Efficient polynomial L ∞-approximations

Proceedings - Symposium on Computer Arithmetic

Volumn , Issue , 2007, Pages 169-176

  Brisebarre, Nicolas ^a,b   Chevillard, Sylvain ^c  

^a UNIVERSITÉ JEAN MONNET   (France)

^b UNIV LYON   (France)

^c UNIVERSITÉ LYON 1   (France)

Author keywords

Absolute error; Closest vector problem; Efficient polynomial approximation; Floating point arithmetic; L norm; Lattice basis reduction; LLL algorithm

Indexed keywords

ABSOLUTE ERROR; CLOSEST VECTOR PROBLEM; EFFICIENT POLYNOMIAL APPROXIMATION; FLOATING-POINT ARITHMETIC;

ALGORITHMS; DIGITAL ARITHMETIC; FUNCTION EVALUATION; PROBLEM SOLVING;

POLYNOMIAL APPROXIMATION;

EID: 36048977318     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1109/ARITH.2007.17     Document Type: Conference Paper

References (24)

1. E. Agrell, T. Eriksson, A. Vardy, and K. Zeger. Closest point search in lattices. IEEE Transactions on Information Theory, 48(8):2201-2214, Aug. 2002.
2. L. Babai. On Lovász' lattice reduction and the nearest lattice point problem. Combinatorica, 6(1):1-13, 1986.
3. N. Brisebarre, J.-M. Muller, and A. Tisserand. Computing machine-efficient polynomial approximations. ACM Transactions on Mathematical Software, 32(2), June 2006.
4. J. W. S. Casseis. An introduction to the geometry of numbers. Classics in Mathematics. SpringerVerlag, Berlin, 1997. Corrected reprint of the 1971 edition.
5. E. W. Cheney. Introduction to approximation theory. AMS Chelsea Publishing, second edition, 1982.
6. M. Cornea, J. Harrison, and P. T. P. Tang. Scientific Computing on Itanium-Based Systems. Intel Press, 2002.
7. W. Gautschi. Numerical analysis. Birkhäuser Boston Inc., Boston, MA, 1997. An introduction.
8. O. Goldreich and S. Goldwasser. On the limits of non-approximability of lattice problems. In Proceedings of 30th Annual ACM Symposium on Theory of Computing (STOC), pages 1-9. ACM, May 1998.
9. P. M. Gruber and C. G. Lekkerkerker. Geometry of numbers, volume 37 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, second edition, 1987.
10. G. Hanrot and D. Stehlé. Tighter analysis of Kannan's enumeration algorithm. 2007. Submitted.
11. Intel corporation. Intel Itanium 2 processor reference manual for software development and optimization. Technical Report 251110-003, Intel, 2004.
12. R. Kannan. Minkowski's convex body theorem and integer programming. Math. Oper. Res., 12(3):415-440, 1987.
13. A. K. Lenstra, H. W. Lenstra, and L. Lovász. Factoring polynomials with rational coefficients. Math. Annalen, 261:515-534, 1982.
14. L. Lovász. An algorithmic theory of numbers, graphs and convexity, volume 50 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1986.
15. P. Markstein. IA-64 and Elementary Functions : Speed and Precision. Hewlett-Packard Professional Books. Prentice Hall, 2000.
16. D. Micciancio. The hardness of the closest vector problem with preprocessing. IEEE Trans. Inform. Theory, 47(3):1212-1215, 2001.
17. J.-M. Müller. Elementary Functions, Algorithms and Implementation. Birkhäuser, Boston, 1997.
18. P. Q. Nguyen and J. Stern. The two faces of lattices in cryptology. In Proceedings of CALC '01, Lecture Notes in Computer Science, volume 2146, pages 146-180. Springer Verlag, 2001.
19. E. Remes. Sur un procédé convergent d'approximations successives pour déterminer les polynômes d'approximation. CR. Acad. Sci Paris, 198:2063-2065, 1934.
20. S. Story and P. T. P. Tang. New algorithms for improved transcendental functions on IA-64. In Koren and Kornerup, editors, Proceedings of the 14th IEEE Symposium on Computer Arithmetic, pages 4-11, Los Alamitos, CA, Apr. 1999. IEEE Computer Society Press.
21. The Arénaire Project. CRlibm, Correctly Rounded mathematical library, July 2006. http://lipforge.ens-lyon.fr/www/ crlibm/.
22. V Shoup. NTL, a library for doing number theory, version 5.4. http://shoup.net/ntl/, 2005.
23. P. van Emde Boas. Another NP-complete problem and the complexity of computing short vectors in a lattice. Technical Report 81-04, Mathematische Instituut, University of Amsterdam, 1981.
24. L. Veidinger. On the numerical determination of the best approximations in the Chebyshev sense. Numerische Mathematik, 2:99-105, 1960.