{"paper":{"title":"Uncertainty Principles for the Number Theoretic Transform","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.CR","cs.DS"],"primary_cat":"math.NT","authors_text":"Alon Rosen, Giulio Malavolta","submitted_at":"2026-06-07T15:05:53Z","abstract_excerpt":"Motivated by polynomial identity testing with exponentials (Li and Wu, ITCS'26), we study uncertainty principles for the number-theoretic transform (NTT). We show that the NTT satisfies strong sparsity tradeoffs: For every fixed prime $q$ and for all but finitely many primes $p \\equiv 1 \\pmod q$ every nonzero $f\\in \\mathbb F_p^{\\mathbb Z_q}$ and its number-theoretic transform $\\hat f$ satisfy \\[ |\\mathrm{Supp}(f)| + |\\mathrm{Supp}(\\hat f)| \\ge q+1. \\] Thus, a $k$-sparse function has transform support at least $q-k+1$. As our main technical contribution, we prove a probabilistic version of the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08662","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.08662/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}