Algebraic number fields and the LLL algorithm

IF 0.6 4区 数学 Q4 COMPUTER SCIENCE, THEORY & METHODS
M.J. Uray
{"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 R-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 Zn to Kn, and give a polynomial upper bound on the running time when the computations in K are performed exactly (as opposed to floating-point approximations).

代数数字字段和LLL算法
在本文中,我们使用精确算术来分析代数数域中各种运算和算法的计算成本。设K是一个代数数域。在论文的前半部分,我们根据K的输入大小和参数来计算K中许多运算的运行时间和输出大小。我们包括了一些关于这些运算的早期结果,但我们比它们走得更远,例如,我们还分析了K中一些特定于R的运算,如小于比较。在论文的后半部分,我们分析了两种算法:Bareiss算法和LLL算法,前者是高斯消去的整数保留版本,后者用于格基约简。在这两种情况下,我们都将算法从Zn扩展到Kn,并在精确执行K中的计算时给出运行时间的多项式上界(与浮点近似相反)。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
Journal of Symbolic Computation
Journal of Symbolic Computation 工程技术-计算机:理论方法
CiteScore
2.10
自引率
14.30%
发文量
75
审稿时长
142 days
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.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信