Factorization of univariate integer polynomials by diophantine approximation and an improved basis reduction algorithm

Abstract

All steps described in the preceding section for fixed m are certainly covered by the rather crude bound of $O(m(n^{5 + \varepsilon } + n^3 (log|f|)^{2 + \varepsilon } ))$ bit operations. For a factor p of degree $k \leqslant \frac{n}{2}$ this bound applies for m=2,4,... until some m with m/2<k≤m is reached. The sum of the corresponding time bounds is therefore $O(k(n^{5 + \varepsilon } + n^3 (log|f|)^{2 + \varepsilon } ))$. Further factors are found in the very same way, dealing with f/p, etc. There is at most one factor of degree k>n/2 (possibly f itself), thus the final time bound $O(n^{6 + \varepsilon } + n^4 (log|f|)^{2 + \varepsilon } ))$ is obtained.

It should be observed that the distinctions between the real and complex case in Lemma 6.1 and in Lemma 6.2 nicely match such that in both cases log B∼6mn+2(m+n) log|f|. Due to the estimation (6.1) sometimes shortcuts in the reduction process may be possible. As soon as |b ^*_m |² becomes greater than 3·2^2m-1|f|², ^bm can be eliminated from the reduction, etc.

It is conceivable that further (mainly theoretical) improvements of our algorithm are possible, for instance by exploiting fast matrix multiplication, or by iterating the block reduction technique.