紹介
First developed in the early 1980s by Lenstra, Lenstra, and Lovasz, the LLL algorithm was originally used to provide a polynomial-time algorithm for factoring polynomials with rational coefficients. It very quickly became an essential tool in integer linear programming problems and was later adapted for use in cryptanalysis. This book provides an introduction to the theory and applications of lattice basis reduction and the LLL algorithm. With numerous examples and suggested exercises, the text discusses various applications of lattice basis reduction to cryptography, number theory, polynomial factorization, and matrix canonical forms.
目次
Introduction to Lattices Euclidean space Rn Lattices in Rn Geometry of numbers Projects Exercises Two-Dimensional Lattices The Euclidean algorithm Two-dimensional lattices Vallee's analysis of the Gaussian algorithm Projects Exercises Gram-Schmidt Orthogonalization The Gram-Schmidt theorem Complexity of the Gram-Schmidt process Further results on the Gram-Schmidt process Projects Exercises The LLL Algorithm Reduced lattice bases The original LLL algorithm Analysis of the LLL algorithm The closest vector problem Projects Exercises Deep Insertions Modifying the exchange condition Examples of deep insertion Updating the GSO Projects Exercises Linearly Dependent Vectors Embedding dependent vectors The modified LLL algorithm Projects Exercises The Knapsack Problem The subset-sum problem Knapsack cryptosystems Projects Exercises Coppersmith's Algorithm Introduction to the problem Construction of the matrix Determinant of the lattice Application of the LLL algorithm Projects Exercises Diophantine Approximation Continued fraction expansions Simultaneous Diophantine approximation Projects Exercises The Fincke-Pohst Algorithm The rational Cholesky decomposition Diagonalization of quadratic forms The original Fincke-Pohst algorithm The FP algorithm with LLL preprocessing Projects Exercises Kannan's Algorithm Basic definitions Results from the geometry of numbers Kannan's algorithm Complexity of Kannan's algorithm Improvements to Kannan's algorithm Projects Exercises Schnorr's Algorithm Basic definitions and theorems A hierarchy of polynomial-time algorithms Projects Exercises NP-Completeness Combinatorial problems for lattices A brief introduction to NP-completeness NP-completeness of SVP in the max norm Projects Exercises The Hermite Normal Form The row canonical form over a field The Hermite normal form over the integers The HNF with lattice basis reduction Systems of linear Diophantine equations Using linear algebra to compute the GCD The HMM algorithm for the GCD The HMM algorithm for the HNF Projects Exercises Polynomial Factorization The Euclidean algorithm for polynomials Structure theory of finite fields Distinct-degree decomposition of a polynomial Equal-degree decomposition of a polynomial Hensel lifting of polynomial factorizations Polynomials with integer coefficients Polynomial factorization using LLL Projects Exercises
