{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:02:02Z","timestamp":1776837722776,"version":"3.51.2"},"reference-count":16,"publisher":"American Mathematical Society (AMS)","issue":"346","license":[{"start":{"date-parts":[[2024,8,9]],"date-time":"2024-08-09T00:00:00Z","timestamp":1723161600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100007601","name":"Horizon 2020","doi-asserted-by":"publisher","award":["grant 780701"],"award-info":[{"award-number":["grant 780701"]}],"id":[{"id":"10.13039\/501100007601","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We exhibit an explicit\n                    <italic>short<\/italic>\n                    basis of the Stickelberger ideal of cyclotomic fields of any conductor\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n                        <mml:semantics>\n                          <mml:mi>m<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , i.e., a basis containing only short elements. An element\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"sigma-summation Underscript sigma element-of upper G Subscript m Endscripts epsilon Subscript sigma Baseline sigma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:munder>\n                              <mml:mo>\n                                \u2211\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03c3\n                                  \n                                <\/mml:mi>\n                                <mml:mo>\n                                  \u2208\n                                  \n                                <\/mml:mo>\n                                <mml:msub>\n                                  <mml:mi>G<\/mml:mi>\n                                  <mml:mi>m<\/mml:mi>\n                                <\/mml:msub>\n                              <\/mml:mrow>\n                            <\/mml:munder>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03b5\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03c3\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mi>\n                              \u03c3\n                              \n                            <\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\sum _{\\sigma \\in G_m} \\varepsilon _{\\sigma }\\sigma<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of the group ring\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper Z left-bracket upper G Subscript m Baseline right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>G<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>m<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {Z}[G_{m}]<\/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=\"upper G Subscript m\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>G<\/mml:mi>\n                            <mml:mi>m<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">G_m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is the Galois group of the field, is said to be short if all of its coefficients\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"epsilon Subscript sigma\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>\n                              \u03b5\n                              \n                            <\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>\n                                \u03c3\n                                \n                              <\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\varepsilon _{\\sigma }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"0\">\n                        <mml:semantics>\n                          <mml:mn>0<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    \u00a0or\u00a0\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1\">\n                        <mml:semantics>\n                          <mml:mn>1<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>\n                  <p>\n                    As a direct practical consequence, we deduce from this short basis an\n                    <italic>explicit<\/italic>\n                    upper bound on the relative class number that is valid for\n                    <italic>any<\/italic>\n                    conductor. This basis also has several concrete applications, in particular for the cryptanalysis of the Shortest Vector Problem on Ideal Lattices.\n                  <\/p>","DOI":"10.1090\/mcom\/3863","type":"journal-article","created":{"date-parts":[[2023,5,17]],"date-time":"2023-05-17T10:16:15Z","timestamp":1684318575000},"page":"887-909","source":"Crossref","is-referenced-by-count":0,"title":["A short basis of the Stickelberger ideal of a cyclotomic field"],"prefix":"10.1090","volume":"93","author":[{"given":"Olivier","family":"Bernard","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Radan","family":"Ku\u010dera","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2023,8,9]]},"reference":[{"issue":"4","key":"1","doi-asserted-by":"publisher","first-page":"Paper No. 4.41, 35","DOI":"10.37236\/10367","article-title":"A survey of the Hadamard maximal determinant problem","volume":"28","author":"Browne, Patrick","year":"2021","journal-title":"Electron. 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