{"paper":{"title":"Improvement Of Barreto-Voloch Algorithm For Computing $r$th Roots Over Finite Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CR","math.NT"],"primary_cat":"cs.SC","authors_text":"Xiao Fan, Zhengjun Cao","submitted_at":"2011-10-19T13:39:05Z","abstract_excerpt":"Root extraction is a classical problem in computers algebra. It plays an essential role in cryptosystems based on elliptic curves. In 2006, Barreto and Voloch proposed an algorithm to compute $r$th roots in ${F}_{q^m} $ for certain choices of $m$ and $q$. If $r\\,||\\,q-1$ and $ (m, r)=1, $ they proved that the complexity of their method is $\\widetilde{\\mathcal {O}}(r(\\log m+\\log\\log q)m\\log q) $. In this paper, we extend the Barreto-Voloch algorithm to the general case that $r\\,||\\,q^m-1$, without the restrictions $r\\,||\\,q-1$ and $(m, r)=1 $. We also specify the conditions that the Barreto-Vol"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.4801","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"}