{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,20]],"date-time":"2025-12-20T09:44:30Z","timestamp":1766223870487,"version":"3.48.0"},"reference-count":28,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"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":[[2025,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Let\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jmc-2024-0017_eq_001.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:mi>p<\/m:mi>\n                          <m:mo>&gt;<\/m:mo>\n                          <m:mn>1<\/m:mn>\n                        <\/m:math>\n                        <jats:tex-math>p\\gt 1<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    be a large prime number, and let\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jmc-2024-0017_eq_002.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:mi>\u03b5<\/m:mi>\n                          <m:mo>&gt;<\/m:mo>\n                          <m:mn>0<\/m:mn>\n                        <\/m:math>\n                        <jats:tex-math>\\varepsilon \\gt 0<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    be a small number. The established unconditional upper bounds of the least primitive root\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jmc-2024-0017_eq_003.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:mi>u<\/m:mi>\n                          <m:mo>\u2260<\/m:mo>\n                          <m:mo>\u00b1<\/m:mo>\n                          <m:mn>1<\/m:mn>\n                          <m:mo>,<\/m:mo>\n                          <m:msup>\n                            <m:mrow>\n                              <m:mi>v<\/m:mi>\n                            <\/m:mrow>\n                            <m:mrow>\n                              <m:mn>2<\/m:mn>\n                            <\/m:mrow>\n                          <\/m:msup>\n                        <\/m:math>\n                        <jats:tex-math>u\\ne \\pm 1,{v}^{2}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    in the prime finite 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-2024-0017_eq_004.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:msub>\n                            <m:mrow>\n                              <m:mi mathvariant=\"double-struck\">F<\/m:mi>\n                            <\/m:mrow>\n                            <m:mrow>\n                              <m:mi>p<\/m:mi>\n                            <\/m:mrow>\n                          <\/m:msub>\n                        <\/m:math>\n                        <jats:tex-math>{{\\mathbb{F}}}_{p}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    have exponential magnitudes\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jmc-2024-0017_eq_005.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:mi>u<\/m:mi>\n                          <m:mo>\u226a<\/m:mo>\n                          <m:msup>\n                            <m:mrow>\n                              <m:mi>p<\/m:mi>\n                            <\/m:mrow>\n                            <m:mrow>\n                              <m:mn>1<\/m:mn>\n                              <m:mo>\u2044<\/m:mo>\n                              <m:mn>4<\/m:mn>\n                              <m:mo>+<\/m:mo>\n                              <m:mi>\u03b5<\/m:mi>\n                            <\/m:mrow>\n                          <\/m:msup>\n                        <\/m:math>\n                        <jats:tex-math>u\\ll {p}^{1\/4+\\varepsilon }<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . This note contributes a new result to the literature. It proves that the upper bound of the least primitive root has polynomial magnitude\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jmc-2024-0017_eq_006.