Algorithm4 Comparison of two values encoded in two input vectors

Input: Encoded $V_X$ as $[{\mathbf{x}}_{0,L},{\mathbf{x}}_{0,G},{\mathbf{x}}_{1,L},{\mathbf{x}}_{1,G},\dots ,{\mathbf{x}}_{D-3,L},{\mathbf{x}}_{D-3,G}]$, encoded $V_Y$ as $[{\mathbf{y}}_0,{\mathbf{y}}_1,\dots ,{\mathbf{y}}_{D-3}]$,     vector size D Ensure: $D\ge 3$ Output: 1 if $V_X\le V_Y$; 0 otherwise. 1:$N\leftarrow D-2$ {Number of vectors} 2: $n\leftarrow 0$ 3:while$n<N$do {$0\le n<N$} 4: $r\leftarrow \langle {\mathbf{x}}_{n,L},{\mathbf{y}}_n\rangle$ 5: if $r=0$ then 6:  return 1 7: end if 8: $r\leftarrow \langle {\mathbf{x}}_{n,G},{\mathbf{y}}_n\rangle$ 9: if $r=0$ then 10:  return 0 11: end if 12: $n\leftarrow n+1$ 13: end while 14:return 1