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

Associative table lookup processing for multioperand residue arithmetic

“…We conclude that, owing to the parallelism of the residue operations, this complexity measure will simply be equal to the summation of the corresponding memory sizes taken over all moduli in the system. This conclusion makes Table III directly applicable to This inequality, easily verifiable using Table III, has been established in somewhat different context in [20]. As an example, if M = 35, Ml = 7, MZ = 5, and D = 2, then from Table III we have L@(35, 2) = 2975, Le(35, 2) = 2886, L@(7, 2) + L@(5, 2) = 63 + 25 = 88, and Le(7, 2) + L@(5, 2) = 54 + 20 = 74.…”

“…It is worth noting in Table III that the required associative memory size for addition is always greater than the one for multiplication mod M. This result was established in a somewhat different context in [20]. 7 63 54 21 1620 1494 8 96 68 28 1792 1452 9 117 102 29 1943 1876 10 150 110 30 2130 1754 11 181 170 31 2325 2250 12 240 172 32 2560 2064 13 286 264 33 2613 2562 14 350 279 34 2822 2418 15 420 360 35 2975 2886 16 512 392 36 3168 2484 17 561 528 37 3330 3240 18 630 504 38 3534 3044 19 703 666 39 3144 3516 20 800 620 40 4000 3256 21 882 840 41 4182 PROOF.…”

“…. , Mi -1); the encoding effect is discussed in [20]. In view of Section 1, the complexity of the proposed multioperand scheme for residue arithmetic should be defined as the total associative memory size of the structure in Figure 8.…”

“…A similar definition holds for multiplication (@) modM. The above residue functions may be depicted by multioperand truth tables or mappings of various forms [20]. An example of 2-operand multiplication tables mod 4 is in Figure 2 Table lookup implementation scheme for an addition and multiplication table mod M by an array of associative memory modules.…”