The Magic of Elliptic Curves and Public-Key Cryptography

Florian Heß; Andreas Stein; Sandra Stein; Manfred Lochter

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The Magic of Elliptic Curves and Public-Key Cryptography

机译：椭圆曲线和公共密钥密码术的魔力

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摘要

Elliptic curves are beautiful mathematical objects that again and again appear in the most surprising places. Their history certainly originates at least in ancient Greece, whereas the study of arithmetic properties of elliptic curves as objects in algebra, geometry, and number theory traces back to the nineteenth century. Curiously, the earliest use of the term "elliptic curve" in the literature seems to have been by James Thomson in 1727 in "A Poem sacred to the Memory of Sir Isaac Newton": "He, first of Men, with awful Wing pursu'd the Comet tro' the long Elliptic Curve." In 1985, Koblitz and Miller independently proposed to use elliptic curves in cryptography which can only be described as a magnificent and practical application of elliptic curves. This paper intends to mostly present a low-brow introduction of elliptic curves and their use in real-world applications of public-key cryptography.

机译：椭圆曲线是美丽的数学对象，它一次又一次出现在最令人惊讶的位置。他们的历史至少可以追溯到古希腊，而椭圆曲线作为代数，几何和数论中的对象的算术特性的研究则可以追溯到19世纪。奇怪的是，最早在文学中使用“椭圆曲线”一词的人似乎是詹姆斯·汤姆森（James Thomson）于1727年在“纪念以撒·牛顿爵士的一首诗”中：“他，第一个人类，怀着可怕的Wing pursu” d彗星长椭圆形曲线。” 1985年，Koblitz和Miller独立提出在加密技术中使用椭圆曲线，这只能描述为椭圆曲线的一种宏伟而实际的应用。本文打算主要介绍椭圆曲线的低端介绍及其在公共密钥密码术的实际应用中的使用。

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来源
《Jahresbericht der Deutschen Mathematiker-Vereinigung》 |2012年第2期|p.59-88|共30页
作者
Florian Heß; Andreas Stein; Sandra Stein; Manfred Lochter;
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作者单位

Institut fur Mathematik, Carl von Ossietzky Universitat Oldenburg, 26111 Oldenburg, Germany;

Institut fur Mathematik, Carl von Ossietzky Universitat Oldenburg, 26111 Oldenburg, Germany;

Institut fur Mathematik, Carl von Ossietzky Universitat Oldenburg, 26111 Oldenburg, Germany;

Bundesamt fur Sicherheit in der Informationstechnik (BSI), Postfach 200363, 53133 Bonn, Germany;

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正文语种 eng
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cryptography; elliptic curves; discrete logarithm problem; public-keycryptography; pairing-based cryptosystem; weil pairing; tate-lichtenbaumpairing; side channel analysismathematics;

机译：密码学椭圆曲线离散对数问题公钥密码术基于配对的密码系统;配对tate-lichtenbaumpairing;旁道分析数学;

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专利

1. RSA and Public-Key Cryptography Introduction to Cryptography Cryptography: Theory and Practice Algebraic Aspects of Cryptography Elliptic Curves: Number Theory and Cryptography Elliptic Curves in Cryptography Modern Cryptography, Probabilistic Proofs, and Pseudorandomness Foundations of Cryptography: Basic Tools The Design of Rijndael: AES --- the Advanced Encryption Standard Handbook of Applied Cryptography [J] . Susan Landau Bulletin of the American Mathematical Society . 2004,第3期

机译：RSA和公钥密码学密码学简介密码学：密码学的理论和实践椭圆曲线的密码学：密码学中的数论和密码学椭圆曲线现代密码学，概率证明和伪随机性基础密码学：基本工具Rijndael的设计： -应用密码学高级加密标准手册
2. Constructing public-key cryptographic schemes based on class group action on a set of isogenous elliptic curves [J] . Stolbunov A. Advances in mathematics of communications . 2010,第2期

机译：基于一组同构椭圆曲线上的类组操作构造公共密钥加密方案
3. RGB based multiple share creation in visual cryptography with aid of elliptic curve cryptography [J] . K. Shankar, P. Eswaran Communications, China . 2017,第2期

机译：借助椭圆曲线密码术在视觉密码术中基于RGB的多个共享创建
4. Study of an Electronic Seal System Based on Elliptic Curve Cryptography and Public-key Infrastructure [C] . Xu Jian, Ma Bin International Conference on Future Computer and Communication;ICFCC . 2010

机译：基于椭圆曲线密码学和公钥基础设施的电子印章系统的研究
5. Tutorial on Elliptic Curve Arithmetic and Introduction to Elliptic Curve Cryptography (ECC) [D] . Bommireddipalli, Nithesh Venkata Ramana Surya. 2017

机译：椭圆曲线算术教程和椭圆曲线密码学（ECC）简介
6. iTrust—A Trustworthy and Efficient Mapping Scheme in Elliptic Curve Cryptography [O] . Hisham Almajed, Ahmad Almogren, Mohammed Alabdulkareem 2020

机译：ITRUST-A Queltworthy且有效的椭圆曲线密码映射方案
7. A Public-Key Infrastructure for Key Distribution in TinyOS Based on Elliptic Curve Cryptography [O] . David J. Malan, Matt Welsh, Michael D. Smith 2004

机译：基于椭圆曲线密码的TinyOs密钥分配公钥基础设施

The Magic of Elliptic Curves and Public-Key Cryptography

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