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:mi>u<\/m:mi>\n                          <m:mo>\u2264<\/m:mo>\n                          <m:msup>\n                            <m:mrow>\n                              <m:mrow>\n                                <m:mo>(<\/m:mo>\n                                <m:mrow>\n                                  <m:mi>log<\/m:mi>\n                                  <m:mi>p<\/m:mi>\n                                <\/m:mrow>\n                                <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                            <\/m:mrow>\n                            <m:mrow>\n                              <m:mn>1<\/m:mn>\n                              <m:mo>+<\/m:mo>\n                              <m:mi>\u03b5<\/m:mi>\n                            <\/m:mrow>\n                          <\/m:msup>\n                        <\/m:math>\n                        <jats:tex-math>u\\le {\\left(\\log p)}^{1+\\varepsilon }<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    unconditionally.\n                  <\/jats:p>","DOI":"10.1515\/jmc-2024-0017","type":"journal-article","created":{"date-parts":[[2025,4,1]],"date-time":"2025-04-01T09:23:45Z","timestamp":1743499425000},"source":"Crossref","is-referenced-by-count":0,"title":["The least primitive roots mod\n                    <i>p<\/i>"],"prefix":"10.1515","volume":"19","author":[{"given":"Nelson","family":"Carella","sequence":"first","affiliation":[{"name":"Department of Mathematics, Fordham University and CUNY , Bronx , NY 10458, New York , United States of America"}]}],"member":"374","published-online":{"date-parts":[[2025,4,1]]},"reference":[{"key":"2025122009205590775_j_jmc-2024-0017_ref_001","doi-asserted-by":"crossref","unstructured":"Burgess DA. On character sums and primitive roots. Proc London Math Soc. 1962;12(3):179\u201392.","DOI":"10.1112\/plms\/s3-12.1.179"},{"key":"2025122009205590775_j_jmc-2024-0017_ref_002","doi-asserted-by":"crossref","unstructured":"Shoup V. Searching for primitive roots in finite fields. Math Comp. 1992;58(197):369\u201380.","DOI":"10.1090\/S0025-5718-1992-1106981-9"},{"key":"2025122009205590775_j_jmc-2024-0017_ref_003","doi-asserted-by":"crossref","unstructured":"Elliott PDTA. The distribution of primitive roots. Canadian J Math. 1969;21:822\u201341.","DOI":"10.4153\/CJM-1969-092-6"},{"key":"2025122009205590775_j_jmc-2024-0017_ref_004","doi-asserted-by":"crossref","unstructured":"Bach E. Comments on search procedures for primitive roots. Math Comp. 1997;66(220):1719\u201327.","DOI":"10.1090\/S0025-5718-97-00890-9"},{"key":"2025122009205590775_j_jmc-2024-0017_ref_005","unstructured":"McGown K, Sorenson J. Computation of the least primitive root. 2022. Arxiv.org\/abs\/2206.14193."},{"key":"2025122009205590775_j_jmc-2024-0017_ref_006","doi-asserted-by":"crossref","unstructured":"Shparlinski IE. On finding primitive roots in finite fields. Theoret Comput Sci. 1996;157:273\u20135.","DOI":"10.1016\/0304-3975(95)00164-6"},{"key":"2025122009205590775_j_jmc-2024-0017_ref_007","unstructured":"Grossman O. Finding primitive roots pseudo-deterministically. Preprint. 2015. eccc.weizmann.ac.il\/report\/2015\/207."},{"key":"2025122009205590775_j_jmc-2024-0017_ref_008","doi-asserted-by":"crossref","unstructured":"Shparlinski IE. On constructing primitive roots in finite fields with advice. IEEE Trans Inform Theory. 2018;64(11):7132\u20136.","DOI":"10.1109\/TIT.2018.2810938"},{"key":"2025122009205590775_j_jmc-2024-0017_ref_009","doi-asserted-by":"crossref","unstructured":"Miller GL. Riemannas hypothesis and tests for primality. J Comput System Sci. 1976;13(3):300\u201317.","DOI":"10.1016\/S0022-0000(76)80043-8"},{"key":"2025122009205590775_j_jmc-2024-0017_ref_010","doi-asserted-by":"crossref","unstructured":"Bach E. Explicit bounds for primality testing and related problems. Math Comp. 1990;55(191):355\u201380.","DOI":"10.1090\/S0025-5718-1990-1023756-8"},{"key":"2025122009205590775_j_jmc-2024-0017_ref_011","doi-asserted-by":"crossref","unstructured":"Bach E, Huelsbergen L. Statistical evidence for small generating sets. Math Comp. 1993;61:69\u201382.","DOI":"10.1090\/S0025-5718-1993-1195432-5"},{"key":"2025122009205590775_j_jmc-2024-0017_ref_012","doi-asserted-by":"crossref","unstructured":"Agrawal M, Kayal N, Saxena N. PRIMES is in P. Ann Math. 2004;160(2):781\u201393.","DOI":"10.4007\/annals.2004.160.781"},{"key":"2025122009205590775_j_jmc-2024-0017_ref_013","doi-asserted-by":"crossref","unstructured":"Lenstra H, Pomerance C. Primality testing with Gaussian periods. J Eur Math Soc. 2019;21(4):1229\u201369.","DOI":"10.4171\/jems\/861"},{"key":"2025122009205590775_j_jmc-2024-0017_ref_014","doi-asserted-by":"crossref","unstructured":"Brier E, Ferradi H, Joye M, Naccache D. New number-theoretic cryptographic primitives. J Math Cryptol. 