Algorithm 1: Random Integer Lattice Generation

Require: Dimension n, large integer M

Ensure: n-dim random integer lattice $\mathcal{L}$ with $det\left(\mathcal{L}\right)\le M$

Step 1: Generate $h_{11},\cdots ,h_{n-1,n-1}$

$D_0=1$

for $i=1$ to $n-1$ do

$j_i=1$, $s_i=1$

choose $y_i\in [0,1]$ uniformly

while $s_i<\zeta (n+1-i)·y_i$ do

$j_i=j_i+1$

$s_i=s_i+j_i^{-(n+1-i)}$

end while

$D_i=D_{i-1}·j_i$

set $h_{ii}=j_i$

end for

Step 2: Generate $h_{nn}$

choose $y\in [0,1]$ uniformly

$z=y^{1/n}$

$z=z·\lfloor \frac{M}{D_{n-1}}\rfloor$

set $h_{nn}=\lfloor z\rceil$

Step 3: Generate $h_{ij}(i\ne j)$

for $j=1$ to n do

for $i=1$ to $j-1$ do

choose $h_{ij}\in [0,h_{jj})$ uniformly

end for

for $i=j+1$ to n do

set $h_{ij}=0$

end for

end for

Step 4: Set $H=\left(h_{ij}\right)$, and output $\mathcal{L}\left(H\right)$