Rotated Z^n-Lattices via Real Subfields of Q(\zeta_2r)

Abstract

A method for constructing rotated Z^n-lattices, with n a power of 2, based on totally real subfields of the cyclotomic field Q(\zeta_{2^r}), where r\geq 4 is an integer, is presented. Lattices exhibiting full diversity in some dimensions n not previously addressed are obtained.

A Construction of Rotated Lattices via Totally Real Subfields of the Cyclotomic Field Q(z_p)

“…Algebraic number theory has recently raised a great interest for their new role in algebraic lattice theory and for application in coding and modulation. The problem of finding algebraic lattices with maximal minimum product distance has been studied in last years and this has motivated special attention of many researchs in considering ideals of certain rings [5], [2] and [1]. Eva Bayer et al [7] and Andrade et al [1] have presented families of rotated Z n -lattices based on algebraic number theory.…”

“…The problem of finding algebraic lattices with maximal minimum product distance has been studied in last years and this has motivated special attention of many researchs in considering ideals of certain rings [5], [2] and [1]. Eva Bayer et al [7] and Andrade et al [1] have presented families of rotated Z n -lattices based on algebraic number theory. We know that totally real algebraic number fields result in the maximum diversity, equal to the dimension of the lattice [3].…”

“…In [1], for any integer r ≥ 4, rotated Z n -lattices, n = 2 r−2 and n = 2 r−3 , were constructed from…”

Theoretical and practical contributions to Ring-LWE cryptography

“…Examples of ideal lattices equivalent to Z n are those obtained from cyclotomic number fields Q(ζ 2 k ) [27], and their maximal real subfields [16], and the maximal real subfields…”

“…In this context, we are interested in algebraic lattices equivalent to Z n , which are known to be constructed from power-of-two cyclotomic number fields [27] and their maximal real subfields [16], and from the maximal real subfields of p-th cyclotomic number fields for any prime p ≥ 5 [26]. Recall from Chapter 4 that lattices equivalent to Z n lead to efficient sampling from spherical Gaussian distributions.…”