Study on Number Theoretic Algorithms and Developments of Systems for Number Theory

数论算法研究及数论系统的发展

• 批准号: 13440013
• 负责人: NAKAMULA Ken
• 金额: $ 4.16万
• 依托单位: Tokyo Metropolitan University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-13440013/
• 关键词: Number Theoretic Algorithms; Systems for Number Theory; Cryptology; Elliptic Curves; JANT; SIMATH; NZMATH; Python; 加法鎖; 64ビット; 円楕円法CEM; 計算機実験; 楕円曲線暗号; 国際研究者交流; 多国籍

As in the references below, many theoretical results were obtained on number theoretic algorithms and their applications related to Scholz's conjecture on addition chains, Iwasawa invariants, elliptic curves, prime ideal decomposition in polynomial rings, hyperplane arrangement, unramified extensions of quadratic fields, several zeta values, class groups and unit groups of number fields, and cryptology, for example.The 4th to 11th meetings were held each with about 50 participants by our main research organization JANT, and, the details are reported in http://ntw.e-one.uec.ac.jp/jant. As a result, we edited two special issues of Transactions of the JSIAM on the topics of the most advanced results and a general survey on number theoretic algorithms, and held an organized session at the yearly meeting of the JSIAM. To exchange with broader areas of algebra, we organized the 4th and 5th meetings of Algebra and Computation' (AC2001 and AC2003) each with almost 150 participants including those from abroad and published the proceedings at ftp://tnt.math.metro-u.ac.jp/pub/ac0[13]/. For wide international exchange of mathematical software research, we help the international Congress of Mathematical Softwares ICMS2002 at Beijin with about 110participants.We took over the system for number theory SIMATH, developed in Germany since 1980's with highly efficient functions for elliptic curves, and have been maintaining and supporting via http://simath.info/. We have succeeded in implementing it on 64 bit machines, have released it as Ver.4.6, and packaged and installed the CEM up to cubic fields. After that, restricting ourselves to maintain SIMATH as it is because of the problems of license and development environment, we started to develop a new system NZMATH by the script language Python, released its alpha version at http://tnt.math.metro-u.ac.jp/nzmath/, and we are going to create a system for number theory constructed together by developers and users of the system.

我们的主要研究机构JANT在第四至第十一次会议上，分别就加法链上的肖尔茨猜想、岩泽不变量、椭圆曲线、多项式环上的素理想分解、超平面排列、二次域的未分支扩张、若干Zeta值、数域的类群和单位群以及密码学等方面的算法及其应用取得了许多理论结果。我们的主要研究机构JANT举行了第四至第十一次会议，每次会议约50人，详细情况已在http://ntw.e-one.uec.ac.jp/jant.上报道因此，我们编辑了两期JSIAM会刊，以最先进的结果和数论算法概述为主题，并在JSIAM年会上组织了一次会议。为了与更广泛的代数领域交流，我们组织了第四次和第五次代数和计算会议(AC2001和AC2003)，每个会议有近150人参加，其中包括来自国外的人，并在ftp://tnt.math.metro-u.ac.jp/pub/ac0[13]/.上发表了会议记录为了扩大数学软件研究的国际交流，我们帮助在北京举行的国际数学软件大会ICMS2002，约110人参加。我们接管了数论SIMATH系统，该系统自1980年S以来在德国开发，具有高效的椭圆曲线函数，并一直通过http://simath.info/.维护和支持我们已经成功地在钻头机上实现了它，并发布了4.6版，并将CEM打包安装到了立方田块。在此之后，由于许可证和开发环境的问题，我们限制自己只能保持SIMATH的现状，我们开始使用脚本语言PYTHON开发一个新的系统NZMATH，并在http://tnt.math.metro-u.ac.jp/nzmath/，上发布了它的阿尔法版本，我们将创建一个由系统的开发人员和用户共同构建的数论系统。

项目成果

Matsui, T.et al.: "SIMATH---recent developments in TMU"Mathematical Software, World Sci.Publishing. 505-506 (2002)

Matsui, T.et al.：“SIMATH --- TMU 的最新进展”数学软件，World Sci.Publishing。

Nakamula, K.: "A Survey of Computational Algebraic Number Theory (in Japanese)"Transactions of JSIAM.. 13. 305-320 (2003)

Nakamula, K.：“计算代数数论综述（日文）”JSIAM 汇刊.. 13. 305-320 (2003)

Tsumura, H.: "On modification of the q-L-series and applications"Nagoya Math. J.. 164. 185-197 (2001)

津村 H.：“关于 q-L 级数的修改和应用”名古屋数学。

Tsumura, H.et al.: "Inductive construction of rapidly convergent series representations for ζ(2n+1)"International J.of Computer Mathematics. 80. 1161-1173 (2003)

Tsumura, H.et al.：“快速收敛级数表示的归纳构造 z(2n+1)”国际计算机数学杂志 80. 1161-1173 (2003)

Kurata, T.: "Intersection and singleton type assignment characterizing finite Bo"hm-tress"Information and Computation. 178. 1-11 (2002)

Kurata, T.：“表征有限 Bo”hm-tress”的交集和单例类型赋值信息和计算。178. 1-11 (2002)