Number Theory Meets Cache Locality – Efficient Implementation of a Small Prime FFT for the GNU Multiple Precision Arithmetic Library

When multiplying really large integer operands, the GNU Multiple Precision Arithmetic Library uses a method based on the Fast Fourier Transform. To make an algorithm execute quickly on a modern computer, data has to be available in the cache memory. If that is not the case, a large portion of the execution time will be spent accessing the main memory. It might pay off to perform much extra work to achieve good cache locality. In extreme cases, 500 primitive operations may be performed in the time of a single memory access. This report describes the implementation of a cache friendly variant of the Fast Fourier Transform and its application to integer multiplication. The variant uses arithmetic modulo primes near machine word-size. The multiplication method is shown to be competitive with its counterpart in version 4.1.4 of the GNU Multiple Precision Arithmetic Library for interesting platforms. Talteori möter cachelokalitet Effektiv implementation av småprimtals-FFT för GNU:s multiprecisionsaritmetikbibliotek