Polynomial Factorization

1
Polynomial Factorization

• Olga Sergeeva
• Ferien-Akademie 2004, September 19 October 1

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Overview

• Univariate Factorization
• Overview of the algorithms and the required
  simplifications
• Factoring over finite fields
• Factorization based on Hensel lifting
• LLL algorithm
• Multivariate Factorization
• Problems overview
• The idea of the algorithm
• Analysis of correctness probability.

3
Univariate Factorization algorithms

• We consider factorization of polynomials over the
  rational integers, Z, and different approaches to
  this problem.

4
Univariate Factorization algorithms

• We consider factorization of polynomials over the
  rational integers, Z, and different approaches to
  this problem.
• Algorithms, solving the problem for univariate
  polynomials
• Kronecker, interpolation algorithm

5
Univariate Factorization algorithms

• We consider factorization of polynomials over the
  rational integers, Z, and different approaches to
  this problem.
• Algorithms, solving the problem for univariate
  polynomials
• Kronecker, interpolation algorithm
• Algorithm, which uses Hensel lifting techniques
  and factorization over finite fields

6
Univariate Factorization algorithms

• We consider factorization of polynomials over the
  rational integers, Z, and different approaches to
  this problem.
• Algorithms, solving the problem for univariate
  polynomials
• Kronecker, interpolation algorithm
• Algorithm, which uses Hensel lifting techniques
  and factorization over finite fields
• A. K. Lenstra, H. W. Lenstra and Lovasz
  polynomial time algorithm using basic reduction
  techniques for lattices.

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Univariate Factorization simplifications

• When factoring a univariate polynomial over Z,
  the following simplifications are effective
• removing the integer content of F(Z)

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Univariate Factorization simplifications

• When factoring a univariate polynomial over Z,
  the following simplifications are effective
• removing the integer content of F(Z)
• computing square free decomposition (with use of
  GCD computations or modular interpolation
  techniques).

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Univariate Factorization simplifications

• When factoring a univariate polynomial over Z,
  the following simplifications are effective
• removing the integer content of F(Z)
• computing square free decomposition (with use of
  GCD computations or modular interpolation
  techniques).
• one could try to monicize F(Z), but this
  increases the size of the coefficients of F and
  in most cases in not worthwhile

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Examples

• Factorization of polynomials over Z will not be
  more fine-grained, but will only be coarser than
  factorization over a .
• For example, has complex roots and
  thus it is irreducible over Z. But it is
  factorizable over any .
• For instance,

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Univariate Factorization over

• Let be a polynomial with coefficients from
• First, we get rid of squares
• 
  
• 
  

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Univariate Factorization over

• Let be a polynomial with coefficients from
• First, we get rid of squares
• 
  

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Factorization over - theoretical basis
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Is there any use of this theorem?

• Let us now understand that the equation
• is in fact equal to a system of linear equations
  over
• Due to the fact that we are over ,
• (because almost all the binomials are divided by
  p).

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And what?
Also, and we get a system of linear equations

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And what?
Also, and we get a system of linear equations
The dimension of its solution space is k,
where k is the number of irreducible factors of
f.
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The last slide about finite fields

• We now know, how many factors there are.
• Let to be a basis.
  If k1 then the f is irreducible
• In the case kgt1, we search for
  , for all .
• As a result, we get a number of divisors of f
• If sltk, we calculate
  and so on.

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The last slide about finite fields

• We now know, how many factors there are.
• Let to be a basis.
  If k1 then the f is irreducible
• In the case kgt1, we search for
  , for all .
• As a result, we get a number of divisors of f
• If sltk, we calculate
  and so on.
• At the end, we will get all the k factors for
  two different factors
• there exists an element from the basis
  such that

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No, this is the last one
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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.

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Univariate Factorization algorithm (UFA)

• The classical univariate factorization algorithm
  consists of three steps
• Choose a good random rational prime p and
  factor into irreducible factors modulo p

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Univariate Factorization algorithm (UFA)

• The classical univariate factorization algorithm
  consists of three steps
• Choose a good random rational prime p and
  factor into irreducible factors modulo p
• Use Newtons iteration to lift the to
  factors modulo

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Univariate Factorization algorithm (UFA)

• The classical univariate factorization algorithm
  consists of three steps
• Choose a good random rational prime p and
  factor into irreducible factors modulo p
• Use Newtons iteration to lift the to
  factors modulo
• Combine the , as needed, into true divisors
  of over Z.

