Compressed Modular Matrix Multiplication

TL;DR

This paper introduces a method for compressing multiple integers modulo a small prime into a single machine word, enabling efficient matrix multiplication on compressed data through specialized arithmetic routines.

Contribution

It presents a novel approach to perform matrix multiplication on compressed modular integers using word-level operations, with bounds and implementation details.

Findings

Efficient modular addition using word-level operations.

Feasibility bounds for prime sizes and matrix dimensions.

Explicit routines for compressed modular arithmetic.

Abstract

We propose to store several integers modulo a small prime into a single machine word. Modular addition is performed by addition and possibly subtraction of a word containing several times the modulo. Modular Multiplication is not directly accessible but modular dot product can be performed by an integer multiplication by the reverse integer. Modular multiplication by a word containing a single residue is a also possible. Therefore matrix multiplication can be performed on such a compressed storage. We here give bounds on the sizes of primes and matrices for which such a compression is possible. We also explicit the details of the required compressed arithmetic routines.