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There have been several algorithms designed to optimise matrixmultiplication. From schoolbook method with complexity $O(n^3)$ to advancedtensor-based tools with time complexity $O(n^{2.3728639})$ (lowest possiblebound achieved), a lot of work has been done to reduce the steps used in therecursive version. Some group-theoretic and computer algebraic estimations alsoconjecture the existence of an $O(n^2)$ algorithm. This article discusses aquadratic-time number-theoretic approach that converts large vectors in theoperands to a single large entity and combines them to make the dot-product.For two $n \times n$ matrices, this dot-product is iteratively used for eachsuch vector. Preprocessing and computation makes it a quadratic time algorithmwith a considerable constant of proportionality. Special strategies forintegers, floating point numbers and complex numbers are also discussed, with atheoretical estimation of time and space complexity.

A new quadratic-time number-theoretic algorithm to solve matrix multiplication problem