{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T14:00:21Z","timestamp":1776866421270,"version":"3.51.2"},"reference-count":30,"publisher":"American Mathematical Society (AMS)","issue":"303","license":[{"start":{"date-parts":[[2017,4,13]],"date-time":"2017-04-13T00:00:00Z","timestamp":1492041600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["Catrel project"],"award-info":[{"award-number":["Catrel project"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this work, we consider the proportion of friable (free of large prime factors) values of a binary form\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper F left-parenthesis upper X 1 comma upper X 2 right-parenthesis element-of bold upper Z left-bracket upper X 1 comma upper X 2 right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>F<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>X<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>X<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"bold\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>X<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>X<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">F(X_1,X_2)\\in \\mathbf {Z}[X_1,X_2]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In the particular case of quadratic forms, we give an asymptotic equivalent for this proportion which depends on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper F\">\n                        <mml:semantics>\n                          <mml:mi>F<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">F<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . This is related to Murphy\u2019s\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b1\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    function, which is known in the cryptologic community, but which has not been studied before from a mathematical point of view. This has consequences on the first step, called polynomial selection, of the Number Field Sieve, the fastest algorithm of integer factorization.\n                  <\/p>","DOI":"10.1090\/mcom\/3112","type":"journal-article","created":{"date-parts":[[2015,10,1]],"date-time":"2015-10-01T12:48:44Z","timestamp":1443703724000},"page":"397-418","source":"Crossref","is-referenced-by-count":3,"title":["Some mathematical remarks on the polynomial selection in NFS"],"prefix":"10.1090","volume":"86","author":[{"given":"Razvan","family":"Barbulescu","sequence":"first","affiliation":[]},{"given":"Armand","family":"Lachand","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2016,4,13]]},"reference":[{"key":"1","unstructured":"[Bai11] S. Bai, Polynomial selection for the number field sieve, Ph.D. thesis, Australian National University, 2011."},{"issue":"296","key":"2","doi-asserted-by":"publisher","first-page":"2987","DOI":"10.1090\/S0025-5718-2015-02940-8","article-title":"Selecting polynomials for the function field sieve","volume":"84","author":"Barbulescu, Razvan","year":"2015","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"3","doi-asserted-by":"publisher","first-page":"639","DOI":"10.4171\/CMH\/264","article-title":"Friable values of binary forms","volume":"87","author":"Balog, Antal","year":"2012","journal-title":"Comment. Math. Helv.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-2571","issn-type":"print"},{"key":"4","unstructured":"[BGI14] C. Bouvier, P. Gaudry, L. Imbert, H. Jeljeli, and E. Thomas, Announcement to the number theory mailing list: Discrete logarithms in GF(p) \u2014 180 digits, 2014. Announcement available at the NMBRTHRY archives, item 004703."},{"key":"5","unstructured":"[Boe96] H. Boender, The number of relations in the quadratic sieve algorithm, Tech. report, Department of Numerical Mathematics, CWI, Amsterdam, 1996."},{"key":"6","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4542-1","volume-title":"Binary quadratic forms","author":"Buell, Duncan A.","year":"1989","ISBN":"https:\/\/id.crossref.org\/isbn\/0387970371"},{"key":"7","unstructured":"[Ded78] R. Dedekind, \u00dcber den Zusammenhang zwischen der Theorie der Ideale und der h\u00f6heren Kongruenzen, Abh. Kgl. Ges. Wiss. G\u00f6ttingen 23 (1878), 1\u201323."},{"issue":"3","key":"8","doi-asserted-by":"publisher","first-page":"449","DOI":"10.1112\/plms\/s3-63.3.449","article-title":"Entiers sans grand facteur premier en progressions arithmetiques","volume":"63","author":"Fouvry, \u00c9.","year":"1991","journal-title":"Proc. London Math. Soc. (3)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6115","issn-type":"print"},{"key":"9","isbn-type":"print","first-page":"267","article-title":"Smooth numbers: computational number theory and beyond","author":"Granville, Andrew","year":"2008","ISBN":"https:\/\/id.crossref.org\/isbn\/9780521808545"},{"issue":"3","key":"10","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1016\/0022-314X(86)90013-2","article-title":"On the number of positive integers \u2264\ud835\udc65 and free of prime factors >\ud835\udc66","volume":"22","author":"Hildebrand, Adolf","year":"1986","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"2","key":"11","doi-asserted-by":"publisher","first-page":"411","DOI":"10.5802\/jtnb.101","article-title":"Integers without large prime factors","volume":"5","author":"Hildebrand, Adolf","year":"1993","journal-title":"J. Th\\'{e}or. Nombres Bordeaux","ISSN":"https:\/\/id.crossref.org\/issn\/1246-7405","issn-type":"print"},{"issue":"1","key":"12","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1112\/plms\/pdm029","article-title":"Moyennes