An L (1/3 + ε ) Algorithm for the Discrete Logarithm Problem for Low Degree Curves
Identifieur interne : 000F55 ( Istex/Corpus ); précédent : 000F54; suivant : 000F56An L (1/3 + ε ) Algorithm for the Discrete Logarithm Problem for Low Degree Curves
Auteurs : Andreas Enge ; Pierrick GaudrySource :
- Lecture Notes in Computer Science [ 0302-9743 ]
Abstract
Abstract: The discrete logarithm problem in Jacobians of curves of high genus g over finite fields $\mathbb {F}_q$ is known to be computable with subexponential complexity $L_{q^g}(1/2, O(1))$ . We present an algorithm for a family of plane curves whose degrees in X and Y are low with respect to the curve genus, and suitably unbalanced. The finite base fields are arbitrary, but their sizes should not grow too fast compared to the genus. For this family, the group structure can be computed in subexponential time of $L_{q^g}(1/3, O(1))$ , and a discrete logarithm computation takes subexponential time of $L_{q^g}(1/3+ \varepsilon, o(1))$ for any positive ε. These runtime bounds rely on heuristics similar to the ones used in the number field sieve or the function field sieve algorithms.
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DOI: 10.1007/978-3-540-72540-4_22
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<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title xml:lang="en">An L (1/3 + ε ) Algorithm for the Discrete Logarithm Problem for Low Degree Curves</title><author><name sortKey="Enge, Andreas" sort="Enge, Andreas" uniqKey="Enge A" first="Andreas" last="Enge">Andreas Enge</name><affiliation><mods:affiliation>INRIA Futurs & Laboratoire d’Informatique (CNRS/UMR 7161), École polytechnique, 91128, Palaiseau Cedex, France</mods:affiliation></affiliation></author><author><name sortKey="Gaudry, Pierrick" sort="Gaudry, Pierrick" uniqKey="Gaudry P" first="Pierrick" last="Gaudry">Pierrick Gaudry</name><affiliation><mods:affiliation>LORIA (CNRS/UMR 7503), Campus Scientifique, BP 239, 54506, Vandœuvre-lés-Nancy Cedex, France</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:421E119C74EB9AE3357CEC6D6152E462DD4BBE76</idno><date when="2007" year="2007">2007</date><idno type="doi">10.1007/978-3-540-72540-4_22</idno><idno type="url">https://api.istex.fr/ark:/67375/HCB-WH9C6QJG-3/fulltext.pdf</idno><idno type="wicri:Area/Istex/Corpus">000F55</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">000F55</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">An <hi rend="italic">L</hi> (1/3 + <hi rend="italic">ε</hi>) Algorithm for the Discrete Logarithm Problem for Low Degree Curves</title><author><name sortKey="Enge, Andreas" sort="Enge, Andreas" uniqKey="Enge A" first="Andreas" last="Enge">Andreas Enge</name><affiliation><mods:affiliation>INRIA Futurs & Laboratoire d’Informatique (CNRS/UMR 7161), École polytechnique, 91128, Palaiseau Cedex, France</mods:affiliation></affiliation></author><author><name sortKey="Gaudry, Pierrick" sort="Gaudry, Pierrick" uniqKey="Gaudry P" first="Pierrick" last="Gaudry">Pierrick Gaudry</name><affiliation><mods:affiliation>LORIA (CNRS/UMR 7503), Campus Scientifique, BP 239, 54506, Vandœuvre-lés-Nancy Cedex, France</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="s" type="main" xml:lang="en">Lecture Notes in Computer Science</title><idno type="ISSN">0302-9743</idno><idno type="eISSN">1611-3349</idno><idno type="ISSN">0302-9743</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0302-9743</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: The discrete logarithm problem in Jacobians of curves of high genus g over finite fields $\mathbb {F}_q$ is known to be computable with subexponential complexity $L_{q^g}(1/2, O(1))$ . We present an algorithm for a family of plane curves whose degrees in X and Y are low with respect to the curve genus, and suitably unbalanced. The finite base fields are arbitrary, but their sizes should not grow too fast compared to the genus. For this family, the group structure can be computed in subexponential time of $L_{q^g}(1/3, O(1))$ , and a discrete logarithm computation takes subexponential time of $L_{q^g}(1/3+ \varepsilon, o(1))$ for any positive ε. 