Polynomial Factorization

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A subset is called a lattice, if there exists a basis in such, that

Slide 38

Asymptotically Good Algoriths: idea

The beginning is the same with the previous algorith: the polynomial f is factored modulo prime number p. Then an irreducible factor h modulo the power of p is computed, using Hensel’s techniques.

Slide 39

Asymptotically Good Algoriths: idea

The beginning is the same with the previous algorith: the polynomial f is factored modulo prime number p. Then an irreducible factor h modulo the power of p is computed, using Hensel’s techniques.

After this an irreducible factor of f in Z[x] such, that

is searched for.

In our terms, will imply that the coefficients of are the points of some lattice

and will imply that the coefficients of are ‘not too large’ (in other words, a short vector in the lattice corresponds to the searched irreducible factor).

Slide 40

Lattices and factorization

Summing up, we need an algorith for constructing an irreducible factor of f given an irreducible factor h modulo p (with lc(h)=1).

It is convenient to generalize the problem:

Given an irreducible factor h modulo of square free polynomial f, with lc(h)=1, find irreducible such that modulo p.

Slide 41

Lattices and factorization

Let n=deg f, l=deg h. Fix some and consider the set S of polynomials over Z[x] with degree not higher than m, dividable by h modulo

Slide 42

Lattices and factorization

Let n=deg f, l=deg h. Fix some and consider the set S of polynomials over Z[x] with degree not higher than m, dividable by h modulo

If , belongs to S.

Slide 43

Lattices and factorization

Let n=deg f, l=deg h. Fix some and consider the set S of polynomials over Z[x] with degree not higher than m, dividable by h modulo

If , belongs to S.

We can think of polynomials of degree less than or equal to m as of points in

Then the polynomials from S form a lattice L with basis

Slide 44

Lattices and factorization: two theorems

Theorem 1. If a polynomial is such that

Slide 45

Lattices and factorization: two theorems

Theorem 1. If a polynomial is such that

Theorem 2. Let

Suppose that .

Then

Suppose that for some (1) Let t be the largest of such j. Then