Sharper bounds on four lattice constants

The Korkine–Zolotareff (KZ) reduction and its generalisations, are widely used lattice strategies in communications cryptography. KZ constant Schnorr’s were defined by Schnorr 1987. can be to quantify some useful properties of reduced matrices. characterize the output quality his block 2k-reduction is define semi 2k-reduction, which was also developed Hermite’s constant, a fundamental has many applications, such as bounding length shortest nonzero vector orthogonality defect lattices. Rankin’s introduced Rankin 1953 generalization constant. It plays an important role characterizing block-Rankin reduction, proposed Gama et al. 2006. In this paper, we first develop linear upper bound on then use it These bounds sharper than those obtained recently authors, ratio new nonlinear bound, Blichfeldt 1929, asymptotically 1.0047. Furthermore, lower improvement over sharpest existing one around 1.7 times asymptotically, 4 asymptotically. Finally, improvements ones, al., exponential parameter defining