Handbook of Elliptic and Hyperelliptic Curve Cryptography最新文献

Handbook of Elliptic and Hyperelliptic Curve Cryptography Pub Date : 2005-07-19 DOI: 10.1201/9781420034981.ch5

G. Frey, T. Lange

引用次数: 11

Background on Weil Descent Weil Descent的背景

Handbook of Elliptic and Hyperelliptic Curve Cryptography Pub Date : 2005-07-19 DOI: 10.1201/9781420034981.ch7

G. Frey, T. Lange

{"title":"Background on Weil Descent","authors":"G. Frey, T. Lange","doi":"10.1201/9781420034981.ch7","DOIUrl":"https://doi.org/10.1201/9781420034981.ch7","url":null,"abstract":"Weil descent — or, as it is alternatively called — scalar restriction, is a well-known technique in algebraic geometry. It is applicable to all geometric objects like curves, differentials, and Picard groups, if we work over a separable field L of degree d of a ground field K. It relates t-dimensional objects over L to td-dimensional objects over K. As guideline the reader should use the theory of algebraic curves over C, which become surfaces over R. This example, detailed in Section 5.1.2, already shows that the structure of the objects after scalar restriction can be much richer: the surfaces we get from algebraic curves carry the structure of a Riemann surface and so methods from topology and Kahler manifolds can be applied to questions about curves over C. This was the reason to suggest that Weil descent should be studied with respect to (constructive and destructive) applications for DL systems [FRE 1998]. We shall come to such applications in Sections 15.3 and 22.3. In the next two sections we give a short sketch of the mathematical properties of Weil descent. The purpose is to provide a mathematical basis for the descent and show how to construct it. For a thorough discussion in the frame of algebraic geometry and using the language of schemes, we refer to [Die 2001]","PeriodicalId":131128,"journal":{"name":"Handbook of Elliptic and Hyperelliptic Curve Cryptography","volume":"103 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121458602","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Background on Pairings 配对的背景

Handbook of Elliptic and Hyperelliptic Curve Cryptography Pub Date : 2005-07-19 DOI: 10.1201/9781420034981.ch6

S. Duquesne, G. Frey

引用次数: 21

Pairing-Based Cryptography 双线性密码

Handbook of Elliptic and Hyperelliptic Curve Cryptography Pub Date : 2005-07-19 DOI: 10.1201/9781420034981.ch24

S. Duquesne, T. Lange

{"title":"Pairing-Based Cryptography","authors":"S. Duquesne, T. Lange","doi":"10.1201/9781420034981.ch24","DOIUrl":"https://doi.org/10.1201/9781420034981.ch24","url":null,"abstract":"Chapter 23 showed us how to build DL systems on the Jacobian of curves. In Chapter 1 we introduced DL systems with bilinear structure. In this chapter we first give more applications of this construction, namely the extension of the tripartite protocol given before to multiparty key exchange, identity-based cryptography, and short signatures. In recent years many systems using this extra structure have been proposed. We include some more references to further work in the respective sections, since giving a complete survey of all these schemes is completely out of the scope of this book. For a collection of results on pairings we refer to the \"Pairing-Based Crypto Lounge\" [BAR]. The second section is devoted to realizations of such systems. In Chapter 6 we gave the mathematical theory for the Tate–Lichtenbaum pairing and Chapter 16 provided algorithms for efficient evaluation of this pairing on elliptic curves and the Jacobian of hyperelliptic curves. There we assumed that the embedding degree (i.e., the degree k of the extension field Fqk to which the pairing maps), is small, so as to guarantee an efficiently computable map as required in a DL system with bilinear structure. In Section 24.2 we explain for which curves and fields these requirements can be satisfied and give constructions.","PeriodicalId":131128,"journal":{"name":"Handbook of Elliptic and Hyperelliptic Curve Cryptography","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115346878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 90

Arithmetic of Special Curves 特殊曲线的算法

Handbook of Elliptic and Hyperelliptic Curve Cryptography Pub Date : 2005-07-19 DOI: 10.1201/9781420034981.ch15

C. Doche, T. Lange

引用次数: 3

Compositeness and Primality Testing Factoring 组合性和素数检验分解

Handbook of Elliptic and Hyperelliptic Curve Cryptography Pub Date : 2005-07-19 DOI: 10.1201/9781420034981.ch25

R. Avanzi, H. Cohen

引用次数: 1

Background on p-adic Numbers p进数的背景知识

Handbook of Elliptic and Hyperelliptic Curve Cryptography Pub Date : 2005-07-19 DOI: 10.1201/9781420034981.ch3

D. Lubicz

引用次数: 0

Arithmetic of p-adic Numbers p进数的算术

Handbook of Elliptic and Hyperelliptic Curve Cryptography Pub Date : 2005-07-19 DOI: 10.1201/9781420034981.ch12

F. Vercauteren

引用次数: 0

Background on Curves and Jacobians 曲线和雅可比矩阵的背景知识

Handbook of Elliptic and Hyperelliptic Curve Cryptography Pub Date : 2005-07-19 DOI: 10.1201/9781420034981.ch4

G. Frey, T. Lange

引用次数: 7

Generic Algorithms for Computing Discrete Logarithms 计算离散对数的通用算法

Handbook of Elliptic and Hyperelliptic Curve Cryptography Pub Date : 2005-07-19 DOI: 10.1201/9781420034981.pt5

R. Avanzi

引用次数: 6