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Development of true orbit computation: analysis of nonlinear phenomena and application to pseudorandom number generation

Research Project

Project/Area Number 15K00342
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Soft computing
Research InstitutionFuture University-Hakodate

Principal Investigator

Saito Asaki  公立はこだて未来大学, システム情報科学部, 准教授 (60344040)

Co-Investigator(Kenkyū-buntansha) 田村 純一  津田塾大学, 数学・計算機科学研究所, 研究員 (90418905)
安富 真一  東邦大学, 理学部, 教授 (60230231)
山口 明宏  福岡工業大学, 情報工学部, 教授 (60281789)
Project Period (FY) 2015-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2015: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Keywords真軌道計算 / カオス / 非線形現象 / 擬似乱数 / 代数的数 / p進数 / 連分数 / Lagrangeの定理 / 乱数検定
Outline of Final Research Achievements

True orbit computation is an errorless simulation method recently proposed by Saito et al. In order to promote its new development, we have conducted two types of researches: (i) Analyses of nonlinear phenomena by using the true orbit computation. (ii) Application of the true orbit computation to pseudorandom number generation. In particular, as for (ii), we have not only proposed algorithms that transform an initial point (seed) into a high-quality pseudorandom sequence but also have established methods for properly selecting initial points. Also, we have conducted related researches on continued fractions (we have constructed several p-adic continued fraction algorithms and also proved their periodicity for all the quadratic elements).

Academic Significance and Societal Importance of the Research Achievements

擬似乱数は,モンテカルロ法を使った数値計算や暗号通信で欠かせないのはもちろん,シミュレーションなどでも活用されており,現代社会を支える基盤技術の1つと言える.本研究で得られた真軌道擬似乱数生成器によって,高品質な(非周期性などの性質が理論的に保証され,また統計性も極めて良好な)擬似乱数列を大量に生成できるようになった.
また,真軌道計算は従来のシミュレーション法とは比較にならないほど高精度なシミュレーションを実現する.しかし,提案されてから日が浅く,とりわけ応用に関しては初歩的な段階にとどまっており,発展の余地が大きい.本研究課題によって,特に擬似乱数生成への応用に関しては研究が大きく進展した.

Report

(5 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (57 results)

All 2019 2018 2017 2016 2015

All Journal Article (10 results) (of which Peer Reviewed: 7 results,  Open Access: 3 results,  Acknowledgement Compliant: 3 results) Presentation (47 results) (of which Int'l Joint Research: 23 results,  Invited: 13 results)

  • [Journal Article] Continued fraction algorithms and Lagrange's theorem in Q_p2019

    • Author(s)
      Asaki Saito, Jun-ichi Tamura, Shin-ichi Yasutomi
    • Journal Title

      Commentarii Mathematici Universitatis Sancti Pauli

      Volume: 印刷中

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Multidimensional p-adic continued fraction algorithms2019

    • Author(s)
      Asaki Saito, Jun-ichi Tamura, Shin-ichi Yasutomi
    • Journal Title

      Mathematics of Computation

      Volume: 印刷中 Issue: 321 Pages: 351-372

    • DOI

      10.1090/mcom/3458

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Pseudorandom number generator based on the Bernoulli map on cubic algebraic integers2018

    • Author(s)
      Asaki Saito, Akihiro Yamaguchi
    • Journal Title

      Chaos: An Interdisciplinary Journal of Nonlinear Science

      Volume: 28 Issue: 10 Pages: 103122-103122

    • DOI

      10.1063/1.5048115

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] テント写像の真軌道計算とその擬似乱数生成への応用2018

    • Author(s)
      斉藤朝輝
    • Journal Title

      北海道大学数学講究録

      Volume: 172 Pages: 18-23

    • Related Report
      2017 Research-status Report
    • Open Access
  • [Journal Article] p進数体上の連分数アルゴリズムとその周期性2018

    • Author(s)
      斉藤朝輝, 田村純一, 安冨真一
    • Journal Title

      京都大学数理解析研究所講究録

      Volume: 掲載予定

    • Related Report
      2017 Research-status Report
    • Open Access
  • [Journal Article] Multidimensional p-adic continued fraction algorithms2018

    • Author(s)
      安冨真一, 斉藤朝輝, 田村純一
    • Journal Title

      京都大学数理解析研究所講究録

      Volume: 掲載予定

    • Related Report
      2017 Research-status Report
    • Open Access
  • [Journal Article] Pseudorandom number generation using chaotic true orbits of the Bernoulli map2016

    • Author(s)
      Asaki Saito and Akihiro Yamaguchi
    • Journal Title

      Chaos

      Volume: 26 Issue: 6 Pages: 063122-063122

    • DOI

      10.1063/1.4954023

    • NAID

      130006710895

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Dual substitutions over R > 0-powered symbols2016

