Multiword matrix multiplication over large finite fields in floating-point arithmetic

TL;DR

This paper introduces a multiword decomposition method for efficient modular matrix multiplication over large finite fields using floating-point arithmetic, overcoming previous size limitations and improving performance on CPU and GPU.

Contribution

The authors propose a novel multiword decomposition technique that extends the prime size limit for modular matrix multiplication in floating-point arithmetic, with rigorous correctness analysis and extensive experimental validation.

Findings

Handles primes with bitsizes up to nearly the full mantissa size.

Outperforms existing single-word methods for smaller bitsizes.

Efficiently processes large primes on both CPU and GPU architectures.

Abstract

This article is concerned with the efficient computation of modular matrix multiplication C=AB mod p, a key kernel in computer algebra. We focus on floating-point arithmetic, which allows for using efficient matrix multiplication libraries. However, the existing approach is limited to primes p with bitsize at most half the mantissa size (e.g., 26 bits with double precision arithmetic), and becomes quite inefficient when p approaches this limit. We present a new approach that overcomes this limitation and can efficiently handle primes with larger bitsizes. The key idea is to use multiword decompositions, which represent A and B as scaled sums of u and v matrices (words) with smaller coefficients. We provide a rigorous analysis that proves the correctness of this approach for suitably chosen scaling parameters. Our analysis determines the maximum bitsize of p that can be handled for a given number of words; in particular, we show that decomposing in two words each input suffices to handle bitsizes almost equal to the full mantissa size (e.g., the 26 bits limit is raised to 52 bits in double precision arithmetic). Moreover, we show that (1,v) decompositions with v>1 are also of interest to handle intermediate bitsizes. We perform an extensive experimental analysis for various matrix shapes and prime bitsizes. Our performance benchmarks on both CPU and GPU architectures confirm the efficiency of the proposed approach, which can outperform the existing single word approach for bitsizes as low as 23, and can handle bitsizes as high as 52 while retaining high performance.