Book contents
- Frontmatter
- Contents
- Introduction
- 1 Cyclohexane, cryptography, codes, and computer algebra
- I Euclid
- II Newton
- III Gauß
- 14 Factoring polynomials over finite fields
- 15 Hensel lifting and factoring polynomials
- 16 Short vectors in lattices
- 17 Applications of basis reduction
- IV Fermat
- V Hilbert
- Appendix
- Sources of illustrations
- Sources of quotations
- List of algorithms
- List of figures and tables
- References
- List of notation
- Index
- The Holy Qur'ān (732)
15 - Hensel lifting and factoring polynomials
from III - Gauß
Published online by Cambridge University Press: 05 May 2013
- Frontmatter
- Contents
- Introduction
- 1 Cyclohexane, cryptography, codes, and computer algebra
- I Euclid
- II Newton
- III Gauß
- 14 Factoring polynomials over finite fields
- 15 Hensel lifting and factoring polynomials
- 16 Short vectors in lattices
- 17 Applications of basis reduction
- IV Fermat
- V Hilbert
- Appendix
- Sources of illustrations
- Sources of quotations
- List of algorithms
- List of figures and tables
- References
- List of notation
- Index
- The Holy Qur'ān (732)
Summary
In this chapter, we present two modular algorithms for factoring in Q[x] and F[x, y] for a field F. The first one uses factorization modulo a “big” prime and is conceptually easier, and the second one uses factorization modulo a “small” prime and then “lifts” it to a factorization modulo a power of that prime. The latter is computationally faster and comprises our most powerful employment of the prime power modular approach introduced in Chapter 5.
Factoring in ℤ[x] and Q[x]: the basic idea
Our first goal is to understand the difference between “factoring in ℤ[x]” and “factoring a polynomial with integer coefficients in Q[x]”. The basic fact is that the latter corresponds to factoring primitive polynomials in ℤ[x], while the former requires in addition the factoring of an integer, namely the polynomial's content. We rely on the following notions which were introduced in Section 6.2.
Let R be a Unique Factorization Domain (our two main applications are, as usual, R = ℤ and R = F[y] for a field F). The content cont (f) of a polynomial f ∈ R[x] is the greatest common divisor of its coefficients (with the convention that the gcd is positive if R = ℤ and monic if R = F[y]).
- Type
- Chapter
- Information
- Modern Computer Algebra , pp. 433 - 472Publisher: Cambridge University PressPrint publication year: 2013