Complexity of Lattice Problems: A Cryptographic Perspective

by Daniele Micciancio, Shafi Goldwasser

Overview

Lattices are geometric objects that can be pictorially described as the set of intersection points of an infinite, regular n-dimensional grid. De­ spite their apparent simplicity, lattices hide a rich combinatorial structure, which has attracted the attention of great mathematicians over the last two centuries. Not surprisingly, lattices have found numerous applications in mathematics and computer science, ranging from number theory and Diophantine approximation, to combinatorial optimization and cryptography. The study of lattices, specifically from a computational point of view, was marked by two major breakthroughs: the development of the LLL lattice reduction algorithm by Lenstra, Lenstra and Lovasz in the early 80's, and Ajtai's discovery of a connection between the worst-case and average-case hardness of certain lattice problems in the late 90's. The LLL algorithm, despite the relatively poor quality of the solution it gives in the worst case, allowed to devise polynomial time solutions to many classical problems in computer science. These include, solving integer programs in a fixed number of variables, factoring polynomials over the rationals, breaking knapsack based cryptosystems, and finding solutions to many other Diophantine and cryptanalysis problems.

Product Details

ISBN-13:	9780792376880
Publisher:	Springer US
Publication date:	03/31/2002
Series:	The Springer International Series in Engineering and Computer Science , #671
Edition description:	2002
Pages:	220
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

1 Basics.- 1 Lattices.- 2 Computational problems.- 3 Notes.- 2. Approximation Algorithms.- 1 Solving SVP in dimension.- 2 Approximating SVP in dimension n.- 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.- 4 Inapproximability of CVP.- 5 CVP with preprocessing.- 6 Notes.- 4. Shortest Vector Problem.- 1 Kannan’s homogenization technique.- 2 The Ajtai-Micciancio embedding.- 3 NP-hardnessofSVP.- 4 Notes.- 5. Sphere Packings.- 1 Packing Points in Small Spheres.- 2 The Exponential Sphere Packing.- 3 Integer Lattices.- 4 Deterministic construction.- 5 Notes.- 6. Low-Degree Hypergraphs.- 1 Sauer’s Lemma.- 2 Weak probabilistic construction.- 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.- 2 Collision resistant hash functions.- 3 Encryption Functions.- 4 Notes.- 9. Interactive Proof Systems.- 1 Closest vector problem.- 2 Shortest vector problem.- 3 Treating other norms.- 4 What does it mean?.- 5 Notes.- References.

Editorial Reviews

Micciancio (U. of California, San Diego) and Goldwasser (Massachusetts Institute of Technology) introduce lattices, deceptively simple geometric objects that can be pictorially depicted as the set of intersection points of an infinite, regular n-dimensional grid. Their rich combinatoratorial structure has attracted mathematicians and computer scientists working on diverse applications from number theory to cryptography. Following Atjai's lead, the authors focus on designing cryptographic functions that are as difficult to break as solving a computationally hard lattice problem. After introducing the basics of point lattices, complexity theory, and Minkowski's theorems, they venture into specific types of algorithmic problems (e.g., shortest vector, closest vector, and basis reduction problems). The final chapters delve into cryptographic functions and interactive proof systems. Annotation c. Book News, Inc., Portland, OR (booknews.com)