{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T07:50:46Z","timestamp":1740124246327,"version":"3.37.3"},"reference-count":12,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2019,7,10]],"date-time":"2019-07-10T00:00:00Z","timestamp":1562716800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,7,10]],"date-time":"2019-07-10T00:00:00Z","timestamp":1562716800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Period Math Hung"],"published-print":{"date-parts":[[2020,3]]},"DOI":"10.1007\/s10998-019-00296-z","type":"journal-article","created":{"date-parts":[[2019,7,10]],"date-time":"2019-07-10T06:49:23Z","timestamp":1562741363000},"page":"81-94","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Some notes on the multiplicative order of $$\\alpha + \\alpha ^{-1}$$ in finite fields of characteristic two"],"prefix":"10.1007","volume":"80","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8092-1883","authenticated-orcid":false,"given":"Simone","family":"Ugolini","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,7,10]]},"reference":[{"key":"296_CR1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2226-0","volume-title":"Applications of Finite Fields","author":"IF Blake","year":"1993","unstructured":"I.F. Blake, X. Gao, A.J. Menezes, R.C. Mullin, S.A. Vanstone, T. Yaghoobian, Applications of Finite Fields (Kluwer Academic Publishers, Boston, 1993)"},{"key":"296_CR2","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1016\/j.jnt.2018.01.003","volume":"188","author":"A Blokhuis","year":"2018","unstructured":"A. Blokhuis, X. Cao, W.-S. Chou, X.-D. Hou, On the roots of certain Dickson polynomials. J. Number Theory 188, 229\u2013246 (2018)","journal-title":"J. Number Theory"},{"key":"296_CR3","volume-title":"Basic Algebra I","author":"N Jacobson","year":"1985","unstructured":"N. Jacobson, Basic Algebra I, 2nd edn. (W. H. Freeman and Company, New York, 1985)","edition":"2"},{"issue":"3","key":"296_CR4","doi-asserted-by":"publisher","first-page":"686","DOI":"10.1109\/18.54892","volume":"36","author":"G Lachaud","year":"1990","unstructured":"G. Lachaud, J. Wolfmann, The weights of the orthogonals of the extended quadratic binary Goppa codes. IEEE Trans. Inf. Theory 36(3), 686\u2013692 (1990)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"296_CR5","volume-title":"Dickson Polynomials","author":"R Lidl","year":"1993","unstructured":"R. Lidl, G.L. Mullen, G. Turnwald, Dickson Polynomials (Longman Scientific & Technical, Harlow, 1993)"},{"key":"296_CR6","volume-title":"Finite Fields","author":"R Lidl","year":"1997","unstructured":"R. Lidl, H. Niederreiter, Finite Fields (Cambridge Univeristy Press, Cambridge, 1997)"},{"key":"296_CR7","unstructured":"F. L\u00fcbeck, in Conway Polynomials for Finite Fields, \nhttp:\/\/www.math.rwth-aachen.de\/~Frank.Luebeck\/data\/ConwayPol\/index.html\n\n. Accessed 12 Sept 2018"},{"key":"296_CR8","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1007\/BF01810846","volume":"1","author":"H Meyn","year":"1990","unstructured":"H. Meyn, On the construction of irreducible self-reciprocal polynomials over finite fields. AAECC 1, 43\u201353 (1990)","journal-title":"AAECC"},{"key":"296_CR9","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1006\/ffta.2000.0292","volume":"7","author":"I Shparlinski","year":"2001","unstructured":"I. Shparlinski, On the multiplicative orders of $$\\gamma $$ and $$\\gamma +\\gamma ^{-1}$$ over finite fields. Finite Fields Appl. 7, 327\u2013331 (2001)","journal-title":"Finite Fields Appl."},{"key":"296_CR10","series-title":"Contemporary Mathematics","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1090\/conm\/579\/11530","volume-title":"Theory and Applications of Finite Fields","author":"S Ugolini","year":"2012","unstructured":"S. Ugolini, Graphs associated with the map $$x \\mapsto x + x^{-1}$$ in finite fields of characteristic two, Theory and Applications of Finite Fields, vol. 579, Contemporary Mathematics (American Mathematical Society, Providence, RI, 2012), pp. 187\u2013204"},{"key":"296_CR11","first-page":"403","volume":"13","author":"RR Varshamov","year":"1969","unstructured":"R.R. Varshamov, G.A. Garakov, On the theory of selfdual polynomials over a Galois field (Russian). Bull. Math. Sci. Math. R. S. Roumanie (N. S.) 13, 403\u2013415 (1969)","journal-title":"Bull. Math. Sci. Math. R. S. Roumanie (N. S.)"},{"key":"296_CR12","series-title":"Finite Fields and Applications (Augsburg, 1999)","first-page":"162","volume-title":"Gauss Periods in Finite Fields","author":"J von zur Gathen","year":"2001","unstructured":"J. von zur Gathen, I. Shparlinski, Gauss Periods in Finite Fields, Finite Fields and Applications (Augsburg, 1999) (Springer, Berlin, 2001), pp. 162\u2013177"}],"container-title":["Periodica Mathematica Hungarica"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10998-019-00296-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10998-019-00296-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10998-019-00296-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,7,8]],"date-time":"2020-07-08T23:16:41Z","timestamp":1594250201000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10998-019-00296-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,10]]},"references-count":12,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,3]]}},"alternative-id":["296"],"URL":"https:\/\/doi.org\/10.1007\/s10998-019-00296-z","relation":{},"ISSN":["0031-5303","1588-2829"],"issn-type":[{"type":"print","value":"0031-5303"},{"type":"electronic","value":"1588-2829"}],"subject":[],"published":{"date-parts":[[2019,7,10]]},"assertion":[{"value":"10 July 2019","order":1,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}