Hermite’s constant and lattice algorithms

Abstract

We introduce lattices and survey the main provable algorithms for solving the shortest vector problem (SVP), either exactly or approximately. In doing so, we emphasize a surprising connection between lattice algorithms and the historical problem of bounding a well-known constant introduced by Hermite in 1850, which is related to sphere packings. For instance, we present Lenstra–Lenstra–Lovász (LLL) as an (efficient) algorithmic version of Hermite’s inequality on Hermite’s constant. Similarly, we present blockwise generalizations of LLL as (more or less tight) algorithmic versions of Mordell’s inequality.