Design of an RNS reverse converter for a new five-moduli special set

Abstract

In this paper, we present a new residue number system (RNS) {2 n − 1, 2 n , 2 n + 1, 2 n+1 + 1, 2 n−1 + 1} of five wellbalanced moduli that are co-prime for odd n. This new RNS complements the 5-moduli RNS system proposed before for even n {2 n − 1, 2 n , 2 n + 1, 2 n+1 − 1, 2 n−1 − 1}. With the new set, we also present a novel approach to designing multimoduli reverse converters that focuses strongly on moving a significant amount of computations off the critical path. The synthesis of the resulting design over the ST Microelectronics 65nm LP library demonstrates that the delay, area, and power characteristics improve the performance and power consumption of the existing complementary 5-moduli set.

Design of Reverse Converters for a New Flexible RNS Five-Moduli Set $$\{ 2^k, 2^n-1, 2^n+1, 2^{n+1}-1, 2^{n-1}-1 \}$$ { 2 k , 2 n - 1 , 2 n + 1 , 2 n + 1 - 1 , 2 n - 1 - 1 } (n Even)

“…In this section, we will evaluate and compare the gate-level complexity of our converters and their closest counterparts: {2 n , 2 n − 1, 2 n + 1, 2 n+1 − 1, 2 n−1 − 1} (n even), proposed in [4] (special case of our moduli set with k = n), and {2 n − 1, 2 n , 2 n + 1, 2 n+1 + 1, 2 n−1 + 1} (n odd), proposed in [18]. The gate-level complexity evaluations of hardware and delay appear in Tables 1, 2 and 3.…”

“…Thus, we first provide an analysis of higher-level representations like CSAs and CPAs, whereas detailed complexity characteristics expressed in terms of primitive gates will be presented subsequently in Table 3. Table 1 details all adders used in our converters and their counterparts of [4] and [18]. As for the number of CPAs, Version 1 of our converter allows us to spare one CPA mod 2 n+1 − 1, one CPA mod 2 n−1 − 1 and one 4n-bit ordinary CPA, whereas Version 2 of our converter requires the same number of CPAs as its counterpart of [4] (or CPAs mod 2 n±1 + 1 as its counterpart of [18]).…”

“…Table 1 details all adders used in our converters and their counterparts of [4] and [18]. As for the number of CPAs, Version 1 of our converter allows us to spare one CPA mod 2 n+1 − 1, one CPA mod 2 n−1 − 1 and one 4n-bit ordinary CPA, whereas Version 2 of our converter requires the same number of CPAs as its counterpart of [4] (or CPAs mod 2 n±1 + 1 as its counterpart of [18]). The comparison of the number of CSAs is slightly more complicated because (i) this number depends on k and (ii) different circuitry is used to carry out the multiplication by multiplicative inverses in …”

“…The four-moduli sets include those with a single parameter n: {2 n − 1, 2 n , 2 n−1 − 1, 2 n−1 + 1} (n odd) [28], {2 n − 1, 2 n , 2 n + 1, 2 n+1 − 1} (n even) [2,3,5,28,30], {2 n , 2 n − 1, 2 n + 1, 2 n−1 − 1} (n even) [5], {2 n −1, 2 n , 2 n +1, 2 n+1 +1} (n odd) [2,29], and {2 n +1, 2 n −1, 2 n , 2 n−1 +1} (n odd) [20], as well as some of their recently proposed generalisations with a flexible even modulus 2 k : {2 k , 2 n − 1, 2 n + 1, 2 n+1 − 1} (n even and arbitrary k such that n ≤ k ≤ 2n) [6], {2 n−1 − 1, 2 n+1 − 1, 2 k , 2 n − 1} (n even and k > 2) [33] and {2 k , 2 n −1, 2 n +1, 2 n+1 −1, 2 n±1 −1} [19] (n even and k > 2). Further reduction of the widths of residue channels for a given dynamic range allow for two special five-moduli sets composed only of low-cost moduli: {2 n −1, 2 n , 2 n +1, 2 n+1 −1, 2 n−1 −1} (n even) [4] and {2 n − 1, 2 n , 2 n + 1, 2 n+1 + 1, 2 n−1 + 1} (n odd) [18]. The dynamic ranges of these two RNSs make it possible to represent all integers up to 5n − 1 bits (n even) and 5n bits (n odd), respectively.…”

“…These two moduli sets are complementary in the sense that altogether they allow periodically for a 4-or 6-bit resolution of the dynamic range. The major limitation of these two moduli sets is that, should a dynamic range other than those listed above be needed, a designer must use a moduli set whose dynamic range is larger than actually needed, with the closest sufficiently large n. If the dynamic range of 20 bits is needed, the moduli set of [18] for n = 5, i.e. {31, 32, 33, 65, 17}, is the best choice currently available, although its 25-bit dynamic range is unnecessarily large and results in too-large residue datapaths.…”