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We present an algorithm for a family of plane curves whose degrees in X and Y are low with respect to the curve genus, and suitably unbalanced. The finite base fields are arbitrary, but their sizes should not grow too fast compared to the genus. For this family, the group structure can be computed in subexponential time of $L_{q^g}(1/3, O(1))$ , and a discrete logarithm computation takes subexponential time of $L_{q^g}(1/3+ \varepsilon, o(1))$ for any positive ε. These runtime bounds rely on heuristics similar to the ones used in the number field sieve or the function field sieve algorithms.</abstract><qualityIndicators><score>8.524</score><pdfWordCount>6946</pdfWordCount><pdfCharCount>32051</pdfCharCount><pdfVersion>1.3</pdfVersion><pdfPageCount>15</pdfPageCount><pdfPageSize>430 x 660 pts</pdfPageSize><refBibsNative>false</refBibsNative><abstractWordCount>127</abstractWordCount><abstractCharCount>793</abstractCharCount><keywordCount>0</keywordCount></qualityIndicators><title>An L (1/3 + ε ) Algorithm for the Discrete Logarithm Problem for Low Degree Curves</title><chapterId><json:string>22</json:string><json:string>Chap22</json:string></chapterId><genre><json:string>conference</json:string></genre><serie><title>Lecture Notes in Computer Science</title><language><json:string>unknown</json:string></language><copyrightDate>2007</copyrightDate><issn><json:string>0302-9743</json:string></issn><eissn><json:string>1611-3349</json:string></eissn><editor><json:item><name>David Hutchison</name><affiliations><json:string>Lancaster University, Lancaster, UK</json:string></affiliations></json:item><json:item><name>Takeo Kanade</name><affiliations><json:string>Carnegie Mellon University, Pittsburgh, PA, USA</json:string></affiliations></json:item><json:item><name>Josef Kittler</name><affiliations><json:string>University of Surrey, Guildford, UK</json:string></affiliations></json:item><json:item><name>Jon M. 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type="institution">Cornell University</orgName><address><settlement>Ithaca</settlement><region>NY</region><country key="US">UNITED STATES</country></address></affiliation></editor><editor><persName><forename type="first">Friedemann</forename><surname>Mattern</surname></persName><affiliation><orgName type="institution">ETH Zurich</orgName><address><settlement>Zurich</settlement><country key="CH">SWITZERLAND</country></address></affiliation></editor><editor><persName><forename type="first">John</forename><forename type="first">C.</forename><surname>Mitchell</surname></persName><affiliation><orgName type="institution">Stanford University</orgName><address><settlement>Stanford</settlement><region>CA</region><country key="US">UNITED STATES</country></address></affiliation></editor><editor><persName><forename type="first">Moni</forename><surname>Naor</surname></persName><affiliation><orgName type="institution">Weizmann Institute of Science</orgName><address><settlement>Rehovot</settlement><country key="IL">ISRAEL</country></address></affiliation></editor><editor><persName><forename type="first">Oscar</forename><surname>Nierstrasz</surname></persName><affiliation><orgName type="institution">University of Bern</orgName><address><settlement>Bern</settlement><country key="CH">SWITZERLAND</country></address></affiliation></editor><editor><persName><forename type="first">C.