Balanced Dense Polynomial Multiplication on Multi-Cores

Abstract

In symbolic computation, polynomial multiplication is a fundamental operation akin to matrix multiplication in numerical computation. We present efficient implementation strategies for FFT-based dense polynomial multiplication targeting multi-cores. We show that {\it balanced input data} can maximize parallel speedup and minimize cache complexity for bivariate multiplication. However, unbalanced input data, which are common in symbolic computation, are challenging. We provide efficient techniques, what we call {\it contraction} and {\it extension}, to reduce multivariate (and univariate) multiplication to {\it balanced bivariate multiplication}. Our implementation in {\tt Cilk++} demonstrates good speed upon multi-cores.