Residue-to-binary converters based on new Chinese remainder theorems

The speed of arithmetic operations depends on the size of the numbers involved. Smaller numbers have faster operations. That is exactly the reason why residue number systems are attractive in computer arithmetic. However, the conversion from residue to binary numbers involves a large number module operations. Several residue-to-binary converters are proposed in this paper. The converters are based on the New Chinese Remainder Theorems (CRT´s) I and II which represent our work. The New CRT´s improve the celebrated CRT. The new algorithms do not use any large size module adders. The numbers involved are much smaller compared to the numbers in the CRT and its alternative, the Mixed Radix Conversion method. Given a moduli set as (P₁, P ₂,, P_n), to convert a residue number (x₁,x₂,...,x_n) to its decimal correspondence, a matrix of numbers bounded by P_i is needed for the New CRT I compared to the large numbers M/P_i for the CRT, where M=P₁P₂...P_n. The New CRT II uses module multipliers of size less than √M. If the condition P _i+1>P₁+P₂+...+P_i is satisfied, only one module operation of size P_n is needed for the conversion. Residue-to-binary conversion based on the New CRT´s presented here will have a significant impact on many algorithms which currently use the CRT, particularly in computer arithmetic such as residue number systems