    • Author(s)
      Jun-ichi Tamura and Shin-ichi Yasutomi
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: B58 Pages: 231-242

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Substitutions over C-powered symbols, and Rauzy fractals for imaginary directions2016

    • Author(s)
      Jun-ichi Tamura and Shin-ichi Yasutomi
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: B58 Pages: 191-229

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] True orbit simulation of piecewise linear and linear fractional maps of arbitrary dimension using algebraic numbers2015

    • Author(s)
      Asaki Saito, Shin-ichi Yasutomi, Jun-ichi Tamura and Shunji Ito
    • Journal Title

      Chaos

      Volume: 25 Issue: 6 Pages: 063103-063103

    • DOI

      10.1063/1.4921938

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] An exact formula related to simultaneous approximations of p-adic numbers2019

    • Author(s)
      田村純一
    • Organizer
      Workshop「数論とエルゴード理論」
    • Related Report
      2018 Annual Research Report
  • [Presentation] Algebraic integers and pseudorandom number generation2018

    • Author(s)
      Asaki Saito
    • Organizer
      Workshop on Fractal Geometry and Related Topics
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 3次代数的整数上のカオス真軌道を利用した擬似乱数生成2018

    • Author(s)
      斉藤朝輝,山口明宏
    • Organizer
      電子情報通信学会NOLTAソサイエティ大会
    • Related Report
      2018 Annual Research Report
  • [Presentation] C.elegansの歩行の調節と神経系のダイナミクス2018

    • Author(s)
      尾ヶ瀬拓哉,斉藤朝輝
    • Organizer
      電子情報通信学会NOLTAソサイエティ大会
    • Related Report
      2018 Annual Research Report
  • [Presentation] Foundation of Rauzy fractals in the p-adic world2018

    • Author(s)
      Jun-ichi Tamura
    • Organizer
      Workshop on Fractal Geometry and Related Topics
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Generation of a set of complex numbers satisfying certain Boolean inequality2018

    • Author(s)
      Jun-ichi Tamura
    • Organizer
      Workshop on Fractal Geometry and Related Topics
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Finite-state approximation of a pseudorandom number generator using true orbits of the Bernoulli map2018

    • Author(s)
      斉藤朝輝
    • Organizer
      Workshop「数論とエルゴード理論」
    • Related Report
      2017 Research-status Report
  • [Presentation] Some multi-dimensional continued fractions and the linear independence measure of related values2018

    • Author(s)
      田村純一
    • Organizer
      Workshop「数論とエルゴード理論」
    • Related Report
      2017 Research-status Report
  • [Presentation] Certain p-adic continued fraction expansion2018

    • Author(s)
      安冨真一
    • Organizer
      Workshop「数論とエルゴード理論」
    • Related Report
      2017 Research-status Report
  • [Presentation] Certain multidimension p-adic continued fraction algorithm and cubic numbers2018

    • Author(s)
      Shin-Ichi Yasutomi
    • Organizer
      Diophantine Analysis and Related Fields 2018
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] 乱数性の劣る2値系列を用いた乱数検定の特徴付けについて2018

    • Author(s)
      山口明宏, 斉藤朝輝
    • Organizer
      日本応用数理学会第14回研究部会連合発表会
    • Related Report
      2017 Research-status Report
  • [Presentation] Bernoulli写像のカオス的真軌道を使った擬似乱数生成2017

    • Author(s)
      斉藤朝輝,山口明宏
    • Organizer
      日本物理学会第72回年次大会
    • Place of Presentation
      大阪大学(大阪府豊中市)
    • Year and Date
      2017-03-19
    • Related Report
      2016 Research-status Report
  • [Presentation] 真軌道計算とカオス2017

    • Author(s)
      斉藤朝輝
    • Organizer
      複雑系数理の新展開
    • Place of Presentation
      北海道大学(北海道札幌市)
    • Year and Date
      2017-02-15
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Algorithms for p-adic continued fractions and Lagrange's theorem2017

    • Author(s)
      斉藤朝輝
    • Organizer
      Workshop「数論とエルゴード理論」
    • Place of Presentation
      金沢大学サテライト・プラザ(石川県金沢市)
    • Year and Date
      2017-02-12
    • Related Report
      2016 Research-status Report
  • [Presentation] p-adic substitutions, convergence theorems, and Rauzy fractals2017

    • Author(s)
      田村純一
    • Organizer
      Workshop「数論とエルゴード理論」
    • Place of Presentation
      金沢大学サテライト・プラザ(石川県金沢市)
    • Year and Date
      2017-02-12
    • Related Report
      2016 Research-status Report
  • [Presentation] On multidimensional p-adic continued fraction algorithms and p-reduced matrices2017

