Handbook of Elliptic and Hyperelliptic Curve Cryptography最新文献

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Implementation of Pairings 配对的实现
Handbook of Elliptic and Hyperelliptic Curve Cryptography Pub Date : 2005-07-19 DOI: 10.1201/9781420034981.ch16
S. Duquesne, G. Frey
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引用次数: 6
Index Calculus for Hyperelliptic Curves 超椭圆曲线的指数演算
Handbook of Elliptic and Hyperelliptic Curve Cryptography Pub Date : 2005-07-19 DOI: 10.1201/9781420034981.ch21
R. Avanzi, N. Thériault
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引用次数: 2
Arithmetic of Hyperelliptic Curves 超椭圆曲线的算法
Handbook of Elliptic and Hyperelliptic Curve Cryptography Pub Date : 2005-07-19 DOI: 10.1201/9781420034981.ch14
S. Duquesne, T. Lange
{"title":"Arithmetic of Hyperelliptic Curves","authors":"S. Duquesne, T. Lange","doi":"10.1201/9781420034981.ch14","DOIUrl":"https://doi.org/10.1201/9781420034981.ch14","url":null,"abstract":"In Chapter 1 we introduced the discrete logarithm problem and showed that the main operation in a public-key cryptosystem is the computation of scalar multiples in a cyclic group. Chapter 9 showed how the computation of scalar multiples can be reduced to a sequence of additions and doublings in the group. Hence, for an efficient system we need to have groups with efficient group laws. In Chapter 13 we detailed the arithmetic on elliptic curves. This chapter deals with hyperelliptic curves, which can be seen as a generalization of elliptic curves. We first give a brief overview of the main properties of hyperelliptic curves repeating the definitions for the convenience of the reader. The details can be found in Chapter 4 . In the applications, group elements must be stored and transmitted. For restricted environments or restricted bandwidth it might be useful to use compression even though recovering the original coordinates needs some efforts. Accordingly, we consider compression techniques. The main emphasis of this chapter is put on the arithmetic properties, i.e., on algorithms to perform the group operation. We state Cantor’s algorithm, which works for arbitrary ground field and genus of the curve. To obtain better performance one needs to fix the genus and develop explicit formulas as for elliptic curves (cf. Chapters 13.2 and 13.3). We first specialize to considering curves of genus 2 , separately over finite fields of odd and then of even characteristic. For both cases we give formulas for different coordinate systems, namely affine, projective, and new coordinates. The latter two systems allow us to avoid inversions in the group operation. For odd characteristic we also state two possible generalizations of Montgomery coordinates (cf. Section 13.2.3); for even characteristic there is no such generalization yet. Also for genus 3 hyperelliptic curves, explicit formulas have been proposed. We give explicit formulas in affine coordinates in Section 14.6. Also nonhyperelliptic curves of genus 3 have been proposed for cryptographic applications. The final section gives references to these publications and also for genus 4 hyperelliptic curves before we conclude with a comparison and timings.","PeriodicalId":131128,"journal":{"name":"Handbook of Elliptic and Hyperelliptic Curve Cryptography","volume":"73 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":"126758126","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: 5
Integer Arithmetic 整数运算
Handbook of Elliptic and Hyperelliptic Curve Cryptography Pub Date : 1900-01-01 DOI: 10.1201/9781420034981.ch10
Christophe Doche
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引用次数: 8
Smart Cards 智能卡
Handbook of Elliptic and Hyperelliptic Curve Cryptography Pub Date : 1900-01-01 DOI: 10.1201/9781420034981.ch27
Bertrand Byramjee, A. Weigl
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引用次数: 0
Algebraic Background 代数背景
Handbook of Elliptic and Hyperelliptic Curve Cryptography Pub Date : 1900-01-01 DOI: 10.1201/9781420034981.pt1
C. Doche, D. Lubicz
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引用次数: 1
Complex Multiplication 复杂的乘法
Handbook of Elliptic and Hyperelliptic Curve Cryptography Pub Date : 1900-01-01 DOI: 10.1201/9781420034981.ch18
G. Frey, T. Lange
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引用次数: 247
Index Calculus 指数计算
Handbook of Elliptic and Hyperelliptic Curve Cryptography Pub Date : 1900-01-01 DOI: 10.1201/9781420034981.ch20
R. Avanzi, N. Thériault
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引用次数: 0
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