Complexity of Lattice Problems

A Cryptographic Perspective

  • Daniele Micciancio
  • Shafi Goldwasser

Table of contents

  1. Front Matter
    Pages i-x
  2. Daniele Micciancio, Shafi Goldwasser
    Pages 1-22
  3. Daniele Micciancio, Shafi Goldwasser
    Pages 23-44
  4. Daniele Micciancio, Shafi Goldwasser
    Pages 45-68
  5. Daniele Micciancio, Shafi Goldwasser
    Pages 69-90
  6. Daniele Micciancio, Shafi Goldwasser
    Pages 91-110
  7. Daniele Micciancio, Shafi Goldwasser
    Pages 111-124
  8. Daniele Micciancio, Shafi Goldwasser
    Pages 125-142
  9. Daniele Micciancio, Shafi Goldwasser
    Pages 143-194
  10. Daniele Micciancio, Shafi Goldwasser
    Pages 195-210
  11. Back Matter
    Pages 211-220

About this book

Introduction

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 struc­ ture, which has attracted the attention of great mathematicians over the last two centuries. Not surprisingly, lattices have found numerous ap­ plications 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.

Keywords

Approximation Hypergraph algorithms combinatorics complexity complexity theory computational complexity cryptography

Authors and affiliations

  • Daniele Micciancio
    • 1
  • Shafi Goldwasser
    • 2
  1. 1.University of CaliforniaSan DiegoUSA
  2. 2.The Massachusetts Institute of TechnologyCambridgeUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4615-0897-7
  • Copyright Information Kluwer Academic Publishers 2002
  • Publisher Name Springer, Boston, MA
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4613-5293-8
  • Online ISBN 978-1-4615-0897-7
  • Series Print ISSN 0893-3405
  • About this book