1,575 research outputs found
Ring-LWE Cryptography for the Number Theorist
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.Comment: 20 Page
Explicit constructions of RIP matrices and related problems
We give a new explicit construction of matrices satisfying the
Restricted Isometry Property (RIP). Namely, for some c>0, large N and any n
satisfying N^{1-c} < n < N, we construct RIP matrices of order k^{1/2+c}. This
overcomes the natural barrier k=O(n^{1/2}) for proofs based on small coherence,
which are used in all previous explicit constructions of RIP matrices. Key
ingredients in our proof are new estimates for sumsets in product sets and for
exponential sums with the products of sets possessing special additive
structure. We also give a construction of sets of n complex numbers whose k-th
moments are uniformly small for 1\le k\le N (Turan's power sum problem), which
improves upon known explicit constructions when (\log N)^{1+o(1)} \le n\le
(\log N)^{4+o(1)}. This latter construction produces elementary explicit
examples of n by N matrices that satisfy RIP and whose columns constitute a new
spherical code; for those problems the parameters closely match those of
existing constructions in the range (\log N)^{1+o(1)} \le n\le (\log
N)^{5/2+o(1)}.Comment: v3. Minor correction
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