    • Author(s)
      安冨真一
    • Organizer
      Workshop「数論とエルゴード理論」
    • Place of Presentation
      金沢大学サテライト・プラザ(石川県金沢市)
    • Year and Date
      2017-02-12
    • Related Report
      2016 Research-status Report
  • [Presentation] Rauzy fractals in the p-adic world2017

    • Author(s)
      Jun-ichi Tamura
    • Organizer
      New Advances in Symbolic Dynamics
    • Place of Presentation
      Marseille (France)
    • Year and Date
      2017-02-03
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Continued fraction algorithms and Lagrange's theorem in Qp2017

    • Author(s)
      Asaki Saito
    • Organizer
      New Advances in Symbolic Dynamics
    • Place of Presentation
      Marseille (France)
    • Year and Date
      2017-02-02
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Pseudorandom number generator based on the binary expansion of algebraic integers and its p-adic analogue2017

    • Author(s)
      Asaki Saito
    • Organizer
      Prime Numbers and Automatic Sequences: Determinism and Randomness
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Continued fractions and pseudorandom numbers based on p-adic chaotic maps2017

    • Author(s)
      Asaki Saito
    • Organizer
      Workshop “Number Theoretical Aspects of Tilings and Dynamics”
    • Related Report
      2017 Research-status Report
  • [Presentation] 代数的整数上のBernoulli写像による擬似乱数生成2017

    • Author(s)
      斉藤朝輝
    • Organizer
      第3回有限体理論とその擬似乱数系列生成への応用ワークショップ
    • Related Report
      2017 Research-status Report
  • [Presentation] Continued fraction algorithms and Lagrange's theorem in the p-adic number field2017

    • Author(s)
      Asaki Saito
    • Organizer
      Analytic Number Theory and Related Areas
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] Convergence theorems of substitutions and Rauzy fractals in the p-adic world2017

    • Author(s)
      Jun-Ichi Tamura
    • Organizer
      Prime Numbers and Automatic Sequences: Determinism and Randomness
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Multidimensional p-adic continued fraction algorithms and p-reduced matrices2017

    • Author(s)
      Shin-Ichi Yasutomi
    • Organizer
      Prime Numbers and Automatic Sequences: Determinism and Randomness
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On p-reduced lattice and multidimensional p-adic continued fraction algorithms2017

    • Author(s)
      Shin-Ichi Yasutomi
    • Organizer
      Analytic Number Theory and Related Areas
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] Multidimensional p-adic continued fraction algorithms and cubic number fields2017

    • Author(s)
      Shin-Ichi Yasutomi
    • Organizer
      Workshop “Number Theoretical Aspects of Tilings and Dynamics”
    • Related Report
      2017 Research-status Report
  • [Presentation] 乱数検定の独立性解析に向けた区分線形写像のカオス真軌道によるマルコフ過程の構成2017

    • Author(s)
      山口明宏, 斉藤朝輝
    • Organizer
      日本応用数理学会2017年度年会
    • Related Report
      2017 Research-status Report
  • [Presentation] NIST検定の検定結果の独立性2017

    • Author(s)
      山口明宏
    • Organizer
      第3回有限体理論とその擬似乱数系列生成への応用ワークショップ
    • Related Report
      2017 Research-status Report
  • [Presentation] p-adic continued fractions and Lagrange's theorem2016

    • Author(s)
      Asaki Saito
    • Organizer
      Analysis on Fractals and Graphs Workshop
    • Place of Presentation
      Sanya (China)
    • Year and Date
      2016-12-29
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] On multidimensional p-adic continued fraction2016

    • Author(s)
      Shin-ichi Yasutomi
    • Organizer
      Analysis on Fractals and Graphs Workshop
    • Place of Presentation
      Sanya (China)
    • Year and Date
      2016-12-29
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] p-adic substitutions and Rauzy fractals2016

    • Author(s)
      Jun-ichi Tamura
    • Organizer
      Analysis on Fractals and Graphs Workshop
    • Place of Presentation
      Sanya (China)
    • Year and Date
      2016-12-27
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Multidimensional p-adic continued fraction algorithm2016

    • Author(s)
      安冨真一
    • Organizer
      研究集会:エルゴード理論とその周辺
    • Place of Presentation
      筑波大学(茨城県つくば市)
    • Year and Date
      2016-11-25
    • Related Report
      2016 Research-status Report
  • [Presentation] カオス真軌道から構成した相関系列に対するNIST乱数検定の判定結果の解析2016

    • Author(s)
      山口明宏,斉藤朝輝
    • Organizer
      日本応用数理学会2016年度年会
    • Place of Presentation
      北九州国際会議場(福岡県北九州市)
    • Year and Date
      2016-09-12
    • Related Report
      2016 Research-status Report
  • [Presentation] Fractional calculi of substitutions, Rauzy fractals, and diophantine approximation of complex numbers II2016

