Algorithm 2 Root selection algorithm in the presence of CFO

Initialization: N, K, I, s = 1, $R_{\mathrm{root}1}\left[1:I\right]=\varnothing$, $R_{\mathrm{root}2}\left[1:I\right]=0$, $R_{\mathrm{root}}\left[1:2I\right]=\varnothing$, $g^{\prime }=1$, r = 1, i = 1, j = 1

1: While $i\le 2I$ do

2:       if $r\ge N$ then $g^{\prime }=g^{\prime }+1$, r = 1, i = 1, j = 1, $R_{\mathrm{root}1}\left[1:I\right]=\varnothing$, $R_{\mathrm{root}2}\left[1:I\right]=0$, $R_{\mathrm{root}}\left[1:2I\right]=\varnothing$

3:       else

4:            if $\gcd \left(r,N\right)\ne 1$ or $\gcd \left(r+sK,N\right)\ne 1$ then r = r + 1

5:            else

6:                    if i=1 then $R_{\mathrm{root}1}\left[j\right]=r$, j = j + 1, $R_{\mathrm{root}}\left[i\right]=r$, i = i + 1, r = r + 1

7:                     else

8:                            if there exists $\mu \in R_{\mathrm{root}}$ such that $\gcd \left(N,r-\mu \right)>g^{\prime }$ then r = r + 1

9:                            else

10:                                   $r_2=r$, index = 0

11:                                   for k = 1 to j-1 do

12:                                         if $R_{\mathrm{root}2}\left[k\right]=0$ then

13:                                         $r_1=R_{\mathrm{root}1}\left[k\right]$

14:                                                 if $Q{V}_{r_1,r_2}\left(n_1\right)\ne Q{V}_{r_1,r_2}\left(n_2\right)$ for all $n_1\ne n_2$ ($n_1$, $n_2\in \left[0,L+1\right]$),

15:                                                 and $Q{V}_{r_1,r_2}\cap A{B}_{r_1,r_2}=\varnothing$ then

16:                                                         $R_{\mathrm{root}2}\left[k\right]=r$, $R_{\mathrm{root}}\left[i\right]=r$, i = i + 1, r = r + 1, index = 1

17:                                                         break

18:                                                 end if

19:                                         end if

20:                                   end for

21:                                   if index = 0 and $j\le I$ then

22:                                   $R_{\mathrm{root}1}\left[j\right]=r$, j = j + 1, $R_{\mathrm{root}}\left[i\right]=r$, i = i + 1, r = r + 1

23:                                   end if

24:                          end if

25:                  end if

26:          end if

27:     end if

28: end while