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計算數論(英文)/國外優秀數學著作原版系列

包郵 計算數論(英文)/國外優秀數學著作原版系列

出版社:哈爾濱工業大學出版社出版時間:2021-11-01
開本: 16開 頁數: 611
本類榜單:自然科學銷量榜
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計算數論(英文)/國外優秀數學著作原版系列 版權信息

  • ISBN:9787560397375
  • 條形碼:9787560397375 ; 978-7-5603-9737-5
  • 裝幀:一般膠版紙
  • 冊數:暫無
  • 重量:暫無
  • 所屬分類:>

計算數論(英文)/國外優秀數學著作原版系列 內容簡介

本書共包含九章及兩個附錄,具體介紹了計算數論的相關知識,內容涉及整數的算術、有限域算術、多項式算術、橢圓曲線算術、整數分解、離散對數、大稀疏線性系統、公鑰密碼學等內容。本書通過很好基礎的整數和多項式的算術來構建計算數論的基礎,討論了橢圓曲線、整數分解算法、計算離散對數以及稀疏線性系統方法,同時展示了數論在密碼學和密碼分析中的應用,還專門針對數論在公鑰密碼學中的應用設計了一個章節,展示了基于配對的密碼學的近期新發展情況。本書可為深入研究這一方向的讀者提供豐富的參考資料。

計算數論(英文)/國外優秀數學著作原版系列 目錄

Preface 1 Arithmetic cf Integers 1.1 Basic Aiithmetic Operations 1.1.1 Representation of Big Integers 1.1.1.1 Inlzut and Output 1.1.2 Schoolbock Arithmetic 1.1.2.1 Addition 1.1.2.2 Subtraction 1.1.2.3 Multiplication 1.1.2.4 Euclidean Division 1.1.3 Fast Arithmetic 1.1.3.1 Karatsuba-Ofman Multiplication 1.1.3.2 Toom-Cook Multiplicaticn 1.1.3.3 FFT-Based Multiplication 1.1.4 An Introduction to GP/PARI 1.2 GCD 1.2.1 Euclidean GCD Algorithm 1.2.2 Extended GCD Algorithm 1.2.3 Binary GCD Algorithm 1.3 Congruences and Modular Arithmetic 1.3.1 Modular Exponentiation 1.3.2 Fast Modular Exponentiation 1.4 Linear Congruences 1.4.1 Chinese Remainder Theorem 1.5 Polynomial Ccngruences 1.5.1 Hensel Lifting 1.6 Quadratic Congruences 1.6.1 Quadratic Residues and Non-Residues 1.6.2 Legendre Symbol 1.6.3 Jaeobi Symbol 1.7 Multiplicative Orders 1.7.1 Primitive Roots 1.7.2 Coml:uting Orders 1.8 Continued Fractions 1.8.1 Finite Continued Fractions 1.8.2 Infinite Continued Fractions 1.9 Prime Number Theorem and Riemann Hypothesis 1.10 Running Times of Arithmetic Algorithms 2 Arithmetic of Finite Fields 2.1 Existence and Uniqueness of Finite Fields 2.2 Representation of Finite Fields 2.2.1 Polynomial-Basis Representation 2.2.2 Working with Finite Fields in GP/PARI 2.2.3 Choice of the Defining Polynomial 2.3 Implementation of Finite Field Arithmetic 2.3.1 Representation of Elements 2.3.2 Polynomial Arithmetic 2.3.2.1 Addition and Subtraction 2.3.2.2 Multiplication 2.3.2.3 Comb Methods 2.3.2.4 Windowed Comb Methods 2.3.2.5 Modular Reduction 2.3.3 Polynomial GCD and Inverse 2.3.3.1 Euclidean Inverse 2.3.3.2 Binary Inverse 2.3.3.3 Almost Inverse 2.4 Some Properties of Finite Fields 2.4.1 Fermat's Little Theorem for Finite Fields 2.4.2 Multiplicative Orders of Elements in Finite Fields 2.4.3 Normal Elements 2.4.4 Minimal Polynomials 2.4.5 Implementing Some Functions in GP/PARI 2.5 Alternative Representations of Finite Fields 2.5.1 Representation with Respect to Arbitrary Bases 2.5.2 Normal and Optimal Normal Bases 2.5.3 Discrete-Log Representation 2.5.4 Representation with Towers of Extensions 2.6 Computing Isomorphisms among Representations 3 Arithmetic of Polynomials 3.1 Polynomials over Finite Fields 3.1.1 Polynomial Arithmetic 3.1.2 Irreducible Polynomials over Finite Fields 3.1.3 Testing Irreducibility of Polynomials 3.1.4 Handling Irreducible Polynomials in GP/PARI 3.2 Finding Roots of Polynomials over Finite Fields 3.2.1 Algorithm for Fields of Odd Characteristics 3.2.2 Algorithm for Fields of Characteristic Two 3.2.3 Root Finding with GP/PARI 3.3 Factoring Polynomials over Finite Fields 3.3.1 Square-Free Factorization 3.3.2 Distinct-Degree Factorization 3.3.3 Equal-Degree Factorization 3.3.4 Factoring Polynomials in GP/PARI 3.4 Properties of Polynomials with Integer Coefficients 3.4.1 Relation with Polynomials with Rational Coefficients. 3.4.2 Height, Resultant, and Discriminant 3.4.3 Hensel Lifting 3.5 Factoring Polynomials with Integer Coefficients 3.5.1 Berlekamp's Factoring Algorithm 3.5.2 Basis Reduction in Lattices 3.5.3 Lenstra-Lenstra-Lov~sz Factoring Algorithm 3.5.4 Factoring in GP/PARI 4 Arithmetic of Elliptic Curves 4.1 What Is an Elliptic Curve? 4.2 Elliptic-Curve Group 4.2.1 Handling Elliptic Curves in GP/PARI 4.3 Elliptic Curves over Finite Fields 4.4 Some Theory of Algebraic Curves 4.4.1 Affine and Projective Curves 4.4.1.1 Affine Curves 4.4.1.2 Projective Curves ……
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