Classification of Moduli Sets for Residue Number System With Special Diagonal Functions

Abstract

The paper presents algorithms for the generation of Residue Number System (RNS) triples with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$SQ=2^{k}-1$ </tex-math></inline-formula> and quadruples with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$SQ=2^{k}$ </tex-math></inline-formula> for some k. Triples and quadruples allow us to design efficient hardware implementations of non-modular operations in RNS such as division, sign detection, comparison of numbers, reverse conversion with using of a diagonal function from requiring division with the remainder by the diagonal module SQ. Division with a remainder in the general case is the most complex arithmetic operation in computer technology. However, the consideration of special cases can significantly simplify this operation and increase the efficiency of hardware implementation. We show that there are 5573 good RNS triples (2301 even and 2372 odd) with elements less than 10 000, as the values of SQ vary from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{5}-1$ </tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{27}-1$ </tex-math></inline-formula> . In contrast, RNS quadruples with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$SQ=2^{k}$ </tex-math></inline-formula> seem to be quite rare. Restricting our search to sums of the elements in a quadruple less than 4000 we find that exactly 31 such quadruples exist. Their values of SQ vary between 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">20</sup> and 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">30</sup> with always even exponent. We suggest the measure of RNS balance and find perfectly balanced RNS among triples according to this measure. We demonstrate the advantages of more balanced quadruples by means of hardware implementation.