  The Hermite-Korkine-Zolotarev reduction plays a central role in strong
lattice reduction algorithms. By building upon a technique introduced by Ajtai,
we show the existence of Hermite-Korkine-Zolotarev reduced bases that are
arguably least reduced. We prove that for such bases, Kannan's algorithm
solving the shortest lattice vector problem requires
$d^{\frac{d}{2\e}(1+o(1))}$ bit operations in dimension $d$. This matches the
best complexity upper bound known for this algorithm. These bases also provide
lower bounds on Schnorr's constants $\alpha_d$ and $\beta_d$ that are
essentially equal to the best upper bounds. Finally, we also show the existence
of particularly bad bases for Schnorr's hierarchy of reductions.
