Algorithm 3: The n-LLL reduction with pivoted reflection

Set $R=B$ and $Q=I_{m\times m}$

(Move the shortest vector to $r_1$)

Find the shortest vector $r_s$ in $R$

Swap($r_i$,$r_s$) of $R$

Calculate $H_1$ for $R$

Set $R=H_1·R$ and $Q=Q·H_1$

Set i = 2

While $i\le m$           (Pivoted reflection and reduction process)

Find the shortest vector $r_s^i$ in $R_i$

Swap($r_i$, $r_{i+s-1}$) of $R$

Set temp = i

For j = i − 1 down to 1

Set $r_k$ = $r_i-{\left[\frac{r_{j,i}}{r_{j,j}}\right]}_{round}·r_j$                (Size reduction)

If $\delta ·r_{j,j}^2>{\Vert b_{temp-j+1}^j\Vert }^2$ and i − j < n then   (Extended Lovász condition)

Swap ($r_j$, $r_{temp}$)                   (Swap process)

Set temp = j

End

End

Calculate $H_{temp}$ for $R$

Set $R=H_{temp}·R$ and $Q=Q·H_{temp}$

If i ! = temp then

i = temp

Else

Set i = i + 1

End

End