Complexity of lattice problems : a cryptographic perspective /
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| Author / Creator: | Micciancio, Daniele. |
|---|---|
| Imprint: | Boston : Kluwer Academic, c2002. |
| Description: | x, 220 p. : ill. ; 25 cm. |
| Language: | English |
| Series: | The Kluwer international series in engineering and computer science ; SECS 671 Kluwer international series in engineering and computer science ; SECS 671. |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/4735002 |
Table of Contents:
- Preface
- 1.. Basics
- 1. Lattices
- 1.1. Determinant
- 1.2. Successive minima
- 1.3. Minkowski's theorems
- 2. Computational problems
- 2.1. Complexity Theory
- 2.2. Some lattice problems
- 2.3. Hardness of approximation
- 3. Notes
- 2.. Approximation Algorithms
- 1. Solving SVP in dimension 2
- 1.1. Reduced basis
- 1.2. Gauss' algorithm
- 1.3. Running time analysis
- 2. Approximating SVP in dimension n
- 2.1. Reduced basis
- 2.2. The LLL basis reduction algorithm
- 2.3. Running time analysis
- 3. Approximating CVP in dimension n
- 4. Notes
- 3.. Closest Vector Problem
- 1. Decision versus Search
- 2. NP-completeness
- 3. SVP is not harder than CVP
- 3.1. Deterministic reduction
- 3.2. Randomized Reduction
- 4. Inapproximability of CVP
- 4.1. Polylogarithmic factor
- 4.2. Larger factors
- 5. CVP with preprocessing
- 6. Notes
- 4.. Shortest Vector Problem
- 1. Kannan's homogenization technique
- 2. The Ajtai-Micciancio embedding
- 3. NP-hardness of SVP
- 3.1. Hardness under randomized reductions
- 3.2. Hardness under nonuniform reductions
- 3.3. Hardness under deterministic reductions
- 4. Notes
- 5.. Sphere Packings
- 1. Packing Points in Small Spheres
- 2. The Exponential Sphere Packing
- 2.1. The Schnorr-Adleman prime number lattice
- 2.2. Finding clusters
- 2.3. Some additional properties
- 3. Integer Lattices
- 4. Deterministic construction
- 5. Notes
- 6.. Low-Degree Hypergraphs
- 1. Sauer's Lemma
- 2. Weak probabilistic construction
- 2.1. The exponential bound
- 2.2. Well spread hypergraphs
- 2.3. Proof of the weak theorem
- 3. Strong probabilistic construction
- 4. Notes
- 7.. Basis Reduction Problems
- 1. Successive minima and Minkowski's reduction
- 2. Orthogonality defect and KZ reduction
- 3. Small rectangles and the covering radius
- 4. Notes
- 8.. Cryptographic Functions
- 1. General techniques
- 1.1. Lattices, sublattices and groups
- 1.2. Discrepancy
- 1.3. Statistical distance
- 2. Collision resistant hash functions
- 2.1. The construction
- 2.2. Collision resistance
- 2.3. The iterative step
- 2.4. Almost perfect lattices
- 3. Encryption Functions
- 3.1. The GGH scheme
- 3.2. The HNF technique
- 3.3. The Ajtai-Dwork cryptosystem
- 3.4. Ntru
- 4. Notes
- 9.. Interactive Proof Systems
- 1. Closest vector problem
- 1.1. Proof of the soundness claim
- 1.2. Conclusion
- 2. Shortest vector problem
- 3. Treating other norms
- 4. What does it mean?
- 5. Notes
- References
- Index
