{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:53:28Z","timestamp":1776848008514,"version":"3.51.2"},"reference-count":10,"publisher":"American Mathematical Society (AMS)","issue":"225","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Gauss periods have been used successfully as a tool for constructing normal bases in finite fields. Starting from a primitive\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r\">\n                        <mml:semantics>\n                          <mml:mi>r<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">r<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    th root of unity, one obtains under certain conditions a normal basis for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper F Subscript q Sub Superscript n\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">F<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:msup>\n                                <mml:mi>q<\/mml:mi>\n                                <mml:mi>n<\/mml:mi>\n                              <\/mml:msup>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {F}_{q^n}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    over\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper F Subscript q\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">F<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mi>q<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {F}_q<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r\">\n                        <mml:semantics>\n                          <mml:mi>r<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">r<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a prime and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n k equals r minus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>r<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">nk=r-1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for some integer\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We generalize this construction by allowing arbitrary integers\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"r\">\n                        <mml:semantics>\n                          <mml:mi>r<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">r<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n k equals phi left-parenthesis r right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>\n                              \u03c6\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>r<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">nk=\\varphi (r)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and find in many cases smaller values of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    than is possible with the previously known approach.\n                  <\/p>","DOI":"10.1090\/s0025-5718-99-00988-6","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"271-290","source":"Crossref","is-referenced-by-count":34,"title":["Normal bases via general Gauss periods"],"prefix":"10.1090","volume":"68","author":[{"given":"Sandra","family":"Feisel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Joachim","family":"von zur Gathen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M.","family":"Shokrollahi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1999]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1016\/0166-218X(89)90001-2","article-title":"Low complexity normal bases","volume":"25","author":"Ash, David W.","year":"1989","journal-title":"Discrete Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0166-218X","issn-type":"print"},{"key":"2","unstructured":"Shuhong Gao, Gauss periods, groups, and normal bases, preprint, 1997."},{"key":"3","doi-asserted-by":"crossref","unstructured":"Shuhong Gao, Joachim von zur Gathen, and Daniel Panario, Gauss periods and fast exponentiation in finite fields, Proc. Latin \u201995, Valparaiso, Chile, Springer Lecture Notes in Computer Science 911, 1995, pp. 311\u2013322.","DOI":"10.1007\/3-540-59175-3_98"},{"key":"4","doi-asserted-by":"crossref","unstructured":"S. Gao, J. von zur Gathen, and D. Panario, Gauss periods: orders and cryptographical applications, Math. Comp. 67 (1998), 343\u2013352.","DOI":"10.1090\/S0025-5718-98-00935-1"},{"issue":"4","key":"5","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1007\/BF00125200","article-title":"Optimal normal bases","volume":"2","author":"Gao, Shuhong","year":"1992","journal-title":"Des. Codes Cryptogr.","ISSN":"https:\/\/id.crossref.org\/issn\/0925-1022","issn-type":"print"},{"key":"6","volume-title":"Algebraic number theory","author":"Lang, Serge","year":"1970"},{"key":"7","unstructured":"Alfred J. Menezes, Ian F. Blake, XuHong Gao, Ronald C. Mullin, Scott A. Vanstone, and Tomik Yaghoobian, Applications of finite fields, Kluwer Academic Publishers, Norwell MA, 1993."},{"issue":"2","key":"8","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1016\/0166-218X(88)90090-X","article-title":"Optimal normal bases in \ud835\udc3a\ud835\udc39(\ud835\udc5d\u207f)","volume":"22","author":"Mullin, R. C.","year":"1988","journal-title":"Discrete Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0166-218X","issn-type":"print"},{"key":"9","doi-asserted-by":"crossref","unstructured":"Emmy Noether, Normalbasis bei K\u00f6rpern ohne h\u00f6here Verzweigung, Journal f\u00fcr die reine und angewandte Mathematik 167 (1932), 147\u2013152.","DOI":"10.1515\/crll.1932.167.147"},{"issue":"44","key":"10","first-page":"147","article-title":"Zur Arithmetik in endlichen K\u00f6rpern","author":"Wassermann, Alfred","year":"1993","journal-title":"Bayreuth. Math. 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