</forename><surname>Pandu Rangan</surname></persName><affiliation><orgName type="institution">Indian Institute of Technology</orgName><address><settlement>Madras</settlement><country key="IN">INDIA</country></address></affiliation></editor><editor><persName><forename type="first">Bernhard</forename><surname>Steffen</surname></persName><affiliation><orgName type="institution">University of Dortmund</orgName><address><settlement>Dortmund</settlement><country key="DE">GERMANY</country></address></affiliation></editor><editor><persName><forename type="first">Madhu</forename><surname>Sudan</surname></persName><affiliation><orgName type="institution">Massachusetts Institute of Technology</orgName><address><region>MA</region><country key="US">UNITED STATES</country></address></affiliation></editor><editor><persName><forename type="first">Demetri</forename><surname>Terzopoulos</surname></persName><affiliation><orgName type="institution">University of California</orgName><address><settlement>Los Angeles</settlement><region>CA</region><country key="US">UNITED STATES</country></address></affiliation></editor><editor><persName><forename type="first">Doug</forename><surname>Tygar</surname></persName><affiliation><orgName type="institution">University of California</orgName><address><settlement>Berkeley</settlement><region>CA</region><country key="US">UNITED STATES</country></address></affiliation></editor><editor><persName><forename type="first">Moshe</forename><forename type="first">Y.</forename><surname>Vardi</surname></persName><affiliation><orgName type="institution">Rice University</orgName><address><settlement>Houston</settlement><region>TX</region><country key="US">UNITED STATES</country></address></affiliation></editor><editor><persName><forename type="first">Gerhard</forename><surname>Weikum</surname></persName><affiliation><orgName type="institution">Max-Planck Institute of Computer Science</orgName><address><settlement>Saarbrücken</settlement><country key="DE">GERMANY</country></address></affiliation></editor><idno type="pISSN">0302-9743</idno><idno type="eISSN">1611-3349</idno><idno type="seriesID">558</idno></series></biblStruct></sourceDesc></fileDesc><profileDesc><abstract xml:lang="en"><head>Abstract</head><p>The discrete logarithm problem in Jacobians of curves of high genus <hi rend="italic">g</hi> over finite fields <formula xml:id="IEq1" notation="TEX"><media mimeType="image" url=""></media>$\mathbb {F}_q$
</formula> is known to be computable with subexponential complexity <formula xml:id="IEq2" notation="TEX"><media mimeType="image" url=""></media>$L_{q^g}(1/2, O(1))$
</formula>. We present an algorithm for a family of plane curves whose degrees in <hi rend="italic">X</hi> and <hi rend="italic">Y</hi> are low with respect to the curve genus, and suitably unbalanced. The finite base fields are arbitrary, but their sizes should not grow too fast compared to the genus. For this family, the group structure can be computed in subexponential time of <formula xml:id="IEq3" notation="TEX"><media mimeType="image" url=""></media>$L_{q^g}(1/3, O(1))$
</formula>, and a discrete logarithm computation takes subexponential time of <formula xml:id="IEq4" notation="TEX"><media mimeType="image" url=""></media>$L_{q^g}(1/3+ \varepsilon, o(1))$
</formula> for any positive <hi rend="italic">ε</hi>. These runtime bounds rely on heuristics similar to the ones used in the number field sieve or the function field sieve algorithms.</p></abstract><textClass ana="subject"><keywords scheme="book-subject-collection"><list><label>SUCO11645</label><item><term>Computer Science</term></item></list></keywords></textClass><textClass ana="subject"><keywords scheme="book-subject"><list><label>I</label><item><term type="Primary">Computer Science</term></item><label>I15033</label><item><term type="Secondary" subtype="priority-1">Data Encryption</term></item><label>I13022</label><item><term type="Secondary" subtype="priority-2">Computer Communication Networks</term></item><label>I14050</label><item><term type="Secondary" subtype="priority-3">Systems and Data Security</term></item><label>I16021</label><item><term type="Secondary" subtype="priority-4">Algorithm Analysis and Problem Complexity</term></item><label>I17028</label><item><term type="Secondary" subtype="priority-5">Discrete Mathematics in 