# Schonhage-Strassen algorithm for multiplication of polynomials over a finite field (additive vs multiplicative complexity)

Trying to understand the Schonhage-Strassen algorithm for multiplying two polynomials $$f(X)$$, $$g(X)$$ of degree $$n$$ over a finite field $$\mathbb{F}_q[X]$$ with $$q$$ a prime such that $$q-1$$ does not have high $$2$$-adicity (which is necessary for the basic FFT). As I understand, the asymptotic complexity is $$O(n\cdot \log(n)\cdot \log(\log(n))).$$

But is the multiplicative complexity (i.e. the number of $$\mathbb{F}_p$$ multiplications) also $$O(n\cdot \log(n)\cdot \log(\log(n)))$$?

Or is it the case that the additive complexity (i.e. the number of $$\mathbb{F}_p$$ multiplications)) is $$O(n\cdot \log(n)\cdot \log(\log(n)))$$ while the multiplicative compexity is $$O(n\cdot \log(n))$$?

The 1971 Schonhage-Strassen paper is in German. The Cantor-Kaltofen algorithm which generalizes it to polynomial products over an arbitrary ring appears to have an additive complexity of $$O(n\cdot \log(n)\cdot \log(\log(n)))$$ and a multiplicative complexity of $$O(n\cdot \log(n)).$$