{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,20]],"date-time":"2025-12-20T22:32:22Z","timestamp":1766269942403,"version":"3.46.0"},"reference-count":19,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2020,11,17]],"date-time":"2020-11-17T00:00:00Z","timestamp":1605571200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,11,17]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    This paper presents an efficient algorithm for computing 11\n                    <jats:sup>th<\/jats:sup>\n                    -power residue symbols in the cyclo-tomic field\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jmc-2020-0077_eq_002.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:mrow>\n                            <m:mi>\u211a<\/m:mi>\n                            <m:mrow>\n                              <m:mo>(<\/m:mo>\n                              <m:mrow>\n                                <m:msub>\n                                  <m:mi>\u03b6<\/m:mi>\n                                  <m:mrow>\n                                    <m:mn>11<\/m:mn>\n                                  <\/m:mrow>\n                                <\/m:msub>\n                              <\/m:mrow>\n                              <m:mo>)<\/m:mo>\n                            <\/m:mrow>\n                            <m:mo>,<\/m:mo>\n                          <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>$ \\mathbb{Q}\\left( {{\\zeta }_{11}} \\right), $<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    where\n                    <jats:sub>11<\/jats:sub>\n                    is a primitive 11\n                    <jats:sup>th<\/jats:sup>\n                    root of unity. It extends an earlier algorithm due to Caranay and Scheidler (Int. J. Number Theory, 2010) for the 7\n                    <jats:sup>th<\/jats:sup>\n                    -power residue symbol. The new algorithm finds applications in the implementation of certain cryptographic schemes.\n                  <\/jats:p>","DOI":"10.1515\/jmc-2020-0077","type":"journal-article","created":{"date-parts":[[2020,11,30]],"date-time":"2020-11-30T15:54:51Z","timestamp":1606751691000},"page":"111-122","source":"Crossref","is-referenced-by-count":6,"title":["The Eleventh Power Residue Symbol"],"prefix":"10.1515","volume":"15","author":[{"given":"Marc","family":"Joye","sequence":"first","affiliation":[{"name":"Zama , Paris , France"}]},{"given":"Oleksandra","family":"Lapiha","sequence":"additional","affiliation":[{"name":"\u00c9cole normale sup\u00e9rieure , Paris , France"}]},{"given":"Ky","family":"Nguyen","sequence":"additional","affiliation":[{"name":"\u00c9cole normale sup\u00e9rieure , Paris , France"}]},{"given":"David","family":"Naccache","sequence":"additional","affiliation":[{"name":"\u00c9cole normale sup\u00e9rieure , Paris , France"}]}],"member":"374","published-online":{"date-parts":[[2020,11,17]]},"reference":[{"key":"2025120600293913449_j_jmc-2020-0077_ref_001","doi-asserted-by":"crossref","unstructured":"William D. Banks, Daniel Lieman and Igor E. Shparlinski, An extremely small and efficient identification scheme, in: Information Security and Privacy (ACISP 2000) (E. Dawson et al., eds.), Lecture Notes in Computer Science 1841, pp. 378\u2013384, Springer, 2000.","DOI":"10.1007\/10718964_31"},{"key":"2025120600293913449_j_jmc-2020-0077_ref_002","unstructured":"\u00c9ric Brier, Houda Ferradi, Marc Joye and David Naccache, New number-theoretic cryptographic primitives, Journal of Mathematical Cryptology (To appear)."},{"key":"2025120600293913449_j_jmc-2020-0077_ref_003","doi-asserted-by":"crossref","unstructured":"Perlas C. Caranay and Renate Scheidler, An efficient seventh power residue symbol algorithm, International Journal of Number Theory 6 (2010), 1831\u20131853.","DOI":"10.1142\/S1793042110003770"},{"key":"2025120600293913449_j_jmc-2020-0077_ref_004","doi-asserted-by":"crossref","unstructured":"Ivan Bjerre Damg\u00e5rd and Gudmund Skovbjerg Frandsen, Eflcient algorithms for the gcd and cubic residuosity in the ring of Eisenstein integers, Journal of Symbolic Computation 39 (2005), 643\u2013652.","DOI":"10.1016\/j.jsc.2004.02.006"},{"key":"2025120600293913449_j_jmc-2020-0077_ref_005","doi-asserted-by":"crossref","unstructured":"Koen de Boer and Carlo Pagano, Calculating the power residue symbol and ibeta: Applications of computing the group structure of the principal units of a p-adic number field completion, in: 42nd International Symposium on Symbolic and Algebraic Computation (M. A. Burr et al., eds.), pp. 117\u2013124, ACM, 2017.","DOI":"10.1145\/3087604.3087637"},{"key":"2025120600293913449_j_jmc-2020-0077_ref_006","doi-asserted-by":"crossref","unstructured":"Shafi Goldwasser and Silvio Micali, Probabilistic encryption, Journal of Computer and System Sciences 28 (1984), 270\u2013299.","DOI":"10.1016\/0022-0000(84)90070-9"},{"key":"2025120600293913449_j_jmc-2020-0077_ref_007","unstructured":"Ernst E. Kummer, Allgemeine Reziprozit\u00e4tsgesetze f\u00fcr beliebig hohe Potenzreste, Monatsberichte der K\u00f6niglichen Preu\u00dfischen Akademie der Wissenschaften zu Berlin (1850), 154\u2013165, Reprinted in [17, pages 345\u2013357]."