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Techniques in Lattice Basis Reduction
- Date Issued:
- 2016
- Summary:
- The mathematical theory of nding a basis of shortest possible vectors in a given lattice L is known as reduction theory and goes back to the work of Lagrange, Gauss, Hermite, Korkin, Zolotarev, and Minkowski. Modern reduction theory is voluminous and includes the work of A. Lenstra, H. Lenstra and L. Lovasz who created the well known LLL algorithm, and many other researchers such as L. Babai and C. P. Schnorr who created signi cant new variants of basis reduction algorithms. The shortest vector (SVP) and closest vector (CVP) problems, presently considered intractable, are algorithmic tasks that lie at the core of many number theoretic problems, integer programming, nding irreducible factors of polynomials, minimal polynomials of algebraic numbers, and simultaneous diophantine approximation. Lattice basis reduction also has deep and extensive connections with modern cryptography, and cryptanalysis particularly in the post-quantum era. In this dissertation we study and compare current systems LLL and BKZ, and point out their strengths and drawbacks. In addition, we propose and investigate the e cacy of new optimization techniques, to be used along with LLL, such as hill climbing, random walks in groups, our lattice di usion-sub lattice fusion, and multistage hybrid LDSF-HC technique. The rst two methods rely on the sensitivity of LLL to permutations of the input basis B, and optimization ideas over the symmetric group Sm viewed as a metric space. The third technique relies on partitioning the lattice into sublattices, performing basis reduction in the partition sublattice blocks, fusing the sublattices, and repeating. We also point out places where parallel computation can reduce runtimes achieving almost linear speedup. The multistage hybrid technique relies on the lattice di usion and sublattice fusion and hill climbing algorithms. Unlike traditional methods, our approach brings in better results in terms of basis reduction towards nding shortest vectors and minimal weight bases. Using these techniques we have published the competitive lattice vectors of ideal lattice challenge on the lattice hall of fame. Toward the end of the dissertation we also discuss applications to the multidimensional knapsack problem that resulted in the discovery of new large sets of geometric designs still considered very rare. The research introduces innovative techniques in lattice basis reduction theory and provides some space for future researchers to contemplate lattices from a new viewpoint.
Title: | Techniques in Lattice Basis Reduction. |
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Name(s): |
Khadka, Bal K., author Magliveras, Spyros S., Thesis advisor Florida Atlantic University, Degree grantor Charles E. Schmidt College of Science Department of Mathematical Sciences |
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Type of Resource: | text | |
Genre: | Electronic Thesis Or Dissertation | |
Date Created: | 2016 | |
Date Issued: | 2016 | |
Publisher: | Florida Atlantic University | |
Place of Publication: | Boca Raton, Fla. | |
Physical Form: | application/pdf | |
Extent: | 146 p. | |
Language(s): | English | |
Summary: | The mathematical theory of nding a basis of shortest possible vectors in a given lattice L is known as reduction theory and goes back to the work of Lagrange, Gauss, Hermite, Korkin, Zolotarev, and Minkowski. Modern reduction theory is voluminous and includes the work of A. Lenstra, H. Lenstra and L. Lovasz who created the well known LLL algorithm, and many other researchers such as L. Babai and C. P. Schnorr who created signi cant new variants of basis reduction algorithms. The shortest vector (SVP) and closest vector (CVP) problems, presently considered intractable, are algorithmic tasks that lie at the core of many number theoretic problems, integer programming, nding irreducible factors of polynomials, minimal polynomials of algebraic numbers, and simultaneous diophantine approximation. Lattice basis reduction also has deep and extensive connections with modern cryptography, and cryptanalysis particularly in the post-quantum era. In this dissertation we study and compare current systems LLL and BKZ, and point out their strengths and drawbacks. In addition, we propose and investigate the e cacy of new optimization techniques, to be used along with LLL, such as hill climbing, random walks in groups, our lattice di usion-sub lattice fusion, and multistage hybrid LDSF-HC technique. The rst two methods rely on the sensitivity of LLL to permutations of the input basis B, and optimization ideas over the symmetric group Sm viewed as a metric space. The third technique relies on partitioning the lattice into sublattices, performing basis reduction in the partition sublattice blocks, fusing the sublattices, and repeating. We also point out places where parallel computation can reduce runtimes achieving almost linear speedup. The multistage hybrid technique relies on the lattice di usion and sublattice fusion and hill climbing algorithms. Unlike traditional methods, our approach brings in better results in terms of basis reduction towards nding shortest vectors and minimal weight bases. Using these techniques we have published the competitive lattice vectors of ideal lattice challenge on the lattice hall of fame. Toward the end of the dissertation we also discuss applications to the multidimensional knapsack problem that resulted in the discovery of new large sets of geometric designs still considered very rare. The research introduces innovative techniques in lattice basis reduction theory and provides some space for future researchers to contemplate lattices from a new viewpoint. | |
Identifier: | FA00004678 (IID) | |
Degree granted: | Dissertation (Ph.D.)--Florida Atlantic University, 2016. | |
Collection: | FAU Electronic Theses and Dissertations Collection | |
Note(s): | Includes bibliography. | |
Subject(s): |
Cryptography. Combinatorial analysis. Group theory. |
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Held by: | Florida Atlantic University Libraries | |
Sublocation: | Digital Library | |
Persistent Link to This Record: | http://purl.flvc.org/fau/fd/FA00004678 | |
Use and Reproduction: | Copyright © is held by the author, with permission granted to Florida Atlantic University to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. | |
Use and Reproduction: | http://rightsstatements.org/vocab/InC/1.0/ | |
Host Institution: | FAU | |
Is Part of Series: | Florida Atlantic University Digital Library Collections. |