TITLE: Karatsuba vs. Schönhage-Strassen for multiplication of polynomials
QUESTION [9 upvotes]: I am wondering how to most effectively multiply two polynomials with several $100$'s of coefficients, each coefficient having $1000$-$2000$ decimal digits.
I know Schönhage-Strassen begins to outperform Karatsuba between $10,000$ and $40,000$ digits. My question is, is SS not worth it because there are much less than $10,000$ coefficients, or is it worth it because a polynomial has more than $100,000$ digits total?

REPLY [2 votes]: If the bit length of the coefficients and the degree of the polynomial don't differ too much*, it is more efficient to directly run the DFT on the polynomial coefficients.
Otherwise, the technique described by François (Kronecker substitution) is faster.
See page 4 of this presentation.
* - they need to be within roughly a factor of 2