    • Author(s)
      Jun-ichi Tamura
    • Organizer
      Substitutions and Continued Fractions (satellite)
    • Place of Presentation
      Lyon (France)
    • Year and Date
      2016-03-16
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] Fractional calculi of substitutions, Rauzy fractals, and diophantine approximation of complex numbers I2016

    • Author(s)
      Jun-ichi Tamura
    • Organizer
      Substitutions and Continued Fractions
    • Place of Presentation
      Paris (France)
    • Year and Date
      2016-03-11
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] Generation of stepped surfaces via modified Jacobi-Perron algorithm and cubic numbers2016

    • Author(s)
      Shin-ichi Yasutomi
    • Organizer
      Substitutions and Continued Fractions
    • Place of Presentation
      Paris (France)
    • Year and Date
      2016-03-11
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] Pseudorandom number generators using true orbits of the 2x modulo 1 map on algebraic integers2016

    • Author(s)
      Asaki Saito
    • Organizer
      Substitutions and Continued Fractions
    • Place of Presentation
      Paris (France)
    • Year and Date
      2016-03-08
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] Pseudorandom number generator with the Bernoulli map on cubic algebraic integers2016

    • Author(s)
      Asaki Saito
    • Organizer
      Workshop Number Theory and Ergodic Theory 2016
    • Place of Presentation
      金沢大学サテライト・プラザ (石川県金沢市)
    • Year and Date
      2016-02-08
    • Related Report
      2015 Research-status Report
  • [Presentation] Fractional calculi of substitutions, Rauzy fractals, and multidimensional complex continued fractions2016

    • Author(s)
      Jun-ichi Tamura
    • Organizer
      Workshop Number Theory and Ergodic Theory 2016
    • Place of Presentation
      金沢大学サテライト・プラザ (石川県金沢市)
    • Year and Date
      2016-02-08
    • Related Report
      2015 Research-status Report
  • [Presentation] Generalization of stepped surfaces2016

    • Author(s)
      Shin-ichi Yasutomi
    • Organizer
      Workshop Number Theory and Ergodic Theory 2016
    • Place of Presentation
      金沢大学サテライト・プラザ (石川県金沢市)
    • Year and Date
      2016-02-08
    • Related Report
      2015 Research-status Report
  • [Presentation] Pseudorandom number generation using true orbits of the Bernoulli map on algebraic integers2015

    • Author(s)
      Asaki Saito
    • Organizer
      Sino-Japanese Workshop on Dynamical Systems and Fractals 2016
    • Place of Presentation
      Ningbo (China)
    • Year and Date
      2015-12-27
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Fractional calculi of substitutions, Rauzy fractals, and continued fractions2015

    • Author(s)
      Jun-ichi Tamura
    • Organizer
      Sino-Japanese Workshop on Dynamical Systems and Fractals 2016
    • Place of Presentation
      Ningbo (China)
    • Year and Date
      2015-12-26
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Dual fractional substitutions2015

    • Author(s)
      Shin-ichi Yasutomi
    • Organizer
      Sino-Japanese Workshop on Dynamical Systems and Fractals 2016
    • Place of Presentation
      Ningbo (China)
    • Year and Date
      2015-12-26
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Fractional calculus of substitutions and Rauzy fractals for imaginary directions2015

    • Author(s)
      Jun-ichi Tamura
    • Organizer
      エルゴード理論とその周辺
    • Place of Presentation
      慶応義塾大学日吉キャンパス (神奈川県・横浜市)
    • Year and Date
      2015-11-07
    • Related Report
      2015 Research-status Report
  • [Presentation] NIST乱数検定におけるP値の一様性に基づく検定結果の判定法について2015

    • Author(s)
      山口明宏,斉藤朝輝
    • Organizer
      日本応用数理学会2015年度年会
    • Place of Presentation
      金沢大学角間キャンパス (石川県・金沢市)
    • Year and Date
      2015-09-09
    • Related Report
      2015 Research-status Report
  • [Presentation] Certain multidimensional continued fraction algorithm2015

    • Author(s)
      Shin-ichi Yasutomi
    • Organizer
      Natural Extention of Arithmetic Algorithms and S-adic System
    • Place of Presentation
      東京大学玉原国際セミナーハウス (群馬県沼田市)
    • Year and Date
      2015-07-24
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] Fractional calculus of substitutions, the Kolakoski word, the Minkowski Question - Mark function and continued fractions2015

    • Author(s)
      Jun-ichi Tamura
    • Organizer
      Natural Extention of Arithmetic Algorithms and S-adic System
    • Place of Presentation
      東京大学玉原国際セミナーハウス (群馬県沼田市)
    • Year and Date
      2015-07-22
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research

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Published: 2015-04-16   Modified: 2020-03-30  

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