## 3 Proposed Multi-Modulus Multiplier Based on Radix-8 Booth Encoding
### Proposed Multi-Modulus Hard Multiple Generator
For the DH block, multiplication by 2 (\(\times 2\)) for the modulo (\(2^{n}-1\)), modulo (\(2^{n}\)), and modulo (\(2^{n}+1\)) multipliers is expressed as:  
\[|+2X|_{m}=\left\{ \begin{array}{ll}(x_{n-2}x_{n-3}\ldots x_{0}x_{n-1})&if\ m=2^{n}-1 \\ (x_{n-2}x_{n-3}\ldots x_{0}0)&if\ m=2^{n}\\ (x_{n-2}x_{n-3}\ldots x_{0}\overline{x_{n-1}})&if\ m=2^{n}+1\end{array} \right.. \tag{17}\]  
Based on Equations (6) and (7) and Equation (17), the DH component is as shown in Figure 6. In Figure 6a, \(i=0\) is shown for the end-around-carry bit in the multi-modulus. Figure 6b is the general circuit implementation for \(i>0\). In the prefix operator block, \(H_{2i-1}{}^{**}\) (\(i=1\), \(2\), \(3\ldots\)) is the modified Ling bit and is expressed at odd positions, which is defined as \(H_{2i-1}^{**}=(G_{2i-1}^{**}\), \(P_{2i-3}^{**})\bullet(G_{2i-5}^{**}\), \(P_{2i-7}^{**})\), where \(H_{-1}^{**}=H_{n-1}^{**}\), \(G_{-i}^{**}=G_{n-i}^{**}\), and \(P_{-i}^{**}=P_{n-i}^{**}\)[22; 23; 24]. Taking \(n=8\) as an example, \(H_{2i-1}{}^{**}\) can be shown as Equation (18). The index of \(H_{2i-1}{}^{**}\) at position 1 and position 5 is different from the index of \(H_{2i-1}{}^{**}\) at position 3 and position 7. Therefore, \(H_{2i-1}{}^{**}\) is separated into two groups: \(H_{4k+1}{}^{**}\) and \(H_{4k+3}{}^{**}\)[24], where \(k=0\), \(1\), \(2\), \(3\), \(\ldots\). That is to say, \(H_{2i-1}{}^{**}\) = (\(H_{1}{}^{**}\), \(H_{3}{}^{**}\), \(P_{1}^{**}\), \(H_{2}{}^{**}\), \(H_{3}{}^{**}\), \(H_{4}{}^{**}\), \(H_{1}{}^{**}\)\(\ldots\)) is divided into two groups: \(H_{4k+1}{}^{**}\) = (\(H_{1}{}^{**}\), \(H_{5}{}^{**}\), \(H_{6}{}^{**}\)\(\ldots\)) and \(H_{4k+3}{}^{**}\) = (\(H_{3}{}^{**}\), \(H_{7}{}^{**}\), \(H_{1}{}^{**}\)\(\ldots\)). The general expressions of \(H_{2i-1}{}^{**}\) for modulo (\(2^{n}-1\)), modulo (\(2^{n}\)), and modulo (\(2^{n}+1\)) are derived from the modulo (\(2^{n}-1\)) multiplier [22], modulo (\(2^{n}\)) multiplier [23], and modulo (\(2^{n}+1\)) multiplier [24], respectively:  
\[H_{1}^{**}=(G_{1}^{**}\), \(\overline{G}_{7}^{**})\bullet(\overline{P}_{5}^{**}\), \(\overline{G}_{3}^{**})\] \[H_{3}^{**}=(G_{3}^{**}\), \(P_{1}^{**})\bullet(\overline{P}_{7}^{**}\), \(\overline{G}_{5}^{**})\] \[H_{5}^{**}=(G_{5}^{**}\), \(P_{3}^{**})\bullet(G_{1}^{**}\), \(\overline{G}_{7}^{**})\] \[H_{7}^{**}=(G_{7}^{**}\), \(P_{5}^{**})\bullet(G_{3}^{**}\), \(P_{1}^{**})\] \[\left\{ \begin{array}{ll}H_{4k+1}{}^{**}=(\underline{G_{4k+1}}{}^ {**},P_{4k-1}{}^{**})\bullet\ldots\bullet(G_{1}{}^{**},P_{-1}{}^{**})\bullet \ldots\bullet(G_{4k-3}{}^{**},P_{4k-5}{}^{**})\\ \\ H_{4k+3}{}^{**}=(\underline{G_{4k+3}}{}^{**},P_{4k+1}{}^{**})\bullet\ldots \bullet(G_{3}{}^{**},P_{1}{}^{**})\bullet\ldots\bullet(G_{4k-1}{}^{**},P_{4k-3 }{}^{**})\end{array} \right.\text{, \ for modulo }(2^{n}-1) \tag{19}\]  
\[\left\{ \begin{array}{ll}H_{4k+1}{}^{**}=(\underline{G_{4k+1}}{}^ {**},P_{4k-1}{}^{**})\bullet\ldots\bullet(G_{1}{}^{**},0)\bullet\ldots\bullet(0,0)\\ \\ H_{4k+3}{}^{**}=(\underline{G_{4k+1}}{}^{**},P_{4k-1}{}^{**})\bullet\ldots \bullet(G_{3}{}^{**},P_{1}{}^{**})\bullet\ldots\bullet(0,0)\\ \\ \end{array} \right.\text{, \ for modulo }2^{n} \tag{20}\]  
The final results of the HMG for the sum bit are expressed as follows. The equations for the even-position sum bit and odd-position sum bit for \(i=0\) are expressed as Equations (22) and (24), respectively, and the equations for the even-position sum bit and odd-position sum bit for \(i>0\) are expressed as Equations (23) and (25), respectively:  
\[\left\{ \begin{array}{l}s_{2i}=h_{2i}\oplus(P_{2i-1}{}^{*}H_{2i-1}{}^{**}),\, if\,i=0\,,\,for\,m=2^{n}-1\\ s_{2i}=h_{2i}\oplus 0,\quad\,if\,i=0\,,\,for\,m=2^{n}\\ s_{2i}=h_{2i}\oplus\left(\overline{P_{2i-1}}{}^{*}+\overline{H_{2i-1}}{}^{*} \right)\,,\,if\,i=0\,,\,for\,m=2^{n}+1\end{array} \right. \tag{22}\]  
\[\left\{