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Finding Reduced Basis for Lattices

Finding Reduced Basis for Lattices

Ido Heskia

Math/Csc 870

Slide 2

Due to:

Due to:

A.K. Lenstra

H.W. Lenstra

L. Lovasz

LLL Algorith

Introduction

Slide 3

A Lattice

A Lattice

1.

2.

Slide 4

Let n be a positive integer. A subset L of the n-dimensional real vector space is

Let n be a positive integer. A subset L of the n-dimensional real vector space is

called a lattice if there exists a basis b1,b2,,bn of such that

The bis span L.

n is the rank of L.

We will consider only

Slide 5

Constructing lattices:

Constructing lattices:

Slide 6

Determinant of L:

Determinant of L:

The bis are written as column

vectors. Apparently, this positive

real number doesnt depend on the

choice of the basis.

Slide 7

Let be linearly independent. Suppose it is a basis for

Let be linearly independent. Suppose it is a basis for

We perform the Gram-Schmidt process:

b1

b2

b2

0

0

L

0

Slide 8

Similarly, define:

Similarly, define:

Forms an orthogonal basis of L

Dividing by shortens our vectors.

Slide 9

A basis b1, ,bn of a lattice is called reduced if :

A basis b1, ,bn of a lattice is called reduced if :

for

* ¾ can be replaced by any ¼<y<1

* | | is Euclidean length.

Slide 10

Applications

Applications

Factoring polynomials with rational coeffecients

For example:

Lives in

Slide 11

An irreducible polynomial over a field is

An irreducible polynomial over a field is

non-constant and cannot be

represented as the product of at-least 2

non-constant Polynomials.

Reducible (over ):

Irreducible:

Slide 12

How to find, for a given non-zero

How to find, for a given non-zero

polynomial in its decomposition into

Irreducibles?

Factor primitive polynomials

(gcd of all coeffecients of f is 1)

Into irreducible factors in

Use LLL

Slide 13

Simultaneous Diophantine approximations

Simultaneous Diophantine approximations

Given , and

Find such that:

Or

Slide 14

Cryptography

Cryptography

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