Factoring polynomials over global fields

Abstract

Let K be a global field and f ∈ K[X] be a polynomial. We present an efficient algorithm which factors f in polynomial time.

Gradual Sub-lattice Reduction and a New Complexity for Factoring Polynomials

“…This matrix has p a−b j > 2 N 2 +N log(A) > 2α 4r+2 B 2 for all j. An (α, B)-reduction of this matrix will solve the recombination problem by a similar argument to the one presented in [4] and refined in [22]. Now we look at the computational complexity of making and reducing this matrix which gives the new result for factoring inside Z[x].…”

“…In [4] it is shown that the problem of factoring a polynomial, f ∈ Z[x], can be accomplished by the reduction of a large knapsack-type lattice. In this subsection we merely apply our algorithm to the lattice suggested in [4].…”

“…For our purposes we choose B := √ r + 1. We will make some minor changes to the All-Coefficients matrix defined in [4] to produce a matrix that looks like:…”

The van Hoeij Algorithm for Factoring Polynomials

“…This is a real restriction because it might be the case that f (a, x) has multiple factors for all a ∈ F p . For the solution of this problem we refer the reader to [BHKS08]. Using Hensel lifting we get from the factorizationf =f 1 · · ·f r ∈ F p [x] a factorization f (t, x) =f 1 · · ·f r in the power series ring…”

“…He was only able to show that his algorithm terminates, but he gave very impressive practical examples of factorizations which have not been possible to compute before. Together with Karim Belabas, Mark van Hoeij, and Allan Steel [BHKS08] the author of this survey simplified the presentation of this algorithm and introduced a variant for factoring bivariate polynomials over finite fields. Furthermore we have been able to show that the new factoring algorithm runs in polynomial time.…”