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 [EHL, ELOS] 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.
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