J. C. Lagarias. Point lattices. In R. L. Graham et al. (eds.), Handbook of Combinatorics, pages 919--966. MIT Press, 1995. 19  Home/Search   Document Not in Database   Summary   Related Articles   Check

....factor rational polynomials in polynomial time (back then, a famous problem) from which the name LLL comes. Further refinements of the LLL algorithm were later proposed, notably by Schnorr [121, 122] Those algorithms have proved invaluable in many areas of mathematics and computer science (see [91, 78, 132, 64, 36, 84]) In particular, their relevance to cryptology was immediately understood, and they were used to break schemes based on the knapsack problem (see [119, 29] which were early alternatives to the RSA cryptosystem [120] The success of reduction algorithms at breaking various cryptographic schemes ....

....We will not discuss Ajtai s worst case average case equivalence [3, 33] which refers to special versions of SVP and SBP (see [30, 31, 14] such as SVP when the lattice gap 2 = 1 is at least polynomial in the dimension. 2. 4 Algorithmic results The main algorithmic results are surveyed in [91, 78, 132, 64, 36, 84, 30, 109]. No polynomial time algorithm is known for approximating either SVP, CVP or SBP to within a polynomial factor in the dimension d. In fact, the existence of such algorithms is an important open problem. The best polynomial time algorithms achieve only slightly subexponential factors, and are based ....

J. C. Lagarias. Point lattices. In R. Graham, M. Grotschel, and L. Lov'asz, editors, Handbook of Combinatorics, volume 1, chapter 19. Elsevier, 1995.

....factor rational polynomials in polynomial time (back then, a famous problem) from which the name LLL comes. Further refinements of the LLL algorithm were later proposed, notably by Schnorr [101, 102] Those algorithms have proved invaluable in many areas of mathematics and computer science (see [75, 64, 109, 52, 30, 69]) In particular, their relevance to The technique is however polynomial time for fixed dimension, which was enough in [74] cryptology was immediately understood, and they were used to break schemes based on the knapsack problem (see [99, 23] which were early alternatives to the RSA ....

....We will not discuss Ajtai s worst case average case equivalence [3, 27] which refers to special versions of SVP and SBP (see [24, 25, 11] such as SVP when the lattice gap 2 = 1 is at least polynomial in the dimension. 2. 4 Algorithmic results The main algorithmic results are surveyed in [75, 64, 109, 52, 30, 69, 24, 97]. No polynomial time algorithm is known for approximating either SVP, CVP or SBP to within a polynomial factor in the dimension d. In fact, the existence of such algorithms is an important open problem. The best polynomial time algorithms achieve only slightly subexponential factors, and are based ....

J. C. Lagarias. Point lattices. In R. Graham, M. Grotschel, and L. Lov'asz, editors, Handbook of Combinatorics, volume 1, chapter 19. Elsevier, 1995.

....types for k strings depends only on k (namely, it is given by the Bell number B(k) k ) Using the column types, Closest String can be formulated as an integer linear program (ILP) having only B(k) k 1) variables. Since ILPs with a constant number of variables can be solved in linear time [5, 7, 8], this is also true for Closest String with constant k. The algorithms, however, lead to huge running times, even for moderate number of variables. For this reason, we present a direct (not using linear programming) and ecient linear time algorithm that solves Closest String for k = 3. We start ....

J. C. Lagarias. Point lattices. In R. L. Graham et al. (eds.) Handbook of Combinatorics, pages 919-966. MIT Press, 1995.

....terminates. 1. Since a b, b g, g a are positive, at most one conorm is negative. 13 7.0.3 Remark. The algorithms described here are theoretical rather than practical. In practice a reduction algorithm such as that of Lenstra, Lenstra and Lovasz 1982 (see also Lagarias 1996) would be used before applying our algorithm. 8. The five parallelohedra In this section we study the Voronoi cells of three dimensional lattices and prove Fedorov s theorem. Let L be an arbitrary three dimensional lattice, with obtuse superbase v 0 , v 1 , v 2 , v 3 and conorms p ij . A ....

