LLL - Lattice Reduced Basis Algorithm question?

Question

I have two related questions:

Version 1: Let $B=\{b_1,b_2,\dots,b_n\}$ be an orthogonal basis for $R^n$. What is the associated reduced basis obtained by applying LLL algorithm to $B$?

I know how to apply LLL algorithm and I can apply it for $R^3$ case. (Since for $n=3$ it takes reasonable time in the exam, this is an exam question by the way.)

But this case is general case for $n$, so I don't know what to do? is there a shortcut to find the reduced basis when the vectors are orthogonal ?

another version of this question is the following:

Version 2: Let $B=\{b_1,b_2,\dots,b_6\}$ be an orthogonal basis for $R^6$ .

Having $||b_1||=||b_3||=1$, $||b_2||^2=||b_6||^2=2$, $||b_4||^2=3,||b_5||^2=4$

What is the associated reduced basis obtained by applying LLL algorithm to this ordered basis $B$?

Answer 1

LLL algorithm has two main parts:

1. the reduction step: which recomputes the basis vectors aiming to reduce the values of the Gram-Schmidt coefficients to some value smaller than one half. This condition is usually written as $|\mu_{i, j}| < \frac{1}{2}$.
2. the swap step: which swaps basis vectors aiming to achieve the Lovász condition, that is $ \delta \Vert \mathbf{b}^*_{i}\Vert^2 \leq \Vert \mathbf{b}^*_{i+1}\Vert^2+ \mu_{i+1,i}^2\Vert \mathbf{b}^*_{i}\Vert^2$.

Notice that the swap step just changes the order of the vectors, so, if somehow the reduction step is not executed, the output is just a reordering of the input basis...

Now, what is the Gram-Schmidt orthogonalization of a basis that is already orthogonal? Answering this will give you a trivial relation between $\mathbf{b}_i$ and the corresponding GS vector $\mathbf{b}_i^*$. And what are the values of $\mu_{i, j}$ given that $\mathbf{b}_i$ and $\mathbf{b}_j$ are orthogonal?

With these information, you can see that, for an orthogonal basis, the reduction step is ignored and the LLL algorithm just reorder the basis vectors according to whether $ \delta \Vert \mathbf{b}_{i}\Vert^2 \leq \Vert \mathbf{b}_{i+1}\Vert^2$ or not.

And to answer your second question, you just have to run this "simplified LLL" that you will have found in the first question and get the correct reordering.