Comments on "Direct Implementation of Discrete and Residue-Based Functions Via Optimal Encoding: A Programmable Array Logic Approach"

An analytic expression for the lower bound on the complexity of residue multiplication is developed. Significant reduction of the required stored logic in a content-addressable memory is noted. Errors in Figs. 5(a), 7, and 8 of the above paper' are corrected. Index Terms -Addition-and multiplication mod M, complexity of residue addition and multiplication, content-addressable memory, optimal residue encoding, positional coding. In an interesting paper, ' Papachristou has analyzed the complexity of direct implementation of truth tables by associative logic processing. This information is valuable in evaluating a number of modem computing structures (both electronic and optical) which are based on truth-table look-up processing. The purpose of this correspondence is to present additional insight into the complexity of residue addition and multiplication and to correct some inadvertent errors that appear in the above paper. The lower bounds on the complexity of residue addition and multiplication, which correspond to the positional coding, are given as (11) in the subject paper. The expression for the multiplication should be changed to L(*M) = ITm2f1t1, since i = 1 (included in the summation in the subject paper) is used to indicate the case of having O at the output and this case does not contribute to the complexity. Another relationship for the lower bound on the complexity of residue multiplication that gives more insight and allows easy calculations can be obtained as follows. If the modulus is prime, the corresponding multiplication table is cyclic and consists of M 1 nonzero rows, each row being a permuation of numbers 0, 1, ..M 1. In positional coding, each nonzero number is represented by a 1 located at a particular bit position, hence L(*M) = (M 1)2. If the modulus is not prime, some of the entries of the multiplication table are integer products of the modulus and, therefore, are represented by zeros. Deducting these entries, the exact value for the lower bound on the multiplication complexity is