Having introduced the LLL reduction, we now provide a more general notions of a reduced basis for a lattice as well as provide bounds for the basis vectors. The key idea behind introducing these definitions is that once we know some basis vector is -reduced, we can bound the sizes of the basis, which is important when algorithms require short vectors in a lattice. For fast algorithms, LLL-reduction is typically the most important notion as it can be computed quickly. Two main definitions appear often when discussing lattice reductions, which we will provide here.
A basisis size-reduced if . Intuitively this captures the idea that a reduced basis being "almost orthogonal".
Let be a lattice, , we define the successive minima as
Intuitively, is the length of the " shortest lattice vector". This intuition is illustrated by the definition of :
However this is not precise as if is the shortest lattice vector, then is also the shortest lattice vector.
Unfortunately, a basisfor where for dimensions and above. This tells us that we can't actually define "the most reduced basis" in contrast to the 2D case (see Lagrange's algorithm) and we would need some other definition to convey this intuition.
An alternate definition ofthat will be helpful is the radius of the smallest ball centered at the origin such that the ball contains at leastlinearly independent vectors in.
1) Show that both definitions of are equivalent
2) Consider the lattice . Show that the successive minima are allbut no basiscan satisfy .