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University</OrgName><OrgAddress><City>Pittsburgh</City><State>PA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff3"><OrgName>University of Surrey</OrgName><OrgAddress><City>Guildford</City><Country>UK</Country></OrgAddress></Affiliation><Affiliation ID="Aff4"><OrgName>Cornell University</OrgName><OrgAddress><City>Ithaca</City><State>NY</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff5"><OrgName>ETH Zurich</OrgName><OrgAddress><City>Zurich</City><Country>Switzerland</Country></OrgAddress></Affiliation><Affiliation ID="Aff6"><OrgName>Stanford University</OrgName><OrgAddress><City>Stanford</City><State>CA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff7"><OrgName>Weizmann Institute of Science</OrgName><OrgAddress><City>Rehovot</City><Country>Israel</Country></OrgAddress></Affiliation><Affiliation ID="Aff8"><OrgName>University of Bern</OrgName><OrgAddress><City>Bern</City><Country>Switzerland</Country></OrgAddress></Affiliation><Affiliation ID="Aff9"><OrgName>Indian Institute of Technology</OrgName><OrgAddress><City>Madras</City><Country>India</Country></OrgAddress></Affiliation><Affiliation ID="Aff10"><OrgName>University of Dortmund</OrgName><OrgAddress><City>Dortmund</City><Country>Germany</Country></OrgAddress></Affiliation><Affiliation ID="Aff11"><OrgName>Massachusetts Institute of Technology</OrgName><OrgAddress><State>MA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff12"><OrgName>University of California</OrgName><OrgAddress><City>Los Angeles</City><State>CA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff13"><OrgName>University of California</OrgName><OrgAddress><City>Berkeley</City><State>CA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff14"><OrgName>Rice University</OrgName><OrgAddress><City>Houston</City><State>TX</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff15"><OrgName>Max-Planck Institute of Computer Science</OrgName><OrgAddress><City>Saarbrücken</City><Country>Germany</Country></OrgAddress></Affiliation></EditorGroup></SeriesHeader><Book Language="En"><BookInfo BookProductType="Proceedings" ContainsESM="No" Language="En" MediaType="eBook" NumberingDepth="2" NumberingStyle="ContentOnly" OutputMedium="All" TocLevels="0"><BookID>978-3-540-72540-4</BookID><BookTitle>Advances in Cryptology - EUROCRYPT 2007</BookTitle><BookSubTitle>26th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Barcelona, Spain, May 20-24, 2007. Proceedings</BookSubTitle><BookVolumeNumber>4515</BookVolumeNumber><BookSequenceNumber>4515</BookSequenceNumber><BookDOI>10.1007/978-3-540-72540-4</BookDOI><BookTitleID>149806</BookTitleID><BookPrintISBN>978-3-540-72539-8</BookPrintISBN><BookElectronicISBN>978-3-540-72540-4</BookElectronicISBN><BookChapterCount>33</BookChapterCount><BookCopyright><CopyrightHolderName>Springer-Verlag Berlin Heidelberg</CopyrightHolderName><CopyrightYear>2007</CopyrightYear></BookCopyright><BookSubjectGroup><BookSubject Code="I" Type="Primary">Computer Science</BookSubject><BookSubject Code="I15033" Priority="1" Type="Secondary">Data Encryption</BookSubject><BookSubject Code="I13022" Priority="2" Type="Secondary">Computer Communication Networks</BookSubject><BookSubject Code="I14050" Priority="3" Type="Secondary">Systems and Data Security</BookSubject><BookSubject Code="I16021" Priority="4" Type="Secondary">Algorithm Analysis and Problem Complexity</BookSubject><BookSubject Code="I17028" Priority="5" Type="Secondary">Discrete Mathematics in Computer Science</BookSubject><BookSubject Code="I24067" Priority="6" Type="Secondary">Management of Computing and Information Systems</BookSubject><SubjectCollection Code="SUCO11645">Computer Science</SubjectCollection></BookSubjectGroup><BookContext><SeriesID>558</SeriesID></BookContext></BookInfo><BookHeader><EditorGroup><Editor AffiliationIDS="Aff16"><EditorName DisplayOrder="Western"><GivenName>Moni</GivenName><FamilyName>Naor</FamilyName></EditorName><Contact><Email>moni.naor@weizmann.ac.il</Email></Contact></Editor><Affiliation ID="Aff16"><OrgDivision>Department of Computer