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Finding short lattice vectors within mordell's inequality

Published: 17 May 2008 Publication History

Abstract

The celebrated Lenstra-Lenstra-Lovász lattice basis reduction algorithm (LLL) can naturally be viewed as an algorithmic version of Hermite's inequality on Hermite's constant. We present a polynomial-time blockwise reduction algorithm based on duality which can similarly be viewed as an algorithmic version of Mordell's inequality on Hermite's constant. This achieves a better and more natural approximation factor for the shortest vector problem than Schnorr's algorithm and its transference variant by Gama, Howgrave-Graham, Koy and Nguyen. Furthermore, we show that this approximation factor is essentially tight in the worst case.

References

[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. Lovasz, 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. Stehle. Worst-case Hermite-Korkine-Zolotarev reduced lattice bases. CoRR, abs/0801.3331, 2008.]]
[12]
C. Hermite. Extraits de lettres de M. Hermite a M. Jacobi sur differents objets de la theorie des nombres, deuxieme 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 NgVa07.]]
[15]
A. Korkine and G. Zolotareff. Sur les formes quadratiques. Math. Ann., 6:336--389, 1873.]]
[16]
J. L. Lagrange. Recherches d'arithmetique. Nouveaux Memoires de l'Academie de Berlin, 1773.]]
[17]
A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovasz. Factoring polynomials with rational coefficients. Mathematische Ann., 261:513--534, 1982.]]
[18]
L. Lovasz. An Algorithmic Theory of Numbers, Graphs and Convexity, volume 50. SIAM Publications, 1986.]]
[19]
CBMS-NSF Regional Conference Series in Applied Mathematics.]]
[20]
J. Martinet. Perfect lattices in Euclidean spaces, volume 327 of Grundlehren der Mathematischen Wissenschaften {Fundamental Principles of Mathematical Sciences}. Springer-Verlag, Berlin, 2003.]]
[21]
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.]]
[22]
J. Milnor and D. Husemoller. Symmetric bilinear forms. Math. Z, 1973.]]
[23]
L. J. Mordell. Observation on the minimum of a positive quadratic form in eight variables. J. London Math. Soc., 19:3--6, 1944.]]
[24]
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.]]
[25]
P. Q. Nguyen and B. Vallee, editors. LLL+25. Information Security and Cryptography. Springer, 2008. To appear.]]
[26]
P. Q. Nguyen and T. Vidick. Sieve algorithms for the shortest vector problem are practical. J. of Mathematical Cryptology, 2008. To appear.]]
[27]
O. Regev. On the complexity of lattice problems with polynomial approximation factors. 2007. To appear in NgVa07.]]
[28]
C.-P. Schnorr. A hierarchy of polynomial lattice basis reduction algorithms. Theoretical Computer Science, 53:201--224, 1987.]]
[29]
C. P. Schnorr and M. Euchner. Lattice basis reduction: improved practical algorithms and solving subset sum problems. Math. Programming, 66:181--199, 1994.]]
[30]
G. Villard. Parallel Lattice Basis Reduction. In Proc. ISSAC '92, pages 269--277. ACM, 1992.]]

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    cover image ACM Conferences
    STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computing
    May 2008
    712 pages
    ISBN:9781605580470
    DOI:10.1145/1374376
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    Published: 17 May 2008

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    Author Tags

    1. lattice reduction
    2. lll
    3. schnorr's algorithm
    4. slide reduction
    5. transference reduction.

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    May 17 - 20, 2008
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