Polynomial Factorization

Page
3

Slide 19

No, this is the last one

Slide 20

Univariate Factorization over Z

Square free decomposition computing:

Let be factorization of over Z.

Then . So over Z

We can divide by and thus get a polynomial free of squares.

From now and on, cont(f)=1 and GCD(f,f’)=1.

Slide 21

Univariate Factorization algorith (UFA)

The classical univariate factorization algorith consists of three steps:

Choose a ‘good’ random rational prime p and factor into irreducible factors modulo p:

Slide 22

Univariate Factorization algorith (UFA)

The classical univariate factorization algorith consists of three steps:

Choose a ‘good’ random rational prime p and factor into irreducible factors modulo p:

Use Newton’s iteration to lift the to factors modulo

Slide 23

Univariate Factorization algorith (UFA)

The classical univariate factorization algorith consists of three steps:

Choose a ‘good’ random rational prime p and factor into irreducible factors modulo p:

Use Newton’s iteration to lift the to factors modulo

Combine the , as needed, into true divisors of over Z.

Slide 24

UFA: step 1

Step 1, ‘choose a ‘good’ random rational prime p and factor into irreducible factors modulo p’:

Slide 25

UFA: step 1

Step 1, ‘choose a ‘good’ random rational prime p and factor into irreducible factors modulo p’:

The best primes in the first step are those for which the factorization of modulo p is as close as possible to the factorization of over Z. This is a reason to try several primes and pick the one that fives the coarsest factorization.

Slide 26

UFA: step 1

Step 1, ‘choose a ‘good’ random rational prime p and factor into irreducible factors modulo p’:

The best primes in the first step are those for which the factorization of modulo p is as close as possible to the factorization of over Z. This is a reason to try several primes and pick the one that fives the coarsest factorization.

Over these prime modulo, we compare square free decompositions

After, apply one of the univariate finite field factorization algoriths.

Slide 27

Hensel techniques reminder