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- Title
Fractional non-norm elements for division algebras, and an application to Cyclic Learning with Errors.
- Authors
Mendelsohn, Andrew; Ling, Cong
- Abstract
Given a cyclotomic field $ K $ and a finite Galois extension $ L $, we discuss the construction of unit-magnitude elements in $ K $ which are not in the image of the field norm map $ N_{L/K}(L^\times) $. We observe that the construction of Elia, Sethuraman, and Kumar extends to all cyclotomic fields whose rings of integers are a principal ideal domain, a fact we have not seen appear elsewhere in the literature. We then prove a number of lemmas concerning non-norm elements, and extend the above results to hold for arbitrary cyclotomic ground fields. We give examples of towers of fields and corresponding non-norm elements in both instances. Finally, we apply this to cryptography, defining a novel variant of Learning with Errors, defined over cyclic division algebras with fractional unit-magnitude non-norm elements, and reduce lattice problems defined over ideals in maximal orders in such algebras to the search problem for this form of LWE.
- Subjects
DIVISION algebras; CYCLOTOMIC fields; RINGS of integers; ALGEBRA; BANACH lattices; CRYPTOGRAPHY
- Publication
Advances in Mathematics of Communications, 2024, Vol 18, Issue 2, p1
- ISSN
1930-5346
- Publication type
Article
- DOI
10.3934/amc.2023043