Finding short lattice vectors within mordell's inequality

Proceedings of the Annual ACM Symposium on Theory of Computing

Volumn , Issue , 2008, Pages 207-216

  Gama, Nicolas ^a   Nguyen, Phong Q ^a  

^a ECOLE NORMALE SUPÉRIEURE   (France)

Author keywords

Lattice reduction; LLL; Schnorr's algorithm; Slide reduction; Transference reduction

Indexed keywords

APPROXIMATION ALGORITHMS; POLYNOMIAL APPROXIMATION;

APPROXIMATION FACTOR; LATTICE BASIS REDUCTION; LATTICE REDUCTION; REDUCTION ALGORITHMS; SCHNORR'S ALGORITHM; SHORTEST VECTOR PROBLEM; SLIDE REDUCTION; TRANSFERENCE REDUCTION;

COMPUTATION THEORY;

EID: 57049097675     PISSN: 07378017     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1145/1374376.1374408     Document Type: Conference Paper

References (29)

1. M. Ajtai. The worst-case behavior of Schnorr's algorithm approximating the shortest nonzero vector in a lattice, in Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, pages 396-406 (electronic), New York, 2003. ACM.
2. M. Ajtai, R. Kumar, and D. Sivakumar. A sieve algorithm for the shortest lattice vector problem, in Proc. 33rd STOC, pages 601-610. ACM, 2001.
3. H. Cohen. A Course in Computational Algebraic Number Theory. Springer-Verlag, 1995. Second edition.
4. H. Cohn and N. Elkies. New upper bounds on sphere packings. I. Ann. of Math. (2), 157(2):689-714, 2003.
5. J. Conway and N. Sloane. Sphere Packings, Lattices and Groups. Springer-Verlag, 1998. Third edition.
6. N. Gama, N. Howgrave-Graham, H. Koy, and P. Q. Nguyen, Rankin's constant and blockwise lattice reduction. In Proceedings of CRYPTO '06, volume 4117 of LNCS, Springer, pages 112-130, 2006.
7. N. Gama, N. Howgrave-Graham, and P. Q. Nguyen, Symplectic Lattice Reduction and NTRU. In Proceedings of EUROCRYPT '06, volume 4004 of LNCS, Springer, pages 233-253, 2006.
8. N. Gama and P. Q. Nguyen. Predicting Lattice Reduction. In Proceedings of EUROCRYPT '08. LNCS, Springer Verlag, pages 31 - 51, 2008.
9. M. Grötschel, L. Lovász, and A. Schrijver. Geometric algorithms and combinatorial optimization, volume 2 of Algorithms and Combinatorics: Study and Research Texts. Springer-Verlag, Berlin, 1988.
10. M. Gruber and C. G. Lekkerkerker. Geometry of Numbers. North-Holland, 1987.
11. G. Hanrot and D. Stehlé. Worst-case Hermite-Korkine-Zolotarev reduced lattice bases. CoR.R., abs/0801.3331, 2008.
12. C. Hermite. Extraits de lettres de M. Hermite à M. Jacobi sur différents objets de la théorie des nombres, deuxiéme lettre. J. Reine Angew. Math., 40:279-290, 1850. Also available in the first volume of Hermite's complete works, published by Gauthier-Villars.
13. R. Kannan. Improved algorithms for integer programming and related lattice problems. In Proc. of 15th STOC, pages 193-206. ACM, 1983.
14. S. Khot. Inapproximability results for computational problems on lattices. 2007. To appear in [24].
15. A. Korkine and G. Zolotareff. Sur les formes quadratiques. Math. Ann., 6:336-389, 1873.
16. J. L. Lagrange. Recherches d'arithmétique. Nouveaux Mémoires de l'Acadeḿie de Berlin, 1773.
17. A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Ann., 261:513-534, 1982.
18. L. Lovász. An Algorithmic Theory of Numbers, Graphs and Convexity, volume 50. SIAM Publications, 1986. CBMS-NSF Regional Conference Series in Applied Mathematics.
19. J. Martinet. Perfect lattices in Euclidean spaces, volume 327 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2003.
20. D. Micciancio and S. Goldwasser. Complexity of lattice problems. The Kluwer International Series in Engineering and Computer Science, 671. Kluwer Academic Publishers, Boston, MA, 2002. A cryptographic perspective.
21. J. Milnor and D. Husemoller. Symmetric bilinear forms. Math. Z, 1973.
22. L. J. Mordell. Observation on the minimum of a positive quadratic form in eight variables. J. London Math. Soc., 19:3-6, 1944.
23. P. Q. Nguyen and J. Stern. The two faces of lattices in cryptology. In Proc. of CALC '01, volume 2146 of LNCS. Springer-Verlag, 2001.
24. P. Q. Nguyen and B. Vallée, editors. LLL+25 Information Security and Cryptography. Springer, 2008. To appear.
25. P. Q. Nguyen and T. Vidiek. Sieve algorithms for the shortest vector problem are practical, J. of Mathematical Cryptology, 2008. To appear.
26. O. Regev. On the complexity of lattice problems with polynomial approximation factors. 2007. To appear in [24].
27. C.-P. Schnorr. A hierarchy of polynomial lattice basis reduction algorithms. Theoretical Computer Science. 53:201-224, 1987.
28. C. P. Schnorr and M. Euchner. Lattice basis reduction: improved practical algorithms and solving subset sum problems. Math. Programming, 66:181-199, 1994.
29. G. Villard. Parallel Lattice Basis Reduction, In Proc. ISSAC '92, pages 269-277. ACM, 1992.