Table I
Residue In Binary with a Bias Notation Based on the Bias Method for the
Numerical Example
Modulus (m_{r})  Residue
(R_{mr})  Residue in binary notation  Bias  Residue in binary with bias notation 

3  0  00  01  01 
1  01  10 
2  10  11 

4  0  00  00  00 
1  01  01 
2  10  10 
3  11  11 

5  0  000  011  011 
1  001  100 
2  010  101 
3  011  110 
4  100  111 
Table II
Ratios as Measures of Operation Time Between Two Cases of Binary and
Residue Number
 m_{N}  w_{mN} 
$${w}_{mN}^{2}/{w}^{2}$$ 

n  50  500  50  500  50  500 
w 
4  13  17  4  5  1. 000  1. 563 
8  19  23  5  5  0. 391  0. 391 
16  37  37  6  6  0. 141  0. 141 
32  59  61  6  6  0. 035  0. 035 
Table III
ResiduetoBinary Conversion by Using a Mixed Radix Number System (an
example for x = 35 and N =
3)
  

χ in decimal notation  χ in residue
notation{R_{m}_{1},
R_{m}_{2},
R_{m}_{3}}   



m_{1} = 3  m_{2} = 4  m_{3} = 5 

  

59  2  3  4   
58  1  2  3   
:  :  :  :   
:  :  :  :   
  
35  2  3  0 

$$\begin{array}{ll}\{1,1,1\}\hfill & \times 2\hfill \\ & \hfill \parallel \\ & \hfill {a}_{1}\end{array}$$ 

34  1  2  4 
33  0  1  3 
32  2  0  2 
:  :  :  :  
:  :  :  : 
$$\begin{array}{ll}\underbrace{\{0,3,3\}}\hfill & \times 3\hfill \\ \hfill \parallel \hfill & \hfill \parallel \hfill \\ \hfill {m}_{1}\hfill & \hfill {a}_{2}\hfill \end{array}$$ 
25  1  1  0 
24  0  0  4 
23  2  3  3 
22  1  2  2  
21  0  1  1  
20  2  0  0  
19  1  3  4  
18  0  2  3  
17  2  1  2  
16  1  0  1  
15  0  3  0  
14  2  2  4  
13  1  1  3  
12  0  0  2 
$$\begin{array}{ll}\underbrace{\{0,0,2\}}\hfill & \times 2\hfill \\ \hfill \parallel \hfill & \hfill \parallel \hfill \\ \hfill {m}_{1}\cdot {m}_{2}\hfill & \hfill {a}_{3}\hfill \end{array}$$ 
11  2  3  1 
10  1  2  0 
9  0  1  4  
8  2  0  3 
$$\begin{array}{rrr}\hfill \{1,1,1\}& \hfill \times 2=& \hfill 2\\ \hfill \{0,3,3\}& \hfill \times 3=& \hfill 9\\ \hfill \{0,0,2\}& \hfill \times 2=& \hfill 24\\ \hfill & \hfill & \hfill 2\\ \hfill & \hfill & \hfill 9\\ \hfill & \hfill & \hfill \frac{+24}{35}\end{array}$$ 
7  1  3  2 
6  0  2  1 
5  2  1  0 
4  1  0  4 
3  0  3  3 
2  2  2  2 
1  1  1  1 
0  0  0  0 
  
Table IV
Approximate Representation of Operation Time for a Matrix
Multiplication
Scheme of operation  Operation time 

(1) Matrix Multiplication in
residue number using Fourier processor and OFFs  Binarytoresidue conversion ≃
w(T_{SLM} +
T_{CO}) 
Multiplication in residue number $\simeq {w}_{{m}_{N}^{2}}\phantom{\rule{0.4em}{0ex}}({T}_{\text{SLM}}+{T}_{\text{co}\phantom{\rule{0.2em}{0ex}}n})$ 
Residuetobinary conversion and
truncation $\simeq 2\phantom{\rule{0.4em}{0ex}}\{\phantom{\rule{0.2em}{0ex}}\text{\u2211}_{r=1}^{N}\phantom{\rule{0.2em}{0ex}}({m}_{r}1)\phantom{\rule{0.2em}{0ex}}\}\phantom{\rule{0.4em}{0ex}}({T}_{\text{SLM}}+{T}_{\text{co}})$ 

(2) Matrix multiplication in binary number
using all electronic system  ≃
w^{2}T_{ce}n 
(
w : maximum digit number of inputted elements in binary
notation,
T_{SLM}: operation tine of SLM,
T_{co} : clock period of system using
Fourier processor and residue number,
w_{mN}
:maximum digit number of residue modulo
m_{N}
in binary notation,
n : order of matrices,
N : number of moduli,
m_{r}:
rth modulus of residue number system,
T_{ce}: clock period of all electronic
system).