Ring-LWE Cryptography for the Number Theorist

TL;DR

This paper surveys attacks on RLWE and PLWE problems, discusses their security implications, and explores open problems in number theory related to algebraic number fields and spectral properties.

Contribution

It extends existing attacks on RLWE/PLWE and connects them to open problems in algebraic number theory, highlighting new research directions.

Findings

Extended attacks on RLWE and PLWE problems.

Identified open problems in number theory related to spectral distortion and Mahler measure.

Surveyed the relationship between algebraic properties and cryptographic security.

Abstract

In this paper, we survey the status of attacks on the ring and polynomial learning with errors problems (RLWE and PLWE). Recent work on the security of these problems [Eisentr\"ager-Hallgren-Lauter, Elias-Lauter-Ozman-Stange] gives rise to interesting questions about number fields. We extend these attacks and survey related open problems in number theory, including spectral distortion of an algebraic number and its relationship to Mahler measure, the monogenic property for the ring of integers of a number field, and the size of elements of small order modulo q.