Given degree $d_1$ and $d_2$ polynomials in $\Bbb Z[x]$ with coefficient sizes of bits $b_1$ and $b_2$ respectively
(1) what is the bit complexity of multiplying the two polynomials?
(2) What is the total number of multiplications and additions needed on $O(b_1+b_2+\log(d_1+d_2))$ bit sized words?
this is what I thought. If you employ fft then you should be able to multiply polynomials in $(d_1+d_2)\log(d_1+d_2)$ multiplications and additions on $O(b_1+b_2+log(d_1+d_2))$ bit sized words. Fürer's algorithm needs $O(c\log c2^{log^∗c})$ bit operations per integer multiplication on $O(c)$ bit sized words. So we need a total of $O(c\log c2^{log^∗c}(d_1+d_2)log(d_1+d_2))$ bit operations where $c=b_1+b_2+log(d_1+d_2)$ holds. So I thought the $2^{log^∗c}$ factor was irrelevant to count total number of integer multiplications needed to perform polynomial multiplication.
