Ring-LWE Cryptography for the Number Theorist
This is an incremental survey paper for cryptographers and number theorists analyzing RLWE/PLWE security.
This paper surveys and extends existing attacks on ring and polynomial learning with errors problems (RLWE and PLWE), focusing on their connections to number theory. It explores open problems including spectral distortion relationships to Mahler measure, monogenic properties of number fields, and element sizes modulo q.
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 [Eisenträger-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.