Big Prime Field FFT on the GPU

Abstract

We consider prime fields of large characteristic, typically fitting on k machine words, where k is a power of 2. When the characteristic of these fields is restricted to a subclass of the generalized Fermat numbers, we show that arithmetic operations in such fields offer attractive performance both in terms of algebraic complexity and parallelism. In particular, these operations can be vectorized, leading to efficient implementation of fast Fourier transforms on graphics processing units.

Big Prime Field FFT on Multi-core Processors

“…Therefore, we consider the so-called generalized Fermat prime numbers. The detailed introduction of generalized Fermat prime numbers can be found in the previous work of our research group [5].…”

“…We consider prime fields of large characteristic, typically fitting on k machine words, where k is a power of 2. When the characteristic of these fields is restricted to a subclass of the generalized Fermat numbers, the authors of [5] have shown, in an ISSAC 2017 paper, that arithmetic operations in such fields offer attractive performance, both in terms of algebraic complexity and parallelism. In particular, these operations can be vectorized, leading to an efficient implementation of fast Fourier transforms on graphics processing units (GPUs), reported in that same paper.…”

“…These features of GPU architectures have been essential in the implementation of arithmetic operations of generalized Fermat prime fields. Hence, the implementation techniques developed in [5] can not be easily ported and applied to the context of multi-core processors.…”

“…Consider a generalized Fermat prime number of the form p = r k + 1, where k is a power of 2 and r is of machine-word size. As mentioned in [5], multiplying by a power of r modulo p can be done in O(k) machine-word operations. However, multiplying two arbitrary elements of Z/pZ is a non-trivial operation.…”

“…Thus, multiplying two arbitrary elements of Z/pZ requires computation of the product of two univariate polynomials in Z[X ], of degree less than k, modulo X k + 1. In [5], this is done by using plain multiplication, thus Θ(k 2 ) machine-word operations. In Section 3, we explain how to multiply two arbitrary elements x, y of Z/pZ via FFT.…”