Lattices in Computer Science Lecture 2 LLL Algorithm

Lattices in Computer Science Lecture 2 LLL Algorithm Lecturer: Oded Regev Scribe: Eyal Kaplan In this lecture1 we describe an approximation algorithm to the Shortest Vector Problem (SVP). This algorithm, developed in 1982 by A. K. Lenstra, H. W. Lenstra, Jr. and L. Lovasz, usually called the LLL algorithm, gives a ( 2 √ 3 ) n approximation ratio, where n is the dimension of the lattice. In many of the applications, this algorithm is applied for a constant n; in such cases, we obtain a constant approximation factor. In 1801, Gauss gave an algorithm that can be viewed as an algorithm for solving SVP in two dimensions. The LLL algorithm is, in some way, a generalization of Gauss’s algorithm to higher dimensions. In 1987, Schnorr presented an improved algorithm for the SVP. This improved algorithm obtains an approximation factor that is slightly subexponential, namely 2O(n(log log n) 2/ log n). The LLL algorithm has many applications in diverse fields of computer science. Some of these will be described in the following lectures. Here is a brief description of some of these applications.