Science and Applied Mathematics</OrgDivision><OrgName>Weizmann Institute of Science</OrgName><OrgAddress><Postcode>76100</Postcode><City>Rehovot</City><Country>Israel</Country></OrgAddress></Affiliation></EditorGroup></BookHeader><Chapter ID="Chap22" Language="En"><ChapterInfo ChapterType="OriginalPaper" ContainsESM="No" NumberingDepth="2" NumberingStyle="ContentOnly" TocLevels="0"><ChapterID>22</ChapterID><ChapterDOI>10.1007/978-3-540-72540-4_22</ChapterDOI><ChapterSequenceNumber>22</ChapterSequenceNumber><ChapterTitle Language="En">An <Emphasis Type="Italic">L</Emphasis> (1/3 + <Emphasis Type="Italic">ε</Emphasis>) Algorithm for the Discrete Logarithm Problem for Low Degree Curves</ChapterTitle><ChapterFirstPage>379</ChapterFirstPage><ChapterLastPage>393</ChapterLastPage><ChapterCopyright><CopyrightHolderName>Springer-Verlag Berlin Heidelberg</CopyrightHolderName><CopyrightYear>2007</CopyrightYear></ChapterCopyright><ChapterGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ChapterGrants><ChapterContext><SeriesID>558</SeriesID><BookID>978-3-540-72540-4</BookID><BookTitle>Advances in Cryptology - EUROCRYPT 2007</BookTitle></ChapterContext></ChapterInfo><ChapterHeader><AuthorGroup><Author AffiliationIDS="Aff17"><AuthorName DisplayOrder="Western"><GivenName>Andreas</GivenName><FamilyName>Enge</FamilyName></AuthorName></Author><Author AffiliationIDS="Aff18"><AuthorName DisplayOrder="Western"><GivenName>Pierrick</GivenName><FamilyName>Gaudry</FamilyName></AuthorName></Author><Affiliation ID="Aff17"><OrgDivision>INRIA Futurs & Laboratoire d’Informatique (CNRS/UMR 7161)</OrgDivision><OrgName>École polytechnique</OrgName><OrgAddress><Postcode>91128</Postcode><City>Palaiseau Cedex</City><Country>France</Country></OrgAddress></Affiliation><Affiliation ID="Aff18"><OrgName>LORIA (CNRS/UMR 7503), Campus Scientifique</OrgName><OrgAddress><Postbox>BP 239</Postbox><Postcode>54506</Postcode><City>Vandœuvre-lés-Nancy Cedex</City><Country>France</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>The discrete logarithm problem in Jacobians of curves of high genus <Emphasis Type="Italic">g</Emphasis> over finite fields <InlineEquation ID="IEq1"><InlineMediaObject><ImageObject FileRef="978-3-540-72540-4_22_Chapter_TeX2GIFIEq1.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$\mathbb {F}_q$</EquationSource></InlineEquation> is known to be computable with subexponential complexity <InlineEquation ID="IEq2"><InlineMediaObject><ImageObject FileRef="978-3-540-72540-4_22_Chapter_TeX2GIFIEq2.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$L_{q^g}(1/2, O(1))$</EquationSource></InlineEquation>. We present an algorithm for a family of plane curves whose degrees in <Emphasis Type="Italic">X</Emphasis> and <Emphasis Type="Italic">Y</Emphasis> are low with respect to the curve genus, and suitably unbalanced. The finite base fields are arbitrary, but their sizes should not grow too fast compared to the genus. For this family, the group structure can be computed in subexponential time of <InlineEquation ID="IEq3"><InlineMediaObject><ImageObject FileRef="978-3-540-72540-4_22_Chapter_TeX2GIFIEq3.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$L_{q^g}(1/3, O(1))$</EquationSource></InlineEquation>, and a discrete logarithm computation takes subexponential time of <InlineEquation ID="IEq4"><InlineMediaObject><ImageObject FileRef="978-3-540-72540-4_22_Chapter_TeX2GIFIEq4.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$L_{q^g}(1/3+ \varepsilon, o(1))$</EquationSource></InlineEquation> for any positive <Emphasis Type="Italic">ε</Emphasis>. These runtime bounds rely on heuristics similar to the ones used in the number field sieve or the function field sieve algorithms.