Finding Reduced Basis for Lattices

Ido Heskia

Math/Csc 870

Due to:

A.K. Lenstra

H.W. Lenstra

L. Lovasz

LLL Algorithm

Introduction

A Lattice

1.

2. 1 1

|n n

i i i ii i

L b rb r

Let n be a positive integer. A subset L of the n-dimensional real vector space iscalled a lattice if there exists a basis b1,b2,…,bn of such that

The bi’s span L. n is the rank of L.We will consider only

1 1

|n n

i i i ii i

L b rb r

n

n

nib

Constructing lattices:

1det , , nd L b b Determinant of L:

The bi’s are written as column

vectors. Apparently, this positive

real number doesn’t depend on the

choice of the basis.

Let be linearly independent. Suppose it is a basis for

We perform the Gram-Schmidt process:

1,n

nb b

b1

b2 b2

2Lproj b0 0

L

0

*2 2 2Lb b proj b

nL

*1b

*1b

Similarly, define:*

1 1b b* *

2 2 21 1b b b

* * *ij i j j jb b b b

1* *

1

i

i i ij jj

b b b

* *1 , , nb b Forms an orthogonal basis of L

Dividing by shortens our vectors.

2*jb

* * *3 3 31 1 32 2b b b b

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

1) for

2)

* ¾ can be replaced by any ¼<y<1

1

2ij

2 2* * *1 1 1

3,1

4i ii i ib b b i n

* | | is Euclidean length.

1 j i n

Applications

Factoring polynomials with rational coeffecients

0

| , 0n

ii i

i

x a x a n

For example:

Lives in

5 2

42 73 2

x xf x x x

x

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:

2 1 1 1x x x 2 1x

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

f x

x

x

Simultaneous Diophantine approximations

Given , and

Find such that:

Or

n 1, n 0 1

1, , ,np p q

,1 ni ip q q

1

1ii n

p

q q

Cryptography

For given positive

Do there exist such that:

(is s a subset sum of the mi’s)?

1, , ,nm m s

1, , 0,1nz z

1 1 n ns z m z m

Sums of squares

Every prime that is 1mod4 can be

written as sum of two squares.

Those squares are found using LLL

abc Conjecture

For define the radical

, ,a b c

p prime

|

, ,

p abc

rad a b c p

(That’s the product of distinct prime factors of a,b,c). suppose gcd(a,b,c)=1.

log

, ,log , ,

cq a b c

rad a b c

abc conjecture: For every x>1 there exists only finitely many a,b,c with gcd(a,b,c) = 1 and a + b = c such that

, ,q a b c x

The search for examples uses LLL

Proposition:

B1,bn are reduced basis for a lattice L in b1*, bn* defined as before. Then:

1.

2.

3.

4.

(i.e. the 1st vector is “reasonably” short).

22 1 *2 ,1ij ib b j i n

1

4

1

2nn n

ii

d L b d L

1 1

41 2

n

nb d L

Reduced basis, what is it good for?

n

2 211 2 , , 0nb x x L x

Algorithm.doc

Example.doc

Algorithm terminates:

det ,1 ,i j ld b b j l i

*

1

,0i

i jj

d b i n

so each is a pos. real number

20 1, nd d d L

1

1

n

ii

D d

D changes only if some bi* is changed, which only occurs at case 1 of the algorithm. The number is reduced by a factor of ¾ since is, while the other

di’s are unchanged. Hence D reduced by factor of ¾ .

1kd *

1kb

di’s are bounded from below which bounds D from below.

2min : , 0m L x x L x

iid m L

So there’s an upper bound for # of times we pass through case 1.

In end of case 1, k = k-1

End of case 2, k = k+1

Start with k = 2, and 1k n

So # of times we pass through case 2

Is at most n-1 more than the # of times we pass through case 1,

Hence the algorithm terminates.

Complexity:

Initialization step with rationales: 3O n

# of times pass through case 1:

# of times pass through case 2:

2 logO n B

2 logO n B

2, 2, iB B b B

Case 1 requires operations

Case 2 we have values of p

Each requires operations

O n

O n

O n

Hence we get a total of

Operations.

Polynomial Time.

4 logO n B

References:Factoring Polynomials with Rational Coeffecients-- A.K. Lenstra, H.W. Lenstra, Jr. and L. LovaszA Course in Convexity-- Alexander BarvinokLattice Basis Reduction Algorithms and Applications-- Matthew C. CarySome Applications of LLL-- http://www.math.ru.nl/~bosma/onderwijs/voorjaar07/compalg8.pdfLinear Algebra with Applications-- Otto BretcherLattices-- www.cs.tau.ac.il/~safra/ACT2/Lattices.ppt