Putting Fürer's Algorithm into Practice with the BPAS Library

Fast algorithms for integer and polynomial multiplication play an important role in scientific computing as well as in other disciplines. In 1971, Sch{\o}nhage and Strassen designed an algorithm that improved the multiplication time for two integers of at most $n$ bits to $\mathcal{O}(\log \log \log n)$. In 2007, Martin F\urer presented a new algorithm that runs in $O \left(n \log n \cdot 2^{O(\log^* n)} \right)$, where $\log^* n$ is the iterated logarithm of $n$. We explain how we can put F\urer's ideas into practice for multiplying polynomials over a prime field $\mathbb{Z} / p \mathbb{Z}$, for which $p$ is a Generalized Fermat prime of the form $p = r^k + 1$ where $k$ is a power of $2$ and $r$ is of machine word size. When $k$ is at least 8, we show that multiplication inside such a prime field can be efficiently implemented via Fast Fourier Transform (FFT). Taking advantage of Cooley-Tukey tensor formula and the fact that $r$ is a $2k$-th primitive root of unity in $\mathbb{Z} / p \mathbb{Z}$, we obtain an efficient implementation of FFT over $\mathbb{Z} / p \mathbb{Z}$. This implementation outperforms comparable implementations either using other encodings of $\mathbb{Z} / p \mathbb{Z}$ or other ways to perform multiplication in $\mathbb{Z} / p \mathbb{Z}$.