Lattice reduction question regarding the capability of LLL and BKZ
Yes, if you instantiate the SVP oracle in BKZ using sieving then the cost is exponential in time and memory. (If you instantiate it using enumeration then the cost is $k^{1/8\,k + o(k)}$). The norm of the output vector is expected to be $\delta_k^{d-1} \cdot \operatorname{Vol}(\Lambda)^{1/d}$ with $\delta_k \approx GH(k)^{1/(k-1)}$. Here $\operatorname{Vol}(\Lambda)$ is the volume of the lattice, and $GH(k) \approx \sqrt{k/(2\pi e)}$ is the Gaussian heuristic for a lattice with volume 1. This does not depend on the norm of the input basis.