#### Degree

Master of Science

#### Program

Computer Science

#### Supervisor

Moreno Maza, Marc

#### Abstract

Fast algorithms for integer and polynomial multiplication play an important role in scientific computing as well as other disciplines. In 1971, Schönhage and Strassen designed an algorithm that improved the multiplication time for two integers of at most n bits to O(log n log log n). In 2007, Martin Fürer presented a new algorithm that runs in O (n log n · 2 ^O(log* n)) , where log*n is the iterated logarithm of n. We explain how we can put Fürer’s ideas into practice for multiplying polynomials over a prime field Z/pZ, which characteristic 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, we obtain an efficient implementation of FFT over Z/pZ. This implementation outperforms comparable implementations either using other encodings of Z/pZ or other ways to perform multiplication in Z/pZ.

#### Recommended Citation

Wang, Linxiao, "Putting Fürer's Algorithm into Practice with the BPAS Library" (2018). *Electronic Thesis and Dissertation Repository*. 5358.

https://ir.lib.uwo.ca/etd/5358