Lattices Research Papers - Academia.edu (original) (raw)

Let g1,...,g k be the true factors of f in Z[x] and let f1,...,fr be the local factors (over the p-adic integers). Current implementations Hensel lift to determine f1,...,fr with a p-adic accuracy a that is guaranteed to be high enough to... more

Let g1,...,g k be the true factors of f in Z[x] and let f1,...,fr be the local factors (over the p-adic integers). Current implementations Hensel lift to determine f1,...,fr with a p-adic accuracy a that is guaranteed to be high enough to recover any potential factor of f in Z[x]. However, the problem is that this p-adic accuracy, a, is often much higher than what was actually necessary to recover all the factors g1,...,g k. This implies that current implementations often waste CPU time on Hensel Lifting. In practice it frequently happens that f has one large factor, say g1, and zero or more small factors, say g2,...,g k. Then, to recover g1,...,g k we do not need p a to be larger than twice the largest coefficient of g1. All we need is that p a is larger than twice the largest coefficient in g2,...,g k. This suffices to reconstruct g2,...,g k ∈ Z[x] from their modular images, after which the remaining factor g1 can be determined by a division in Z[x]. It is easy to give examples wh...