## 5 Method Based on Reference Points  
In [18], a method of reference points based on the use of LUT tables is proposed for translation from a residue number system.  
For a given set of moduli \(\{p_{1},p_{2},\ldots,p_{n}\}\) is chosen by the reference point modulo \(p_{n}\), which to reduce the number of values in memory it is more efficient to choose the largest modulo RNS.  
Then the contents of the LUT will consist of \(P_{h}=\prod_{i=1}^{n-1}p_{i}\) reference points, which correspond to numbers \(X=(x_{1},x_{2},\ldots,x_{n-1},0)\), which are completely divided by the modulo \(p_{n}\).  
In order to obtain an arbitrary number \(Y=(y_{1},y_{2},\ldots,y_{n})\) that is completely divided by \(p_{n}\), you must subtract \(y_{n}\) from the number \(Y\), that is, calculate \(x_{i}=\left|y_{i}-y_{n}\right|_{p_{i}}\), \(i=1,\ldots,n\).  
For example, for a RNS {7,8,9} and an initial number \(Y=10=(3,2,1)\), the reference point is \(X(|3-1||_{Y,2}|2-1|_{8,\left|1-1\right|_{9}})=(2,1,0)\).  
Thus, by analogy with New CRT II and the MRC for two moduli, the reference point will be \(X=\left|X_{h}\cdot\left|p_{n}^{-1}\right|_{p_{h}}\right|_{p_{h}}\cdot p_{n}\), where \(X_{h}\) is the positional number obtained from \((x_{1},x_{2},\ldots,x_{n-1})\) for the set of moduli \(\{p_{1},p_{2},\ldots,p_{n-1}\}\).  
For example, for the number \(Y=228=(4,4,3)\), the reference point would be the value \(X(|4-3|_{7,\left|4-3|_{8,\left|3-3\right|_{9}}\right)}=(1,1,0)\). Since (1,1) for the set of moduli {7,8} is 1, and \(\left|9^{-1}\right|_{7,8}=25\), the reference point is \(X=\left|1\cdot 25\right|_{56}\cdot 9=225\), which corresponds to the subtraction from \(Y=228\) of residue \(y_{n}=3\).  
It can be seen that the value of the reference point depends on the number \((x_{1},x_{2},\ldots,x_{n-1})\) with moduli \(\{p_{1},p_{2},\ldots,p_{n-1}\}\) and range \(P_{h}=\prod_{i=1}^{n-1}p_{i}\). Then the computation can be replaced by a memory access, which by the value \(X_{h}=(x_{1},x_{2},\ldots,x_{n-1})\) will give the value \(\left|X_{h}\cdot\left|p_{n}^{-1}\right|_{P_{h}}\right|_{P_{h}}\cdot p_{n}\).  
In this case, the reverse converter based on the reference points can have a hierarchical structure, for example, as shown in Figure 2. In this case, based on the differences of the residues \(\left|x_{1}-x_{3}\right|_{p_{1}}\) and \(\left|x_{2}-x_{3}\right|_{p_{2}}\) by moduli \(p_{1}\) and \(p_{2}\), respectively, using LUT with the values of the reference points, which are \(\left|X_{12}\cdot\left|p_{3}^{-1}\right|_{p_{1}p_{2}}\right|_{p_{1}p_{2}} \cdot p_{3}\), where \(X_{12}\) is--the restored value over the moduli \(p_{1},p_{2}\), and adding with the remainder \(x_{3}\) modulo \(p_{3}\) find the restored number \(X_{123}\) modulo \(p_{1},p_{2}\), \(p_{3}\).  
Then similarly are the differences of the residues by moduli \(p_{4}\) and \(p_{5}\) and the reconstructed number \(X_{123}\), and since the value of \(X_{123}\) can greatly exceed the values of modules \(p_{4}\) and \(p_{5}\), forward conversion blocks are used, which can be implemented both arithmetically and using memory. Similarly, the values in the LUT of reference points are found, where the pre-calculated values \(\left|X_{45}\cdot\left|\left(P_{123}\right)^{-1}\right|_{P_{45}}\right|_{P_{4 5}}\cdot P_{123}\) for \(P_{123}=p_{1}\cdot p_{2}\cdot p_{3}\) and \(P_{45}=p_{4}\cdot p_{5}\).  
Thus, by varying the number of LUT inputs from \(1\) to \(n-1\) and the number of reverse conversion layers, a solution with better time or area can be obtained.