},{"key":"2025120600293913449_j_jmc-2020-0077_ref_008","doi-asserted-by":"crossref","unstructured":"Ernst E. Kummer, \u00dcber die Erg\u00e4nzungss\u00e4tze zu den allgemeinen Reciprocit\u00e4tsgesetzen, Journal f\u00fcr die reine und angewandte Mathematik 56 (1859), 270\u2013279, Reprinted in [17, pages 688\u2013697].","DOI":"10.1515\/crll.1859.56.270"},{"key":"2025120600293913449_j_jmc-2020-0077_ref_009","unstructured":"Franz Lemmermeyer, The Euclidean algorithm in algebraic number fields, Expositiones Mathematic\u00e6 13 (1995), 385\u2013416, Updated version, February 14, 2004."},{"key":"2025120600293913449_j_jmc-2020-0077_ref_010","doi-asserted-by":"crossref","unstructured":"Franz Lemmermeyer, Reciprocity Laws: From Euler to Eisenstein Springer Monographs in Mathematics, Springer, 2000.","DOI":"10.1007\/978-3-662-12893-0"},{"key":"2025120600293913449_j_jmc-2020-0077_ref_011","doi-asserted-by":"crossref","unstructured":"Hendrik W. Lenstra, Jr., Euclid\u2019s algorithm in cyclotomic fields, Journal of the London Mathematical Society (2) 10 (1975), 457\u2013465.","DOI":"10.1112\/jlms\/s2-10.4.457"},{"key":"2025120600293913449_j_jmc-2020-0077_ref_012","doi-asserted-by":"crossref","unstructured":"Jean Monnerat and Serge Vaudenay, Short undeniable signatures based on group homomorphisms, Journal of Cryptology 24 (2011), 545\u2013587.","DOI":"10.1007\/s00145-010-9070-1"},{"key":"2025120600293913449_j_jmc-2020-0077_ref_013","doi-asserted-by":"crossref","unstructured":"Renate Scheidler, A public-key cryptosystem using purely cubic fields, Journal of Cryptology 11 (1998), 109\u2013124.","DOI":"10.1007\/s001459900038"},{"key":"2025120600293913449_j_jmc-2020-0077_ref_014","doi-asserted-by":"crossref","unstructured":"Renate Scheidler and Hugh C. Williams, A public-key cryptosystem utilizing cyclotomic fields, Designs, Codes and Cryptography 6 (1995), 117\u2013131.","DOI":"10.1007\/BF01398010"},{"key":"2025120600293913449_j_jmc-2020-0077_ref_015","unstructured":"Henry J. S. Smith, Report on the Theory of Numbers (Part II) Collected Mathematical Papers (J. W. L. Glaisher, ed.), 1, The Clarendon Press, 1894, pp. 93\u2013162."},{"key":"2025120600293913449_j_jmc-2020-0077_ref_016","unstructured":"The Sage developers, SageMath, the Sage Mathematics Software System (Version 8.6) 2019."},{"key":"2025120600293913449_j_jmc-2020-0077_ref_017","unstructured":"Andr\u00e9 Weil (ed.), Collected Papers I: Contributions to Number Theory Springer-Verlag, 1975."},{"key":"2025120600293913449_j_jmc-2020-0077_ref_018","doi-asserted-by":"crossref","unstructured":"Andr\u00e9 Weilert, Fast computation of the biquadratic residue symbol, Journal of Number Theory 96 (2002), 133\u2013151.","DOI":"10.1016\/S0022-314X(02)92783-6"},{"key":"2025120600293913449_j_jmc-2020-0077_ref_019","doi-asserted-by":"crossref","unstructured":"Hugh C. Williams, An M3 public-key encryption scheme, in: Advances in Cryptology \u2013 CRYPTO\u201985 (H. C. Williams, ed.), Lecture Notes in Computer Science 218, pp. 358\u2013368, Springer, 1986.","DOI":"10.1007\/3-540-39799-X_26"}],"container-title":["Journal of Mathematical Cryptology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/jmc\/15\/1\/article-p111.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jmc-2020-0077\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jmc-2020-0077\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T00:30:53Z","timestamp":1764981053000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jmc-2020-0077\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,17]]},"references-count":19,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,11,17]]},"published-print":{"date-parts":[[2020,11,17]]}},"alternative-id":["10.1515\/jmc-2020-0077"],"URL":"https:\/\/doi.org\/10.1515\/jmc-2020-0077","relation":{},"ISSN":["1862-2984"],"issn-type":[{"type":"electronic","value":"1862-2984"}],"subject":[],"published":{"date-parts":[[2020,11,17]]}}}