{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/rlwe-and-plwe-over-cyclotomic-fields-are-not","title":"RLWE and PLWE over cyclotomic fields are not equivalent","arxiv_id":"2201.04365","date":"2022-01-12","proceeding":null,"authors":["Antonio J. Di Scala","Carlo Sanna","Edoardo Signorini"],"abstract":"We prove that the Ring Learning With Errors (RLWE) and the Polynomial Learning With Errors (PLWE) problems over the cyclotomic field $\\mathbb{Q}(\\zeta_n)$ are not equivalent. Precisely, we show that reducing one problem to the other increases the noise by a factor that is more than polynomial in $n$. We do so by providing a lower bound, holding for infinitely many positive integers $n$, for the condition number of the Vandermonde matrix of the $n$th cyclotomic polynomial.","url_abs":"https://arxiv.org/abs/2201.04365v1","url_pdf":"https://arxiv.org/pdf/2201.04365v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"rlwe-and-plwe-over-cyclotomic-fields-are-not","repo_url":"https://github.com/edoars/cyclovandermonde","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}