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The complexity of class polynomial computation via floating point approximations

hal.structure.identifierAlgorithmic number theory for cryptology [TANC]
hal.structure.identifierLaboratoire d'informatique de l'École polytechnique [Palaiseau] [LIX]
hal.structure.identifierLithe and fast algorithmic number theory [LFANT]
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorENGE, Andreas
dc.date.issued2009
dc.identifier.issn0025-5718
dc.descriptionTo appear in Mathematics of Computation.
dc.description.abstractEnWe analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest one of the presented approaches uses a technique devised by Dupont to evaluate modular functions by Newton iterations on an expression involving the arithmetic-geometric mean. It runs in time $O (|D| \log^5 |D| \log \log |D|) = O (|D|^{1 + \epsilon}) = O ( h^{2 + \epsilon})$ for any $\epsilon > 0$, where $D$ is the CM discriminant and $h$ is the degree of the class polynomial. Another fast algorithm uses multipoint evaluation techniques known from symbolic computation; its asymptotic complexity is worse by a factor of $\log |D|$. Up to logarithmic factors, this running time matches the size of the constructed polynomials. The estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary-quadratic order and on a rigorously proven upper bound for the height of class polynomials.
dc.language.isoen
dc.publisherAmerican Mathematical Society
dc.title.enThe complexity of class polynomial computation via floating point approximations
dc.typeArticle de revue
dc.subject.halInformatique [cs]/Analyse numérique [cs.NA]
dc.subject.halInformatique [cs]/Calcul formel [cs.SC]
dc.subject.halMathématiques [math]/Théorie des nombres [math.NT]
dc.subject.halInformatique [cs]/Complexité [cs.CC]
dc.identifier.arxivcs/0601104
bordeaux.journalMathematics of Computation
bordeaux.page1089-1107
bordeaux.volume78
bordeaux.issue266
bordeaux.peerReviewedoui
hal.identifierinria-00001040
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//inria-00001040v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Mathematics%20of%20Computation&rft.date=2009&rft.volume=78&rft.issue=266&rft.spage=1089-1107&rft.epage=1089-1107&rft.eissn=0025-5718&rft.issn=0025-5718&rft.au=ENGE,%20Andreas&rft.genre=article


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