{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T03:09:40Z","timestamp":1787022580183,"version":"3.56.0"},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2014,10,1]],"date-time":"2014-10-01T00:00:00Z","timestamp":1412121600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2015,3,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    In this paper we present a new method of choosing primitive elements for Brezing\u2013Weng families of pairing-friendly elliptic curves with small rho-values, and we improve on previously known best rho-values of families [J. Cryptology 23 (2010), 224\u2013280], [Lecture Notes in Comput. Sci. 5209, Springer (2008), 126\u2013135]\nfor the embedding degrees\n                    <jats:inline-formula id=\"eq1_w2aab3b7b5b1b6b1aab1c13b1b1Aa\">\n                      <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:mrow>\n                            <m:mi>k<\/m:mi>\n                            <m:mo>=<\/m:mo>\n                            <m:mn>16<\/m:mn>\n                          <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>$k=16$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , 22, 28, 40 and 46. We consider elements of the form\n                    <jats:inline-formula id=\"eq2_w2aab3b7b5b1b6b1aab1c13b1b3Aa\">\n                      <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:mrow>\n                            <m:mrow>\n                              <m:mo>(<\/m:mo>\n                              <m:mi>a<\/m:mi>\n                              <m:mo>+<\/m:mo>\n                              <m:mi>b<\/m:mi>\n                              <m:msqrt>\n                                <m:mrow>\n                                  <m:mo>-<\/m:mo>\n                                  <m:mi>D<\/m:mi>\n                                <\/m:mrow>\n                              <\/m:msqrt>\n                              <m:mo>)<\/m:mo>\n                            <\/m:mrow>\n                            <m:msub>\n                              <m:mi>\u03b6<\/m:mi>\n                              <m:mi>k<\/m:mi>\n                            <\/m:msub>\n                          <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>$(a+b\\sqrt{-D})\\zeta _k$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    for rational numbers\n                    <jats:italic>a<\/jats:italic>\n                    ,\n                    <jats:italic>b<\/jats:italic>\n                    and a square-free positive integer\n                    <jats:italic>D<\/jats:italic>\n                    , where\n                    <jats:inline-formula id=\"eq3_w2aab3b7b5b1b6b1aab1c13b1c11Aa\">\n                      <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:msub>\n                            <m:mi>\u03b6<\/m:mi>\n                            <m:mi>k<\/m:mi>\n                          <\/m:msub>\n                        <\/m:math>\n                        <jats:tex-math>$\\zeta _k$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is a primitive\n                    <jats:italic>k<\/jats:italic>\n                    -th root of unity. We also investigate the conditions for an element of the proposed form to be suitable for our construction. Our construction uses fixed discriminants.\n                  <\/jats:p>","DOI":"10.1515\/jmc-2013-0017","type":"journal-article","created":{"date-parts":[[2014,10,1]],"date-time":"2014-10-01T13:02:33Z","timestamp":1412168553000},"page":"1-9","source":"Crossref","is-referenced-by-count":5,"title":["A new method of choosing primitive elements for Brezing\u2013Weng families of pairing-friendly elliptic curves"],"prefix":"10.1515","volume":"9","author":[{"given":"Kisoon","family":"Yoon","sequence":"first","affiliation":[{"name":"NSHC Inc. 55, Gwangjinmal-gil, Uiwang-si, Gyeonggi-do, 437-060, Republic of Korea"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2014,10,1]]},"container-title":["Journal of Mathematical Cryptology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jmc-2013-0017\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jmc-2013-0017\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T00:16:50Z","timestamp":1764980210000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jmc-2013-0017\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,10,1]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2014,7,26]]},"published-print":{"date-parts":[[2015,3,1]]}},"alternative-id":["10.1515\/jmc-2013-0017"],"URL":"https:\/\/doi.org\/10.1515\/jmc-2013-0017","relation":{},"ISSN":["1862-2984","1862-2976"],"issn-type":[{"value":"1862-2984","type":"electronic"},{"value":"1862-2976","type":"print"}],"subject":[],"published":{"date-parts":[[2014,10,1]]}}}