We call a module  essentially retractable if HomR for all essential submodules N of M. For a right FBN ring R, it is shown that: (i)  A non-zero module  is retractable (in the sense that HomR for all non-zero ) if and only if certain factor modules of M are essentially retractable nonsingular modules over R modulo their annihilators. (ii)  A non-zero module  is essentially retractable if and on...

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 me...

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 me...

This is a review of several results related to distribution of powers and combinations of powers modulo 1. We include a proof that given any sequence of real numbers n , it is possible to get an (given 6 = 0), or a (given > 1) such that n is close to n modulo 1. We also prove that in a number eld, if a combination of powers 1 n 1 + + m n m has bounded v-adic absolute value (where v is any non-A...

We study variants of the Erd˝ os distance problem and the dot products problem in the setting of the integers modulo q, where q = p is a power of an odd prime.