Algorithm 1 LLL algorithm.

Input:  lattice basis $\mathbf{B}=[{\mathbf{b}}_1,{\mathbf{b}}_2,\dots ,{\mathbf{b}}_n]\in {\mathbb{R}}^{m\times n}$, a reduction parameter $\delta$. Output:  a $\delta$-LLL reduced basis  1:Calculate the Gram–Schmidt orthogonalization ${\mathbf{B}}^{\ast }=[{\mathbf{b}}_1^{\ast },{\mathbf{b}}_2^{\ast },\dots ,{\mathbf{b}}_n^{\ast }]$.  2:for i = 2, 3,…, n do  3:   for j = i − 1, i − 2,…, 1 do  4:     ${\mathbf{b}}_i={\mathbf{b}}_i-c_{i,j}{\mathbf{b}}_j$, where $c_{i,j}=\lceil \langle {\mathbf{b}}_i,{\mathbf{b}}_j^{\ast }\rangle /\langle {\mathbf{b}}_j^{\ast },{\mathbf{b}}_j^{\ast }\rangle \rfloor$;  5:   end for  6:end for  7:if $\exists i$, s.t.$\delta \parallel {\mathbf{b}}_{i-1}^{\ast }{\parallel }^2>{\parallel {\mu }_{i,i-1}{\mathbf{b}}_{i-1}^{\ast }+{\mathbf{b}}_i^{\ast }\parallel }^2$ then  8:   Swap ${\mathbf{b}}_{i-1}$ and ${\mathbf{b}}_i$;  9:   Go to Step 1.  10:end if  11:return $\mathbf{B}$.