Algorithm 3 Babai’s Nearest Plane algorithm.

Input:  lattice basis ${\mathbf{B}}^{\prime }=[{\mathbf{b}}_1^{\prime },{\mathbf{b}}_2^{\prime },\dots ,{\mathbf{b}}_m^{\prime }]\in {\mathbb{R}}^{m\times m}$, target vector $\mathbf{t}\in {\mathbb{Z}}^m$ Output:  vector $\mathbf{x}\in \mathcal{L}({\mathbf{B}}^{\prime })$, which satisfies $\parallel \mathbf{x}-\mathbf{t}\parallel \le 2^{m/2}dist(\mathbf{t},\mathcal{L}({\mathbf{B}}^{\prime }))$  1:Perform the LLL reduction on ${\mathbf{B}}^{\prime }$ with parameter $\delta =3/4$.  2:Use the Gram–Schmidt orthogonalization on the reduced basis and obtain ${\mathbf{B}}^{\ast }=[{\mathbf{b}}_1^{\ast },{\mathbf{b}}_2^{\ast },\dots ,{\mathbf{b}}_m^{\ast }]$.  3:$\mathbf{b}=\mathbf{t}$.  4:for j = m, m − 1,…, 1 do  5:   $\mathbf{b}=\mathbf{b}-u_j{\mathbf{b}}_j^{\prime }$, where $u_j=\lceil \langle \mathbf{b},{\mathbf{b}}_j^{\ast }\rangle /\langle {\mathbf{b}}_j^{\ast },{\mathbf{b}}_j^{\ast }\rangle \rfloor$;  6:end for  7:return $\mathbf{t}-\mathbf{b}$