It is known that the half-GCD algorithm (https://www.csd.uwo.ca/~mmorenom/CS424/Lectures/FastDivisionAndGcd.html/node6.html) can be used for computing the Bezout's coefficients $a,b$ so that $ap + bq \equiv 1$ for two univariate polynomials $p$ and $q$.
If we know in advance the roots of $p$, i.e., $p = \prod^{n}_{i=1} (x-a_i)$ where $a_i\neq a_j$ if $i\neq j$. Also we know in advance that $GCD(p,q) \equiv 1$. Is there any known algorithm for computing the Bezout's coefficients with lower complexity than the half-GCD?
