# HKZ reduced

## Definition

Let $$\pi\_i$$as the projection to the orthogonal complement of $$\left{b\_j\right}\_{j=1}^{i-1}$$.Then the basis is **HKZ-reduced** if it is size-reduced and $$||b\_i^\*||=\lambda\_1\left(\pi\_i(L)\right)$$. This definition gives us a relatively simple way to compute a HKZ-reduced basis by iteratively finding the shortest vector in orthogonal projections.

## Bounds

$$
\frac4{i+3}\leq\left(\frac{||b\_i||}{\lambda\_i(L)}\right)^2\leq\frac{i+3}4
$$