Polynomial Factorization

Page
4

We will use this factorization to get the factorization of f

modulo

Slide 28

Hensel techniques reminder

We will use this factorization to get the factorization of f

modulo

More precisely, if we have

we will call Hensel continuation of this factorization a factorization

Slide 29

Hensel techniques reminder

Lemma (Hensel)

If then for any factorization , satisfying the above conditions, there exists its Hensel continuation

, and the polynomials are

defined uniquely modulo

Slide 30

UFA: step 2

Step 2, ‘Use Newton’s iteration to lift the to factors modulo ’.

We choose l considering the bounds on the coefficients of the factors.

Slide 31

UFA: step 2

Step 2, ‘Use Newton’s iteration to lift the to factors modulo ’.

We choose l considering the bounds on the coefficients of the factors.

Theorem (Mignotte) Let

Slide 32

UFA: step 2

We have an upper bound for the coefficients factors of f, say M. We then choose l such that

Let be a factor of f.

Slide 33

UFA: step 3

Step 3, ‘Combine the , as needed, into true divisors of over Z’

Slide 34

UFA: step 3

Step 3, ‘Combine the , as needed, into true divisors of over Z’

This is the most time consuming step. We need:

once we have a potential factor of modulo , to convert it to a factor over Z

do a test division to see if it is actually a factor

Slide 35

UFA: step 3

Step 3, ‘Combine the , as needed, into true divisors of over Z’

This is the most time consuming step. We need:

once we have a potential factor of modulo , to convert it to a factor over Z

do a test division to see if it is actually a factor

Trick letting not to perform excessive trial divisions:

If the check failed for integers, there is no need to perform it for polynomials.

Slide 36

Asymptotically Good Algoriths

Lenstra, Lenstra, Lovasz. Factoring polynomials with rational coefficients. 1982

Algorith takes operations.

Slide 37

Asymptotically Good Algoriths: definitions