Abstract
The paper presents algorithms for the generation of Residue Number System (RNS) triples with $SQ=2^{k}-1$ and quadruples with $SQ=2^{k}$ 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 $2^{5}-1$ to $2^{27}-1$ . In contrast, RNS quadruples with $SQ=2^{k}$ 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 220 and 230 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.
Highlights
The current level of computer technology requires the development of parallel computing architectures and methods for organizing calculations on them
Division with the remainder by 2n, costs nothing, unlike division by M in Chinese remainder theorem (CRT) or multiplication by M, as in CRT with fractional values (CRTf) or different operations on the modules m1, m2, . . . , mn, as in mixed radix conversion (MRC). This is confirmed by our previous studies, so in [7] there is an example of the implementation of non-modular operations of comparison and reverse conversion for triples and quadruples, which demonstrates the advantage of our proposals for the hardware implementation of systems based on Residue Number System (RNS) in FPGA
It could be quite replicated for classification results for triples with SQ = 2k + 1
Summary
The current level of computer technology requires the development of parallel computing architectures and methods for organizing calculations on them. It was observed that diagonal modulus of special binary representations (very low or very high Hamming weight) are useful in the construction of RNS with good performance. This point was described in [7]. Remark: It is clear from the proof that the conclusion v2 (a) = v2 (b) of Lemma 1 is true whenever max{v2 (a) , v2 (b)} < k (i.e., the condition for a and b both being positive integers is dropped) We will use this fact once in our analysis of RNS quadruples.
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