In-place fast polynomial modular remainder

We consider the simultaneously fast and in-place computation of the Euclidean polynomial modular remainder <Formula format="inline"><TexMath><?TeX R(X)≡ A(X) B(X)?></TexMath><AltText>Math 1</AltText><File name="issac24-2-inline1" type="svg"/></Formula> with A and B of respective degrees n and m ≤ n. Fast algorithms for this usually come at the expense of a linear amount of extra temporary space. In particular, they require to first compute and store the whole quotient Q(X) such that A = BQ + R.