Rectangular Matrix Multiplication Revisited

Fast Rectangular Matrix Multiplication and Applications

First we study asymptotically fast algorithms for rectangular matrix multiplication. We begin with new algorithms for multiplication of an n_n matrix by an n_n matrix in arithmetic time O(n), |=3.333953..., which is less by 0.041 than the previous record 3.375477... . Then we present fast multiplication algorithms for matrix pairs of arbitrary dimensions, estimate the asymptotic running time as...

Area-Time Tradeoff for Rectangular Matrix Multiplication in VLSI Models

Graph Expansion Analysis for Communication Costs of Fast Rectangular Matrix Multiplication

Graph expansion analysis of computational DAGs is useful for obtaining communication cost lower bounds where previous methods, such as geometric embedding, are not applicable. This has recently been demonstrated for Strassen’s and Strassen-like fast square matrix multiplication algorithms. Here we extend the expansion analysis approach to fast algorithms for rectangular matrix multiplication, o...

Improved Rectangular Matrix Multiplication using Powers of the Coppersmith-Winograd Tensor

In the past few years, successive improvements of the asymptotic complexity of square matrix multiplication have been obtained by developing novel methods to analyze the powers of the Coppersmith-Winograd tensor, a basic construction introduced thirty years ago. In this paper we show how to generalize this approach to make progress on the complexity of rectangular matrix multiplication as well,...

Double-base scalar multiplication revisited

This paper reduces the number of field multiplications required for scalar multiplication on conservative elliptic curves. For an average 256-bit integer n, this paper’s multiply-by-n algorithm takes just 7.47M per bit on twisted Edwards curves −x + y = 1 + dxy with small d. The previous record, 7.62M per bit, was unbeaten for seven years. Unlike previous record-setting algorithms, this paper’s...