Brian A. LaMacchia, "Basis Reduction Algorithms and Subset Sum problems ", SM Thesis, Dept. of Elect. Eng. and Comp. Sci., Massachusetts Institute of Technology, Cambridge, MA  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Schnorr[13] does suggest a family of lattice reduction algorithms that are theoretically superior to Lov asz s. Lagarias, Lenstra and Schnorr[8] also discuss a number of issues relevant to the problem of finding near optimal solutions to the nearest lattice point problem. See also LaMacchia[9] for a comparison of Lov asz and Seysen basis reduction. Another option for small n is brute force search. Experiments showed that with clever pruning, exhaustive search can find the optimum when, say, n 20, but the running time can be huge and the payoff is usually small. 4. l 2 spline with ....

....these problems as well. Acknowledgements We would like thank David Gay for his n2f nonlinear least squares program, Norm Schryer for his ssaf program, and Henry Baird and Ruby Jane Elliott for their implementation of Wall Danielsson. We thank Jeff Lagarias and Andrew Odlyzko for pointing us to [1, 8, 9, 15, 19]. Ken Thompson s extensive processing of the tiger database[18] made it possible to extract sample curves with moderate effort. Norm Schryer and Margaret Wright and an anonymous referee made helpful comments on the manuscript. PostScript is a registered trademark of Adobe Systems Incorporated. ....

B. A. LaMacchia, Basis reduction algorithms and subset sum problems, Master's thesis, EECS Dept., Massachusetts Institute of Technology, 1991.

....Euclidian norm shortest nonzero vector in a lattice (the job of the LLL algorithm) It is a very hard problem in general. The theoretical worst case bounds for the LLL algorithm and its variants are not encouraging. However, these techniques tend to perform much better in practice than in theory [7]. A lot of work has been done in this area, and many different approaches tried. One can point out two main research directions in applying LLL to solving Subset Sum problems. One is the improvement of the LLL algorithm itself. Many different heuristics were invented, and different definitions of ....

Brian A. LaMacchia, "Basis Reduction Algorithms and Subset Sum problems ", SM Thesis, Dept. of Elect. Eng. and Comp. Sci., Massachusetts Institute of Technology, Cambridge, MA

....degree of the polynomial that bounds the running time. Finding short vectors in lattices may be very hard in general. On the other hand, published algorithms, such as the L 3 one, perform much better in practice than is guaranteed by their worst case bounds, especially when they are modified [11, 12, 17], and new algorithms are being invented [18, 19, 20] Thus it is possible that on average, the problem of finding short vectors in lattices is easy, even if it is hard in the worst case. Therefore it seems worthwhile to separate the issues of efficiency of lattice basis reduction algorithms from ....

....to polynomial time solutions of almost all problems of density 0:9408 : Empirical tests show that this modification also leads to dramatic improvement in the performance of practical algorithms. We present some results on this in Section 4. More data and fuller comparisons will be given in [12]. In Section 2 we derive the Lagarias Odlyzko bound using the approach in [7] We show in Section 3 that this bound may be increased to 0:9408 : using a simple modification of the Lagarias Odlyzko attack. Finally, Section 4 discusses possible improvements on the new bound and practical ....

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B. A. LaMacchia, Basis Reduction Algorithms and Subset Sum Problems, SM Thesis, Dept. of Elect. Eng. and Comp. Sci., Massachusetts Institute of Technology, Cambridge, MA (1991).

....the degree of the polynomial that bounds the running time. Finding short vectors in lattices may be very hard in general. On the other hand, published algorithms, such as the L 3 one, perform much better in practice than is guaranteed by their worst case bounds, especially when they are modified [13, 14, 19, 22], and new algorithms are being invented [20, 21, 23] Thus it is possible that on average, the problem of finding short vectors in lattices is easy, even if it is hard in the worst case. Therefore it seems worthwhile to separate the issues of efficiency of lattice basis reduction algorithms from ....

....to polynomial time solutions of almost all problems of density 0:9408 : Empirical tests show that these modifications also lead to dramatic improvements in the performance of practical algorithms. We present some results on this in Section 5. More data and fuller comparisons are given in [14]. In Section 2 we derive the Lagarias Odlyzko bound using the approach in [8] We show in Section 3 that this bound may be increased to 0:9408 : using a simple modification of the Lagarias Odlyzko attack. Section 4 sketches the other modification, which appears to be quite different, but ....

[Article contains additional citation context not shown here]

B. A. LaMacchia, Basis Reduction Algorithms and Subset Sum Problems, SM Thesis, Dept. of Elect. Eng. and Comp. Sci., Massachusetts Institute of Technology, Cambridge, MA, 1991. Also available as AI Technical Report 1283, MIT Artificial Intelligence Laboratory, Cambridge, MA, 1991.

....the degree of the polynomial that bounds the running time. Finding short vectors in lattices may be very hard in general. On the other hand, published algorithms, such as the L 3 one, perform much better in practice than is guaranteed by their worst case bounds, especially when they are modified [13, 14, 19, 22], and new algorithms are being invented [20, 21, 23] Thus it is possible that on average, the problem of finding short vectors in lattices is easy, even if it is hard in the worst case. Therefore it seems worthwhile to separate the issues of efficiency of lattice basis reduction algorithms from ....

....to polynomial time solutions of almost all problems of density 0:9408 : Empirical tests show that these modifications also lead to dramatic improvements in the performance of practical algorithms. We present some results on this in Section 5. More data and fuller comparisons are given in [14]. In Section 2 we derive the Lagarias Odlyzko bound using the approach in [8] We show in Section 3 that this bound may be increased to 0:9408 : using a simple modification of the Lagarias Odlyzko attack. Section 4 sketches the other modification, which appears to be quite different, but ....

[Article contains additional citation context not shown here]

B. A. LaMacchia, Basis Reduction Algorithms and Subset Sum Problems, SM Thesis, Dept. of Elect. Eng. and Comp. Sci., Massachusetts Institute of Technology, Cambridge, MA, 1991. Also available as AI Technical Report 1283, MIT Artificial Intelligence Laboratory, Cambridge, MA, 1991.

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Brian A. LaMacchia, "Basis Reduction Algorithms and Subset Sum problems ", SM Thesis, Dept. of Elect. Eng. and Comp. Sci., Massachusetts Institute of Technology, Cambridge, MA

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