Vectorizing and distributing number-theoretic transform to count Goldbach partitions on Arm-based supercomputers

In this article, we explore the usage of scalable vector extension (SVE) to vectorize number-theoretic transforms (NTTs). In particular, we show that 64-bit modular arithmetic operations, including modular multiplication, can be efficiently implemented with SVE instructions. The vectorization of NTT loops and kernels involving 64-bit modular operations was not possible in previous Arm-based single instruction multiple data architectures since these architectures lacked crucial instructions to efficiently implement modular multiplication. We test and evaluate our SVE implementation on the A64FX processor in an HPE Apollo 80 system. Furthermore, we implement a distributed NTT for the computation of large-scale exact integer convolutions. We evaluate this transform on HPE Apollo 70, Cray XC50, HPE Apollo 80, and HPE Cray EX systems, where we demonstrate good scalability to thousands of cores. Finally, we describe how these methods can be utilized to count the number of Goldbach partitions of all even numbers to large limits. We present some preliminary results concerning this problem, in particular a histogram of the number of Goldbach partitions of the even numbers up to 2 40.</p

Investigating applications on the A64FX

The A64FX processor from Fujitsu, being designed for computational simulation and machine learning applications, has the potential for unprecedented performance in HPC systems. In this paper, we evaluate the A64FX by benchmarking against a range of production HPC platforms that cover a number of processor technologies. We investigate the performance of complex scientific applications across multiple nodes, as well as single node and mini-kernel benchmarks. This paper finds that the performance of the A64FX processor across our chosen benchmarks often significantly exceeds other platforms, even without specific application optimisations for the processor instruction set or hardware. However, this is not true for all the benchmarks we have undertaken. Furthermore, the specific configuration of applications can have an impact on the runtime and performance experienced