{"title":"Algebraic number fields and the LLL algorithm","authors":"M.J. Uray","doi":"10.1016/j.jsc.2023.102261","DOIUrl":null,"url":null,"abstract":"<div><p><span>In this paper we analyze the computational costs of various operations and algorithms in algebraic number fields using exact arithmetic. Let </span><em>K</em> be an algebraic number field. In the first half of the paper, we calculate the running time and the size of the output of many operations in <em>K</em> in terms of the size of the input and the parameters of <em>K</em>. We include some earlier results about these, but we go further than them, e.g. we also analyze some <span><math><mi>R</mi></math></span>-specific operations in <em>K</em> like less-than comparison.</p><p><span>In the second half of the paper, we analyze two algorithms: the Bareiss algorithm, which is an integer-preserving version of the Gaussian elimination, and the LLL algorithm, which is for lattice basis reduction. In both cases, we extend the algorithm from </span><span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> to <span><math><msup><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, and give a polynomial upper bound on the running time when the computations in <em>K</em> are performed exactly (as opposed to floating-point approximations).</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"121 ","pages":"Article 102261"},"PeriodicalIF":0.6000,"publicationDate":"2023-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Symbolic Computation","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0747717123000755","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 1
Abstract
In this paper we analyze the computational costs of various operations and algorithms in algebraic number fields using exact arithmetic. Let K be an algebraic number field. In the first half of the paper, we calculate the running time and the size of the output of many operations in K in terms of the size of the input and the parameters of K. We include some earlier results about these, but we go further than them, e.g. we also analyze some -specific operations in K like less-than comparison.
In the second half of the paper, we analyze two algorithms: the Bareiss algorithm, which is an integer-preserving version of the Gaussian elimination, and the LLL algorithm, which is for lattice basis reduction. In both cases, we extend the algorithm from to , and give a polynomial upper bound on the running time when the computations in K are performed exactly (as opposed to floating-point approximations).
期刊介绍:
An international journal, the Journal of Symbolic Computation, founded by Bruno Buchberger in 1985, is directed to mathematicians and computer scientists who have a particular interest in symbolic computation. The journal provides a forum for research in the algorithmic treatment of all types of symbolic objects: objects in formal languages (terms, formulas, programs); algebraic objects (elements in basic number domains, polynomials, residue classes, etc.); and geometrical objects.
It is the explicit goal of the journal to promote the integration of symbolic computation by establishing one common avenue of communication for researchers working in the different subareas. It is also important that the algorithmic achievements of these areas should be made available to the human problem-solver in integrated software systems for symbolic computation. To help this integration, the journal publishes invited tutorial surveys as well as Applications Letters and System Descriptions.