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UFA step 1

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

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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.

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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 algorithms.

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Hensel techniques reminder

• We will use this factorization to get the
  factorization of f
• modulo

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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

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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

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UFA step 2

• Step 2, Use Newtons iteration to lift the
  to factors modulo .
• We choose l considering the bounds on the
  coefficients of the factors.

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UFA step 2

• Step 2, Use Newtons iteration to lift the
  to factors modulo .
• We choose l considering the bounds on the
  coefficients of the factors.
• Theorem (Mignotte) Let

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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.

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UFA step 3

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

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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

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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.

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Asymptotically Good Algorithms

• Lenstra, Lenstra, Lovasz. Factoring polynomials
  with rational coefficients. 1982
• Algorithm takes
  operations.

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Asymptotically Good Algorithms definitions

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

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Asymptotically Good Algorithms idea

• The beginning is the same with the previous
  algorithm the polynomial f is factored modulo
  prime number p. Then an irreducible factor h
  modulo the power of p is computed, using Hensels
  techniques.

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Asymptotically Good Algorithms idea

• The beginning is the same with the previous
  algorithm the polynomial f is factored modulo
  prime number p. Then an irreducible factor h
  modulo the power of p is computed, using Hensels
  techniques.
• After this an irreducible factor of f in
  Zx 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).

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Lattices and factorization

• Summing up, we need an algorithm 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.

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Lattices and factorization

• Let ndeg f, ldeg h. Fix some and
  consider the set S of polynomials over Zx with
  degree not higher than m, dividable by h modulo

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Lattices and factorization

• Let ndeg f, ldeg h. Fix some and
  consider the set S of polynomials over Zx with
  degree not higher than m, dividable by h modulo
• If , belongs to S.

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Lattices and factorization

• Let ndeg f, ldeg h. Fix some and
  consider the set S of polynomials over Zx 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

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Lattices and factorization two theorems

• Theorem 1. If a polynomial is such that
  

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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

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Auxiliary algorithm

• With fixed m, the algorithm checks if
  
• If it is, the algorithm calculates
• Input f of degree n prime p natural k h such
  that lc(h)1 and
• , also h(mod p)is
  irreducible and f(mod p) is not divided by
  
• natural such that

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Auxiliary algorithm

• With fixed m, the algorithm checks if
  
• If it is, the algorithm calculates
• Input f of degree n prime p natural k h such
  that lc(h)1 and
• , also h(mod p)is
  irreducible and f(mod p) is not divided by
  
• natural such that
• Work For the lattice with basis
• find reduced basis
• If then
  and the algorithm stops
• Otherwise, and

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The main algorithm

• Calculation of .
• ldeg h lt deg fn.
• Work
• Calculate the least k for which
  is held with mn-1.
• For the factorization
  calculate its Hensel lifting
• 
  ,
• Let u be the greatest integer
• Run the auxiliary algorithm for
• until we get
• And if we dont get it, deg gt n-1 and
  is equal to f.

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Multivariate factorization

• The reductions and simplifications, which were
  used in the case of univariate polynomials, are
  not proper when dealing with multivariate ones.
• Performing this type of square free decomposition
  before factoring F leads to exponential
  intermediate expression swell.

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Multivariate factorization idea

• The basic approach used to factor multivariate
  polynomials is much the same as the exponential
  time algorithm for u.p.
• Rouphly speaking, we reduce the problem of
  factoring a polynomial of n variables to the case
  of polynomial of n-1 variables, pointing at one
  (or two) variables at the end.

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Hilbert irreducibility theorem

• Let be an
  irreducible polynomial over Q and let R(N) denote
  the number of n-tuples over Z with xiltN such
  that is reducible.
  Then
• , where
  c depends only on the degree of F.

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Hilbert theorem disadvantages

• There is no upper bound on the number of random
  points needed.
• The approach can not be applied when working over
  finite field.

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Bertinis theorem

• Let be an irreducible
  polynomial of RZ, where
• and is an
  intergal domain. Let the degree of in
  be d,
• Let the total degree of the in
  be . Let L be a subset of of
  cardinality .
• Then
  
  is irreducible over