1 research outputs found
Sharper Bounds on Four Lattice Constants
The Korkine--Zolotareff (KZ) reduction, and its generalisations, are widely
used lattice reduction strategies in communications and cryptography. The KZ
constant and Schnorr's constant were defined by Schnorr in 1987. The KZ
constant can be used to quantify some useful properties of KZ reduced matrices.
Schnorr's constant can be used to characterize the output quality of his block
-reduction and is used to define his semi block -reduction, which was
also developed in 1987. Hermite's constant, which is a fundamental constant
lattices, has many applications, such as bounding the length of the shortest
nonzero lattice vector and the orthogonality defect of lattices. Rankin's
constant was introduced by Rankin in 1953 as a generalization of Hermite's
constant. It plays an important role in characterizing the output quality of
block-Rankin reduction, proposed by Gama et al. in 2006. In this paper, we
first develop a linear upper bound on Hermite's constant and then use it to
develop an upper bound on the KZ constant. These upper bounds are sharper than
those obtained recently by the authors, and the ratio of the new linear upper
bound to the nonlinear upper bound, developed by Blichfeldt in 1929, on
Hermite's constant is asymptotically 1.0047. Furthermore, we develop lower and
upper bounds on Schnorr's constant. The improvement to the lower bound over the
sharpest existing one developed by Gama et al. is around 1.7 times
asymptotically, and the improvement to the upper bound over the sharpest
existing one which was also developed by Gama et al. is around 4 times
asymptotically. Finally, we develop lower and upper bounds on Rankin's
constant. The improvements of the bounds over the sharpest existing ones, also
developed by Gama et al., are exponential in the parameter defining the
constant.Comment: to appear in Designs, Codes and Cryptography. arXiv admin note: text
overlap with arXiv:1904.0939