</Para></Abstract></ChapterHeader><NoBody></NoBody></Chapter></Book></Series></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>An L (1/3 + ε ) Algorithm for the Discrete Logarithm Problem for Low Degree Curves</title></titleInfo><titleInfo type="alternative" contentType="CDATA"><title>An L (1/3 + ε ) Algorithm for the Discrete Logarithm Problem for Low Degree Curves</title></titleInfo><name type="personal"><namePart type="given">Andreas</namePart><namePart type="family">Enge</namePart><affiliation>INRIA Futurs & Laboratoire d’Informatique (CNRS/UMR 7161), École polytechnique, 91128, Palaiseau Cedex, France</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Pierrick</namePart><namePart type="family">Gaudry</namePart><affiliation>LORIA (CNRS/UMR 7503), Campus Scientifique, BP 239, 54506, Vandœuvre-lés-Nancy Cedex, France</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre displayLabel="OriginalPaper" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" type="conference" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-BFHXPBJJ-3">conference</genre><originInfo><publisher>Springer Berlin Heidelberg</publisher><place><placeTerm type="text">Berlin, Heidelberg</placeTerm></place><dateIssued encoding="w3cdtf">2007</dateIssued><copyrightDate encoding="w3cdtf">2007</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract lang="en">Abstract: The discrete logarithm problem in Jacobians of curves of high genus g over finite fields $\mathbb {F}_q$ is known to be computable with subexponential complexity $L_{q^g}(1/2, O(1))$ . We present an algorithm for a family of plane curves whose degrees in X and Y are low with respect to the curve genus, and suitably unbalanced. The finite base fields are arbitrary, but their sizes should not grow too fast compared to the genus. For this family, the group structure can be computed in subexponential time of $L_{q^g}(1/3, O(1))$ , and a discrete logarithm computation takes subexponential time of $L_{q^g}(1/3+ \varepsilon, o(1))$ for any positive ε. These runtime bounds rely on heuristics similar to the ones used in the number field sieve or the function field sieve algorithms.</abstract><relatedItem type="host"><titleInfo><title>Advances in Cryptology - EUROCRYPT 2007</title><subTitle>26th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Barcelona, Spain, May 20-24, 2007. Proceedings</subTitle></titleInfo><name type="personal"><namePart type="given">Moni</namePart><namePart type="family">Naor</namePart><affiliation>Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, 76100, Rehovot, Israel</affiliation><affiliation>E-mail: moni.naor@weizmann.ac.il</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><genre type="book-series" authority="ISTEX" authorityURI="https://publication-type.data.istex.fr" valueURI="https://publication-type.data.istex.fr/ark:/67375/JMC-0G6R5W5T-Z">book-series</genre><originInfo><publisher>Springer</publisher><copyrightDate encoding="w3cdtf">2007</copyrightDate><issuance>monographic</issuance></originInfo><subject><genre>Book-Subject-Collection</genre><topic authority="SpringerSubjectCodes" authorityURI="SUCO11645">Computer Science</topic></subject><subject><genre>Book-Subject-Group</genre><topic authority="SpringerSubjectCodes" authorityURI="I">Computer Science</topic><topic authority="SpringerSubjectCodes" authorityURI="I15033">Data Encryption</topic><topic authority="SpringerSubjectCodes" authorityURI="I13022">Computer Communication Networks</topic><topic authority="SpringerSubjectCodes" authorityURI="I14050">Systems and Data Security</topic><topic authority="SpringerSubjectCodes" authorityURI="I16021">Algorithm Analysis and Problem Complexity</topic><topic authority="SpringerSubjectCodes" authorityURI="I17028">Discrete Mathematics in Computer Science</topic><topic authority="SpringerSubjectCodes" authorityURI="I24067">Management of Computing and Information Systems</topic></subject><identifier type="DOI">10.1007/978-3-540-72540-4</identifier><identifier type="ISBN">978-3-540-72539-8</identifier><identifier type="eISBN">978-3-540-72540-4</identifier><identifier type="ISSN">0302-9743</identifier><identifier type="eISSN">1611-3349</identifier><identifier type="BookTitleID">149806</identifier><identifier type="BookID">978-3-540-72540-4</identifier><identifier type="BookChapterCount">33</identifier><identifier type="BookVolumeNumber">4515</identifier><identifier type="BookSequenceNumber">4515</identifier><part><date>2007</date><detail type="volume"><number>4515</number><caption>vol.