Preconditioning Integer Multiplication via Similarity and Centered Difference Decompositions

This document explores a family of algebraic decompositions for integer multiplication that explicitly separate shared structure between operands from their residual differences before performing the multiplication. Two closely related methods are developed: Similarity–Difference Decomposition and Centered-Difference Decomposition. The core insight is that when two numbers share a common component, this shared part can be isolated and squared once, while the remaining product could be reduced to smaller terms. This transforms the multiplication problem into a combination of one or more squarings, additions, and a smaller residual multiplication. The centered variant further reduces the size of the residual terms by choosing a common reference value near the midpoint of the operands, with an adjustment for parity constraints. The document provides: a formal derivation of both decompositions, a cost model that distinguishes additions, squarings, and general multiplications, a comparison to the Karatsuba algorithm, clarifying that the proposed methods act as preconditioning transformations rather than replacement algorithms, a unified interpretation that relates these decompositions to binary structure, algebraic identities, and classical multiplication techniques. The primary contribution is a reframing of multiplication as a process of structural extraction followed by reduced core computation. This framing is especially relevant for hardware-aware arithmetic and energy-efficient computation, where squaring and addition may be substantially cheaper than general multiplication.