efficient distributed monitoring of thresholded counts , 2006, SIGMOD Conference.

[6]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[7]  Alon Orlitsky,et al.  Coding for computing , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[8]  W. Fischer,et al.  Sphere Packings, Lattices and Groups , 1990 .

[9]  Vinay A. Vaishampayan,et al.  Communication cost of transforming a nearest plane partition to the Voronoi partition , 2017, 2017 IEEE International Symposium on Information Theory (ISIT).

[10]  Avishek Adhikari,et al.  Introduction to Mathematical Cryptography , 2014 .

[11]  Munther A. Dahleh,et al.  Information Theoretic Bounds for Distributed Computation Over Networks of Point-to-Point Channels , 2010, IEEE Transactions on Information Theory.

[12]  A. Razborov.  Communication Complexity , 2011 .

[13]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[14]  M. Hellman.  The Mathematics of Public-Key Cryptography , 1979 .

[15]  Thomas C. Hales.  Sphere packings, I , 1997, Discret. Comput. Geom..

[16]  Sean A. Ramprashad,et al.  Cellular and Network MIMO architectures: MU-MIMO spectral efficiency and costs of channel state information , 2009, 2009 Conference Record of the Forty-Third Asilomar Conference on Signals, Systems and Computers.

[17]  László Babai,et al.  On Lovász’ lattice reduction and the nearest lattice point problem , 1986, Comb..

[18]  Prakash Ishwar,et al.  Some Results on Distributed Source Coding for Interactive Function Computation , 2011, IEEE Transactions on Information Theory.