On Nearly Orthogonal Lattice Bases and ... - Researcher - IBM

More documents

Recommendations

Info

[4] L. Babai, “On Lovász’ lattice reduction and the nearest lattice point problem,” Combinatorica, vol. 6, pp. 1–14,1986.[5] H. H. Bauschke, C. H. Hamilton, M. S. Macklem, J. S. McMichael, and N. R. Swart, “Recompression of JPEGimages by requantization,” IEEE Trans. Image Processing, vol. 12, pp. 843–849, Jul. 2003.[6] E. Candès and T. Tao, “Near optimal signal recovery from random projections and universal encoding strategies,”IEEE Trans. Inform. Theory, 2004. Submitted.[7] O. Damen, A. Chkeif, and J. Belfiore, “Lattice code decoder for space-time codes,” IEEE Commun. Lett.,pp. 161–163, May 2000.[8] S. Dash and R. Neelamani, “Some properties of SVP in random lattices”, 2006. Manuscript in preparation.[9] H. Daudé and B. Vallée, “An upper bound on the average number of iterations of the LLL algorithm,” TheoreticalComp. Sci., vol. 123, pp. 95-115, 1994.[10] I. Dinur, G. Kindler, R. Raz, and S. Safra, “Approximating CVP to within almost-polynomial factors is NP-hard,”Combinatorica, vol. 23, no. 2, pp. 205–243, 2003.[11] J. L. Donaldson, “Minkowski reduction of integral matrices,” Mathematics of Computation, vol. 33, no. 145,pp. 201—216, 1979.[12] N. El-Karoui, “Recent results about the largest eigenvalue of random covariance matrices and statistical applications,”Acta Physica Polonica B, vol. 36, no. 9, pp. 2681–2697, 2005.[13] C. F. Gauss, Disquisitiones Arithmeticae. New York: Springer-Verlag, English edition translated by A. A. Clark,1986.[14] R.A. Horn and C.R. Johnson, “Matrix Analysis,” Cambridge University Press, Cambridge, UK, 1985.[15] R. Kannan, “Algorithmic geometry of numbers,” Annual Review of Comp. Sci. vol. 2, pp. 231–267, 1987.[16] S. Khot, “Hardness of approximating the shortest vector problem in lattices,” J. ACM, vol. 52, pp. 789–808, Sept.2005. Preliminary version in FOCS 2004.[17] A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovász, “Factoring polynomials with rational coefficients,” MathematischeAnnalen, vol. 261, pp. 515–534, 1982.[18] A.E. Litvak, A. Pajor, M. Rudelson and N. Tomczak-Jaegermann, “Singular values of random matrices andgeometry of random polytopes,” Advances in Mathematics, vol. 195, pp. 491–523, Aug. 2005.[19] V. A. Marchenko and L. A. Pastur, “Distribution of eigenvalues in certain sets of random matrices,” Mat. Sb.(N.S.), vol. 72, pp. 407–535, 1967. (in Russian).[20] D. Micciancio, “The shortest vector problem is NP-hard to approximate to within some constant,” SIAM J.Comp., vol. 30, pp. 2008–2035, 2001. Preliminary version in FOCS 1998.[21] D. Micciancio and S. Goldwasser, Complexity of lattice problems: A cryptographic perspective. Boston: KluwerAcademic, 2002.[22] R. Neelamani, Inverse Problems in Image Processing. Ph.D. dissertation, ECE Dept., Rice University, 2003.www.dsp.rice.edu/∼neelsh/publications/.[23] R. Neelamani, R. de Queiroz, Z. Fan, S. Dash, and R. G. Baraniuk, “JPEG compression history estimation forcolor images,” IEEE Trans. Image Processing, May 2006. To appear.[24] P. Nguyen and J. Stern, “Lattice reduction in cryptology: An update,” in Lecture notes in Comp. Sci., vol. 1838,pp. 85–112, Springer-Verlag, 2000.[25] P. Nguyen and D. Stehlé, “Low-dimensional lattice basis reduction revisited,” in Lecture Notes in Comp. Sci.,vol. 3076, pp. 338–357, Springer-Verlag, 2004.20
[26] W. Pennebaker and J. Mitchell, JPEG, Still Image Data Compression Standard. Van Nostrand Reinhold, 1993.[27] C. Poynton, A Technical Introduction to Digital Video. New York: Wiley, 1996.[28] O. Regev, “On lattices, learning with errors, random linear codes, and cryptography,” in Proc. 37th annual ACMSymp. Theory of Computing, (New York, NY, USA), pp. 84–93, ACM Press, 2005.[29] G. Sharma and H. Trussell, “Digital color imaging,” IEEE Trans. Image Processing, vol. 6, pp. 901–932, July1997.[30] J. W. Silverstein, “The smallest eigenvalue of a large dimensional Wishart matrix,” Ann. Probab., vol. 13,pp. 1364–1368, 1985.[31] G. Sorkin. Personal Communication. 2006.[32] V. V. Vazirani, Approximation Algorithms. Berlin: Springer-Verlag, 2001.21
Page 1 and 2: On Nearly Orthogonal Lattice Bases
Page 3 and 4: The closest vector problem (CVP) an
Page 5 and 6: (1,2)(1,1.5)90 deg60 deg(0,0)(a)(0,
Page 7 and 8: Invoking Theorem 1 for 2-D lattices
Page 9 and 10: l ∈ {1,2,... ,k}. Then⎡⎤u 11
Page 11 and 12: new properties of random bases and
Page 13 and 14: 5.4 Proof of Results on Random Latt
Page 15 and 16: We now focus on proving the second
Page 17 and 18: 6.5 Our ApproachOur approach is to
Page 19: 6.5.4 Experimental ResultsWe tested

On Nearly Orthogonal Lattice Bases and ... - Researcher - IBM

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?