de certaines fonctions multiplicatives sur les entiers friables. II","volume":"96","author":"Hanrot, Guillaume","year":"2008","journal-title":"Proc. Lond. Math. Soc. (3)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6115","issn-type":"print"},{"issue":"242","key":"13","doi-asserted-by":"publisher","first-page":"953","DOI":"10.1090\/S0025-5718-02-01482-5","article-title":"Improvements to the general number field sieve for discrete logarithms in prime fields. A comparison with the Gaussian integer method","volume":"72","author":"Joux, Antoine","year":"2003","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"14","isbn-type":"print","doi-asserted-by":"publisher","first-page":"326","DOI":"10.1007\/11818175_19","article-title":"The number field sieve in the medium prime case","author":"Joux, Antoine","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540374329"},{"key":"15","isbn-type":"print","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1007\/978-3-642-14623-7_18","article-title":"Factorization of a 768-bit RSA modulus","author":"Kleinjung, Thorsten","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9783642146220"},{"issue":"256","key":"16","doi-asserted-by":"publisher","first-page":"2037","DOI":"10.1090\/S0025-5718-06-01870-9","article-title":"On polynomial selection for the general number field sieve","volume":"75","author":"Kleinjung, Thorsten","year":"2006","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"17","unstructured":"[Kle08] Thorsten Kleinjung, Polynomial selection, 2008, CADO workshop on integer factorization. Slides available online at \\url{http:\/\/cado.gforge.inria.fr\/workshop\/slides\/kleinjung.pdf}."},{"key":"18","unstructured":"[Lac14a] A. Lachand, Fonctions arithm\u00e9tiques et formes binaires irr\u00e9ductibles de degr\u00e9 3, \\url{https:\/\/hal.archives-ouvertes.fr\/hal-01053649}, 2014."},{"issue":"1","key":"19","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1093\/qmath\/hau029","article-title":"Valeurs friables d\u2019une forme quadratique et d\u2019une forme lin\u00e9aire","volume":"66","author":"Lachand, Armand","year":"2015","journal-title":"Q. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0033-5606","issn-type":"print"},{"key":"20","isbn-type":"print","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1007\/BFb0091537","article-title":"The number field sieve","author":"Lenstra, A. K.","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/3540570136"},{"key":"21","first-page":"409","article-title":"Effective versions of the Chebotarev density theorem","author":"Lagarias, J. C.","year":"1977"},{"key":"22","unstructured":"[Mur99] B. A. Murphy, Polynomial selection for the number field sieve integer factorisation algorithm, Ph.D. thesis, Australian National University, 1999."},{"key":"23","unstructured":"[Nag21] T. Nagell, G\u00e9n\u00e9ralisation d\u2019un th\u00e9or\u00e8me de Tchebycheff, J. Math. Pures Appl. (8) 4 (1921), no. 4, 343\u2013356."},{"key":"24","unstructured":"[Oes79] J. Oesterl\u00e9, Versions effectives du th\u00e9or\u00e8me de Chebotarev sous l\u2019hypoth\u00e8se de Riemann g\u00e9n\u00e9ralis\u00e9e, Ast\u00e9risque 61 (1979), 165\u2013167."},{"key":"25","isbn-type":"print","doi-asserted-by":"publisher","first-page":"4","DOI":"10.1007\/BFb0091536","article-title":"Factoring with cubic integers","author":"Pollard, J. M.","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/3540570136"},{"key":"26","first-page":"64","article-title":"Approximate formulas for some functions of prime numbers","volume":"6","author":"Rosser, J. Barkley","year":"1962","journal-title":"Illinois J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0019-2082","issn-type":"print"},{"issue":"1","key":"27","doi-asserted-by":"publisher","first-page":"78","DOI":"10.1016\/0022-314X(89)90099-1","article-title":"Sur le nombre des entiers sans grand facteur premier","volume":"32","author":"Saias, \u00c9ric","year":"1989","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"1676","key":"28","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1098\/rsta.1993.0139","article-title":"Discrete logarithms and local units","volume":"345","author":"Schirokauer, Oliver","year":"1993","journal-title":"Philos. Trans. Roy. Soc. London Ser. A","ISSN":"https:\/\/id.crossref.org\/issn\/0962-8428","issn-type":"print"},{"key":"29","isbn-type":"print","first-page":"221","article-title":"Sur un probl\u00e8me d\u2019Erd\u0151s et Alladi","author":"Tenenbaum, G\u00e9rald","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0817634932"},{"key":"30","unstructured":"[Win] B. Winckler, Th\u00e9or\u00e8me de Chebotarev effectif, preprint available at \\url{http:\/\/hal.archives-ouvertes.fr\/docs\/00\/90\/74\/10\/PDF\/chebotarev.pdf}."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2017-86-303\/S0025-5718-2016-03112-9\/S0025-5718-2016-03112-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2017-86-303\/S0025-5718-2016-03112-9\/S0025-5718-2016-03112-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:59:52Z","timestamp":1776797992000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2017-86-303\/S0025-5718-2016-03112-9\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,4,13]]},"references-count":30,"journal-issue":{"issue":"303","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["S0025-5718-2016-03112-9"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3112","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2016,4,13]]}}}