Complexity analysis of Polynomial Factorization

We present a comprehensive complexity analysis of an e cient algorithm for polynomial factorization in Z[x]. ? Supported by NSF 0728853 Complexity of Factoring A. Novocin, M. van Hoeij, W. Hart 1 This document is a work in progress. The central problem we address is the factorization of a square-free and monic polynomial in Z[x]. Forgoing for the moment any introduction to the method or explanation of the reasoning we present the algorithm essentially as we have implemented it in FLINT. We have a couple notations,M refers to the fullM which may or may not be virtual depending on the boolean virtual (if it is virtual then pertinent data is in M and G). Also we'll use ba/be to refer to rounding but not into Z but rather Z/2r. Algorithm 1 The main algorithm Input: Square-free, monic, polynomial f ∈ Z[x] of degree N Output: The irreducible factors of f over Z 1. Choose a prime, p, such that gcd(f, f ′) ≡ 1 modulo p. 2. Modular Factorization: Factor f modulo p ≡ f1 · · · fr. 3. if r ≤ Zbound := 15 return Zassenhaus(f) 4. Compute rst target precision a with Algorithm 5 5. until solved: (a) Hensel Lifting: Hensel lift f1 · · · fr to precision pa. (b) Recombination: Algorithm 2(f, f1, . . . , fr, p a) (c) if not solved: a := 2a Algorithm 2 Attempt Reconstruction Input: f , f1, . . . , fr the lifted factors, their precision p a, and possibly M ∈ Zs×(r+c). Output: If solved then the irreducible factors of f over Z otherwise an updated M . 1. If this is the rst call let M := Ir×r 2. Choose a k heuristically (see below for details) 3. For j ∈ {0, . . . , k − 1, N − k − 1, . . . , N − 1} do: (a) Compute CLD bound, Xj , for x j using Algorithm 4 (b) If √ (Ebound := 1.6r)·Xj ≤ pa/2(`:=1.5)r then compute new column vector xj := (x1,j , . . . , xr,j) where xi,j is the coe cient of x j in f · f ′ i/fi 4. For each computed xj do: (a) justified:= True; While justified is True do: i. Decide if LLL is justi ed using Algorithm 3 which augments M ii. If justified then: if virtual run G-LLL(M,G) else run LLL(M) both with parameters δ = .99, η = .51 iii. Compute G-S lengths of rows of M iv. Decrease the number of rows ofM until the nal Gram-Schmidt norm≤ √ (B(r) := r + 2) v. Use Algorithm 6 to test if solved Algorithm 3 Decide if column is worth calling LLL Input: M ∈ Zs×(r+c), data vector xj, pa, Xj the CLD bound for xj, boolean virtual Output: A potentially updated M and/or G and a boolean justified, and boolean virtual 1. Let B := r + 2 and s be the number of rows of M 2. If pa < Xj ·B · √ (Ebound = 1.6r2) · 2(`=1.5)r justified := False and exit 3. Find U the rst r columns of M 4. Compute yj := U · xj 2 Hart, Hoeij, and Novocin 5. If ‖ yj ‖∞< Xj ·B · √ (Ebound = 1.6r2) · 2(NoVecbound:=.937r−.1) then justified := False and exit 6. Find new column scaling k := dlog2 ( pa 2(`=1.5)r )e. 7. Embed xj/2 k and pa/2k into Z/2r by rounding and denote results as x̃j and P̃ 8. Compute ỹj := U · x̃j 9. If P̃− √ B(r) ‖ỹj‖∞ > √ B(r)( ∑s−1 i=0 (1 + η) i) then: