How quickly can we perform base extension in a residue number system?
Check out Lemma 4.4 in HesseAllenderBarrington - it may not be terribly useful for sequential complexity but says essentially that CRR (Chinese Remainder Representation) basis extension can be done in very uniform $\mathsf{TC}^0$. The exact bound is $\mathsf{FOM + POW} = \mathsf{FOM}$ (see also Corollary 6.2 of the same paper).