Efficient Reverse Converter Designs for the New 4-Moduli Sets $\{2^{n} -1, 2^{n}, 2^{n} +1, 2^{2n + 1}-1\}$ and $\{2^{n} -1, 2^{n} +1, 2^{2n}, 2^{2n} +1\}$ Based on New CRTs

In this paper, we introduce two new 4-moduli sets {2
ⁿ
-1, 2
ⁿ
, 2
ⁿ
+1, 2
²ⁿ⁺¹
-1} and {2
ⁿ
-1, 2
ⁿ
+1, 2
²ⁿ
, 2
²ⁿ
+1} for developing efficient large dynamic range (DR) residue number systems (RNS). These moduli sets consist of simple and well-formed moduli which can result in efficient implementation of the reverse converter as well as internal RNS arithmetic circuits. The moduli set {2
ⁿ
-1, 2
ⁿ
, 2
ⁿ
+1, 2
²ⁿ⁺¹
-1} has 5
n
-bit DR and it can result in a fast RNS arithmetic unit, while the 6
n
-bit DR moduli set {2
ⁿ
-1, 2
ⁿ
+1, 2
²ⁿ
, 2
²ⁿ
+1} is a conversion friendly moduli set which can lead to a high-speed and low-cost reverse converter design. Next, efficient reverse converters for the proposed moduli sets based on new Chinese remainder theorems (New CRTs) are presented. The converter for the moduli set {2
ⁿ
-1, 2
ⁿ
, 2
ⁿ
+1, 2
²ⁿ⁺¹
-1} is derived by New CRT-II with better performance compared to the reverse converter for the latest introduced 5
n
-bit DR moduli set {2
ⁿ
-1, 2
ⁿ
, 2
ⁿ
+1, 2
^n-1
-1, 2
ⁿ⁺¹
-1}. Also, New CRT-I is used to achieve a high-performance reverse converter for the moduli set {2
ⁿ
-1, 2
ⁿ
+1, 2
²ⁿ
, 2
²ⁿ
+1}. This converter has less conversion delay and lower hardware requirements than the reverse converter for a recently suggested 6
n
-bit DR moduli set {2
ⁿ
-1, 2
ⁿ
+1, 2
²ⁿ
-2, 2
²ⁿ⁺¹
-3} .