Factor refinement (in \(\mathbb Z\)) refers to the transition from \(m=m_ 1 m_ 2\) to \(m=\prod n_ i^{e_ i}\) where the \(n_ i\) are not necessarily prime but \(\gcd(n_ i n_ j)=1\). The method is to use only the gcd algorithm starting with \(d=\gcd(m_ 1,m_ 2)\) and \(m=(m_ 1/d)d^ 2 (m_ 2/d)\). Then proceed by induction. The process is of complexity \(O(\log m)^ 2\). The corresponding process for \(\mathbb F[x]\) (\(\mathbb F\) a finite field) also ensures squarefree factors, since the derivatives can be used, all in logarithmic complexity. [Reference is made to the authors' joint work in Proc. First Annual ACM--SIAM Symp. Discrete algorithms 1990, 210--211 (1990)].