[Paper Review] Near Quadratic Matrix Multiplication Modulo Composites

Vince Grolmusz|ArXiv.org|Jan 8, 2003

Matrix Theory and Algorithms3 citations

TL;DR

This paper presents a novel algorithm for matrix multiplication modulo composite numbers that achieves near-quadratic complexity, using only n²⁺ᵒ⁽¹⁾ multiplications. By leveraging 1-a-strong polynomial representations over composite moduli—particularly products of distinct primes—the method exploits algebraic structure to reduce the number of required multiplications below the classical n³ threshold, offering a significant asymptotic improvement for modular matrix computations.

ABSTRACT

We show how one can use non-prime-power, composite moduli for computing representations of the product of two $n imes n$ matrices using only $n^{2+o(1)}$ multiplications.

Motivation & Objective

• Develop an efficient algorithm for matrix multiplication modulo composite numbers, specifically non-prime-power composites such as 6.
• Address the computational bottleneck of classical O(n³) matrix multiplication by reducing the number of multiplications required.
• Explore the use of alternative polynomial representations (1-a-strong) over composite rings to enable faster computation than over fields or prime moduli.
• Demonstrate that matrix product computation modulo composite m can be achieved with n²⁺ᵒ⁽¹⁾ multiplications, approaching quadratic complexity.
• Investigate the feasibility of using surplus monomial properties in 1-a-strong representations to maintain correctness under addition and multiplication operations.

Proposed method

• Utilize 1-a-strong representations of polynomials over composite moduli m = p₁p₂⋯pᵣ, where coefficients are correct modulo at least one prime power divisor of m.
• Construct a 1-a-strong representation of the dot product of two n-dimensional vectors using exp(O(√(log n log log n))) multiplications for m with two prime factors.
• Extend the dot product representation to matrix multiplication by applying it n² times to compute each entry of the product matrix.
• Apply bilinear ΣΠΣ circuits to implement the representations efficiently, ensuring the number of multiplications remains within n²⁺ᵒ⁽¹⁾.
• Leverage the property that 1-a-strong representations preserve correctness under multiplication and, under compatible surplus conditions, under addition.
• Ensure symmetry in coefficients (aᵢⱼ = aⱼᵢ) for the dot product representation to maintain consistency in matrix product computation.

Experimental results

Research questions

• RQ1 Can matrix multiplication modulo composite numbers be performed with significantly fewer than n³ multiplications?
• RQ2 What polynomial representation techniques over composite rings enable sub-cubic complexity in matrix multiplication?
• RQ3 How do 1-a-strong representations over composite moduli compare to standard representations in terms of computational efficiency?
• RQ4 Under what conditions can the sum or product of 1-a-strong representations remain 1-a-strong, ensuring correctness in iterative computations?
• RQ5 Is it possible to achieve n²⁺ᵒ⁽¹⁾ complexity for matrix multiplication modulo composite m using only bilinear circuits?

Key findings

• For composite moduli m with two distinct prime factors, a 1-a-strong representation of the matrix product AB modulo m can be computed using n²⁺ᵒ⁽¹⁾ multiplications, specifically n² × exp(O(√(log n log log n))) multiplications.
• When m has r distinct prime factors, the number of multiplications required increases to n² × exp(O(r√(log n(log log n)ʳ⁻¹))), still within the n²⁺ᵒ⁽¹⁾ complexity bound.
• The algorithm is implemented via bilinear ΣΠΣ circuits, ensuring efficient and structured computation of the representations.
• The 1-a-strong representation of the dot product of two n-vectors can be computed with exp(O(√(log n log log n))) multiplications, forming the core building block of the matrix multiplication algorithm.
• The method achieves a significant asymptotic improvement over classical O(n³) matrix multiplication by exploiting algebraic structure in composite rings.
• The sum of two 1-a-strong representations remains 1-a-strong if their surplus monomials are either disjoint or compatible, preserving correctness under addition operations.