On the Optimality of Gauss’s Algorithm over Euclidean Imaginary Quadratic Fields

Abstract

In this paper, we continue our previous work on the reduction of algebraic lattices over imaginary quadratic fields for the special case when the lattice is spanned over a two dimensional basis. In particular, we show that the algebraic variant of Gauss's algorithm returns a basis that corresponds to the successive minima of the lattice in polynomial time if the chosen ring is Euclidean.

Unit Reducible Fields and Perfect Unary Forms

“…Unit reducibility is a useful property in the study of lattice-based cryptography [9]. The security of many lattice-based cryptosystems is underpinned by the so-called "shortest vector problem" (SVP), which asks the adversary to find a shortest nonzero vector of a lattice given an arbitrary lattice basis.…”