Algorithm 2: The LLL reduction with pivoted reflection

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

For i = 1 to m − 1         (Pivoted Householder Reflection)

Find the shortest vector $b_s^i$ in $B_i$

Swap($b_i$, $b_{i+s-1}$) of $B$

Calculate $H_i$ for $B$

Set $R=H_i·R$ and $Q=Q·H_i$

End

Set k = 2

While $k\le m$

For j = k-1 down to 1

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

End

If $\delta ·r_{k-1,k-1}^2>r_{k,k}^2+r_{k-1,k}^2$ then      (Lovász condition)

Swap ($r_{k-1}$,$r_k$)               (Swap process)

Calculate $H_{k-1}$ for $R$

Set $R=H_{k-1}·R$ and $Q=Q·H_{k-1}$

Set k = max(k − 1, 2)

Else

Set k = k + 1

End

End