Conditions for the distributivity of multiplication with respect to set addition and their effect on the design of array multipliers

Some conditions for distributivity of multiplication with respect to set addition are determined. In particular, distributivity holds if positive numbers are used, but does not hold for negative numbers in radix complement form. Use of simple recoding methods to achieve distributivity is not helpful, since costs of elementary multipliers and summing networks are increased. Distributivity does hold for the digit sets used for signed-digit arithmetic, and the requirement for distributivity provides guidance for the design of elementary multipliers.