2020;14(1):224\u201335.","DOI":"10.1515\/jmc-2019-0035"},{"key":"2025122009205590775_j_jmc-2024-0017_ref_015","doi-asserted-by":"crossref","unstructured":"Erdos P, Shapiro H. On the least primitive root of a prime number. Pacific J Math. 1957:7(1):861\u20135.","DOI":"10.2140\/pjm.1957.7.861"},{"key":"2025122009205590775_j_jmc-2024-0017_ref_016","doi-asserted-by":"crossref","unstructured":"Winterhof A. Character sums, primitive elements, and powers in finite fields. J Number Theory. 2001;91(1):153\u201363.","DOI":"10.1006\/jnth.2001.2675"},{"key":"2025122009205590775_j_jmc-2024-0017_ref_017","doi-asserted-by":"crossref","unstructured":"McGown K, Trudgian T. Explicit upper bounds on the least primitive root. Proc Amer Math Soc. 2020;148(3):1049\u201361.","DOI":"10.1090\/proc\/14800"},{"key":"2025122009205590775_j_jmc-2024-0017_ref_018","unstructured":"Landau E. Vorlesungen uuuuber Zahlentheorie: Vol.: 2. Aus der analytischen und geometrischen Zahlentheorie. New York: Chelsea Publishing Co.; 1969, [1927]."},{"key":"2025122009205590775_j_jmc-2024-0017_ref_019","unstructured":"Lidl R, Niederreiter H. Finite fields. Second edition. Encyclopedia of Mathematics and its Applications. vol. 20. Cambridge: Cambridge University Press; 1997."},{"key":"2025122009205590775_j_jmc-2024-0017_ref_020","unstructured":"Vinogradov IM. On the least primitive root. Doklady Akad Nauk SSSR. 1930;1:7\u201311."},{"key":"2025122009205590775_j_jmc-2024-0017_ref_021","doi-asserted-by":"crossref","unstructured":"Erdos P. On the least primitive root of a prime number. Bull Amer Math Soc. 1945;51(11):131\u20132.","DOI":"10.1090\/S0002-9904-1945-08291-3"},{"key":"2025122009205590775_j_jmc-2024-0017_ref_022","doi-asserted-by":"crossref","unstructured":"Martin G. Uniform bounds for the least almost-prime primitive root. Mathematika. 1998;45:19\u2013207.","DOI":"10.1112\/S0025579300014121"},{"key":"2025122009205590775_j_jmc-2024-0017_ref_023","unstructured":"Carella N. Upper bound of the least quadratic nonresidues. 2021. Arxiv.org\/abs\/2106.00544."},{"key":"2025122009205590775_j_jmc-2024-0017_ref_024","doi-asserted-by":"crossref","unstructured":"Apostol Tom M. Introduction to analytic number theory. Undergraduate Texts in Mathematics. New York-Heidelberg: Springer-Verlag; 1976.","DOI":"10.1007\/978-1-4757-5579-4"},{"key":"2025122009205590775_j_jmc-2024-0017_ref_025","doi-asserted-by":"crossref","unstructured":"Montgomery HL, Vaughan RC. Multiplicative number theory. I. Classical theory. Cambridge: Cambridge University Press; 2007.","DOI":"10.1017\/CBO9780511618314"},{"key":"2025122009205590775_j_jmc-2024-0017_ref_026","doi-asserted-by":"crossref","unstructured":"Dusart P. Estimates of some functions over primes, without R.H. Math Comp. 2016;85(298):875\u201388.","DOI":"10.1090\/mcom\/3005"},{"key":"2025122009205590775_j_jmc-2024-0017_ref_027","doi-asserted-by":"crossref","unstructured":"Erdos P, Pomerance C. The normal number of prime factors of \u03c6(n). Rocky Mtn J Math. 1985;15:343\u201352.","DOI":"10.1216\/RMJ-1985-15-2-343"},{"key":"2025122009205590775_j_jmc-2024-0017_ref_028","unstructured":"Davenport H. Multiplicative number theory. Graduate Texts in Mathematics. Vol. 74. New York: Springer-Verlag; 2000."}],"container-title":["Journal of Mathematical Cryptology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jmc-2024-0017\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jmc-2024-0017\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,20]],"date-time":"2025-12-20T09:40:07Z","timestamp":1766223607000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/jmc-2024-0017\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,1]]},"references-count":28,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,4,14]]},"published-print":{"date-parts":[[2025,4,14]]}},"alternative-id":["10.1515\/jmc-2024-0017"],"URL":"https:\/\/doi.org\/10.1515\/jmc-2024-0017","relation":{},"ISSN":["1862-2984"],"issn-type":[{"type":"electronic","value":"1862-2984"}],"subject":[],"published":{"date-parts":[[2025,1,1]]},"article-number":"20240017"}}