Faster Walsh-Hadamard Transform and Matrix Multiplication over Finite Fields using Lookup Tables

TL;DR

This paper introduces lookup table-based algorithms that significantly speed up Walsh-Hadamard transforms and matrix multiplication over finite fields, reducing bit operations and providing new insights into their complexities.

Contribution

It presents novel algorithms leveraging lookup tables to accelerate algebraic computations over finite fields, improving known bounds for transforms and matrix multiplication.

Findings

Walsh-Hadamard transform computed in O(N log N / log log N) bit operations

Matrix multiplication algorithms converted to use fewer bit operations, e.g., Strassen's algorithm

Provides bounds and open problems on implementation constants for fast matrix algorithms

Abstract

We use lookup tables to design faster algorithms for important algebraic problems over finite fields. These faster algorithms, which only use arithmetic operations and lookup table operations, may help to explain the difficulty of determining the complexities of these important problems. Our results over a constant-sized finite field are as follows. The Walsh-Hadamard transform of a vector of length $N$ can be computed using $O(N \log N / \log \log N)$ bit operations. This generalizes to any transform defined as a Kronecker power of a fixed matrix. By comparison, the Fast Walsh-Hadamard transform (similar to the Fast Fourier transform) uses $O(N \log N)$ arithmetic operations, which is believed to be optimal up to constant factors. Any algebraic algorithm for multiplying two $N \times N$ matrices using $O(N^\omega)$ operations can be converted into an algorithm using $O(N^\omega / (\log N)^{\omega/2 - 1})$ bit operations. For example, Strassen's algorithm can be converted into an algorithm using $O(N^{2.81} / (\log N)^{0.4})$ bit operations. It remains an open problem with practical implications to determine the smallest constant $c$ such that Strassen's algorithm can be implemented to use $c \cdot N^{2.81} + o(N^{2.81})$ arithmetic operations; using a lookup table allows one to save a super-constant factor in bit operations.