{"paper":{"title":"Taking Roots over High Extensions of Finite Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Eric Schost, Javad Doliskani","submitted_at":"2011-10-19T18:33:28Z","abstract_excerpt":"We present a new algorithm for computing $m$-th roots over the finite field $\\F_q$, where $q = p^n$, with $p$ a prime, and $m$ any positive integer. In the particular case $m=2$, the cost of the new algorithm is an expected $O(\\M(n)\\log (p) + \\CC(n)\\log(n))$ operations in $\\F_p$, where $\\M(n)$ and $\\CC(n)$ are bounds for the cost of polynomial multiplication and modular polynomial composition. Known results give $\\M(n) = O(n\\log (n) \\log\\log (n))$ and $\\CC(n) = O(n^{1.67})$, so our algorithm is subquadratic in $n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.4350","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":""},"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"}