Lagarias, J. C. 1996. "Point lattices," in Handbook of Combinatorics, ed. R. L. Graham et al., North-Holland, Amsterdam, 1996, pp. 919-966.

....to obtain an order k Gamma 1 square matrix M . The row vectors b 1 ; b k Gamma1 of M can be thought of as defining a k Gamma 1 dimensional lattice, whose fundamental parallelohedron is the set of all points P k Gamma1 i=1 ff i b i , where 0 ff i 1. As is well known (see for example [21], Section 2) jDet(M)j is the volume of the above parallelohedron, and the volume of the simplex K is jDet(M)j= k Gamma 1) More generally, the k vertices of the simplex K may be expressed as points in L dimensional space, where L k Gamma 1. We can associate with K a k by L matrix MK and ....

J. Lagarias. "Point lattices". In Handbook of Combinatorics, Volume 1, R. Graham, M. Grotschel, L. Lovasz (editors), MIT Press, 1995.

....factor rational polynomials in polynomial time (back then, a famous problem) from which the name LLL comes. Further refinements of the LLL algorithm were later proposed, notably by Schnorr [101, 102] Those algorithms have proved invaluable in many areas of mathematics and computer science (see [75, 64, 109, 52, 30, 69]) In particular, their relevance to 1 The technique is however polynomial time for fixed dimension, which was enough in [74] 2 cryptology was immediately understood, and they were used to break schemes based on the knapsack problem (see [99, 23] which were early alternatives to the RSA ....

....We will not discuss Ajtai s worst case average case equivalence [3, 27] which refers to special versions of SVP and SBP (see [24, 25, 11] such as SVP when the lattice gap 2 = 1 is at least polynomial in the dimension. 2. 4 Algorithmic results The main algorithmic results are surveyed in [75, 64, 109, 52, 30, 69, 24, 97]. No polynomial time algorithm is known for approximating either SVP, CVP or SBP to within a polynomial factor in the dimension d. In fact, the existence of such algorithms is an important open problem. The best polynomial time algorithms achieve only slightly subexponential factors, and are based ....

J. C. Lagarias. Point lattices. In R. Graham, M. Grotschel, and L. Lov'asz, editors, Handbook of Combinatorics, volume 1, chapter 19. Elsevier, 1995.

No context found.

J. C. Lagarias. Point lattices. In R. L. Graham et al. (eds.), Handbook of Combinatorics, pages 919--966. MIT Press, 1995. 19

No context found.

J. C. Lagarias. Point lattices. In R. L. Graham et al. (eds.) Handbook of Combinatorics, pages 919-966. MIT Press, 1995.

No context found.

J. C. Lagarias. Point lattices. In R. Graham, M. Gr otschel, and L. Lovasz, editors, Handbook of Combinatorics, volume I, chapter 19, pages 919--966. Elsevier Science Publishers B.V., 1995.

No context found.

Je#rey C. Lagarias. Point lattices. In R. Graham, M. Gr otschel, and L. Lovasz, editors, Handbook of Combinatorics, volume I, chapter 19, pages 919--966. Elsevier Science Publishers B.V., 1995.

No context found.

J.C. Lagarias, Point lattices, in: Handbook of Combinatorics, Vol. 1, 2, 919--966, Elsevier, Amsterdam, 1995.

No context found.

J. Lagarias, Point lattices, in "Handbook of Combinatorics" (R. Graham, M. Grotschel and L. Lovasz, eds.), MIT Press, Cambridge, 1995, p. 919-966.

No context found.

J. Lagarias, Point lattices, in "Handbook of Combinatorics" (R. Graham, M. Grotschel and L. Lov'asz, eds.), MIT Press, Cambridge, 1995, p. 919-966.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC