Speeding up decimal multiplication

TL;DR

This paper introduces optimized algorithms for decimal multiplication using NTT, achieving significant speedups, and presents a new cache-efficient matrix transposition method, with insights on prime modulus usage for improved performance.

Contribution

It provides a portable, faster implementation of decimal multiplication with a novel in-place matrix transposition algorithm and analysis of prime modulus strategies.

Findings

3x-5x speedup over mpdecimal library

New cache-efficient in-place matrix transposition algorithm

Using two prime moduli simplifies answer recovery

Abstract

Decimal multiplication is the task of multiplying two numbers in base $10^N.$ Specifically, we focus on the number-theoretic transform (NTT) family of algorithms. Using only portable techniques, we achieve a 3x-5x speedup over the mpdecimal library. In this paper we describe our implementation and discuss further possible optimizations. We also present a simple cache-efficient algorithm for in-place $2n \times n$ or $n \times 2n$ matrix transposition, the need for which arises in the "six-step algorithm" variation of the matrix Fourier algorithm, and which does not seem to be widely known. Another finding is that use of two prime moduli instead of three makes sense even considering the worst case of increasing the size of the input, and makes for simpler answer recovery.