HKZ reduced

Definition

Let πi\pi_ias the projection to the orthogonal complement of {bj}j=1i1\left\{b_j\right\}_{j=1}^{i-1}.Then the basis is HKZ-reduced if it is size-reduced and bi=λ1(πi(L))||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

4i+3(biλi(L))2i+34\frac4{i+3}\leq\left(\frac{||b_i||}{\lambda_i(L)}\right)^2\leq\frac{i+3}4