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This article presents an approach to area optimization of arithmetic datapaths at register-transfer level (RTL). The focus is on those designs that perform polynomial computations (add, mult) over finite word-length operands (bit-vectors). We model such polynomial computations over m-bit vectors as algebra over finite integer rings of residue classes Z2m. Subsequently, we use the number-theoretic and algebraic properties of such rings to transform a given datapath computation into another, bit-true equivalent computation. We also derive a cost model to estimate, at RTL, the area cost of the computation. Using the transformation procedure along with the cost model, we devise algorithmic procedures to search for a lower-cost implementation. We show how these theoretical concepts can be applied to RTL optimization of arithmetic datapaths within practical CAD settings. Experiments conducted over a variety of benchmarks demonstrate substantial optimizations using our approach.