</caption></detail><extent unit="pages"><start>379</start><end>393</end></extent></part><recordInfo><recordOrigin>Springer-Verlag Berlin Heidelberg, 2007</recordOrigin></recordInfo></relatedItem><relatedItem type="series"><titleInfo><title>Lecture Notes in Computer Science</title></titleInfo><name type="personal"><namePart type="given">David</namePart><namePart type="family">Hutchison</namePart><affiliation>Lancaster University, Lancaster, UK</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Takeo</namePart><namePart type="family">Kanade</namePart><affiliation>Carnegie Mellon University, Pittsburgh, PA, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Josef</namePart><namePart type="family">Kittler</namePart><affiliation>University of Surrey, Guildford, UK</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Jon</namePart><namePart type="given">M.</namePart><namePart type="family">Kleinberg</namePart><affiliation>Cornell University, Ithaca, NY, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Friedemann</namePart><namePart type="family">Mattern</namePart><affiliation>ETH Zurich, Zurich, Switzerland</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">John</namePart><namePart type="given">C.</namePart><namePart type="family">Mitchell</namePart><affiliation>Stanford University, Stanford, CA, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Moni</namePart><namePart type="family">Naor</namePart><affiliation>Weizmann Institute of Science, Rehovot, Israel</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Oscar</namePart><namePart type="family">Nierstrasz</namePart><affiliation>University of Bern, Bern, Switzerland</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">C.</namePart><namePart type="family">Pandu Rangan</namePart><affiliation>Indian Institute of Technology, Madras, India</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Bernhard</namePart><namePart type="family">Steffen</namePart><affiliation>University of Dortmund, Dortmund, Germany</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Madhu</namePart><namePart type="family">Sudan</namePart><affiliation>Massachusetts Institute of Technology, MA, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Demetri</namePart><namePart type="family">Terzopoulos</namePart><affiliation>University of California, Los Angeles, CA, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Doug</namePart><namePart type="family">Tygar</namePart><affiliation>University of California, Berkeley, CA, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Moshe</namePart><namePart type="given">Y.</namePart><namePart type="family">Vardi</namePart><affiliation>Rice University, Houston, TX, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Gerhard</namePart><namePart type="family">Weikum</namePart><affiliation>Max-Planck Institute of Computer Science, Saarbrücken, Germany</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><originInfo><publisher>Springer</publisher><copyrightDate encoding="w3cdtf">2007</copyrightDate><issuance>serial</issuance></originInfo><identifier type="ISSN">0302-9743</identifier><identifier type="eISSN">1611-3349</identifier><identifier type="SeriesID">558</identifier><recordInfo><recordOrigin>Springer-Verlag Berlin Heidelberg, 2007</recordOrigin></recordInfo></relatedItem><identifier type="istex">421E119C74EB9AE3357CEC6D6152E462DD4BBE76</identifier><identifier type="ark">ark:/67375/HCB-WH9C6QJG-3</identifier><identifier type="DOI">10.1007/978-3-540-72540-4_22</identifier><identifier type="ChapterID">22</identifier><identifier type="ChapterID">Chap22</identifier><accessCondition type="use and reproduction" contentType="copyright">Springer-Verlag Berlin Heidelberg, 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