Research on public-key cryptosystems from hyperelliptic-curves

基于超椭圆​​曲线的公钥密码系统研究

• 批准号: 11558033
• 负责人: SAKURAI Kouichi
• 金额: $ 3.97万
• 依托单位: Kyushu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11558033/
• 关键词: cryptography; information security; hyperelliptic curve; public-key encryption; algorithm; cryptanalysis; elliptic curve; fast computation; モンゴメリー型; 梗塞演算

We have designed hyperelliptic curve cryptosystems with considering security and efficiency.We have implemented our designed hyperelliptic curve cryptosystems both over software and over hardware, and confirm their practical performance.We consider the performance of hyperelliptic curve cryptosystems over GF(p) vs. over GF(2^n).We analyze the complexity of the group law of Jacobians and make comparison of their performance between over over GF(p) vs. over GF(2^n). with considering the effectiveness of the word size (32-bit or 64-bit) of the applied CPU (Alpha and Pentium) on the arithmetic on the definition field.We also develop efficient algorithms for the jacobian of the hyperelliptic curve defined by the equation $y^2 = x^p-x+1$ over a finite field GF(p^n) of odd characteristic p. We first determine the zeta function of the curve which yields the order of the jacobian. And we investigate the Frobenius operator and use it to show that, for field extensions GF(p^n) of degree n prime to p, the jacobian has a cyclic group structure. We furthermore propose a method for faster scalar multiplication in the jacobian by using efficient operators other than the Frobenius that have smaller eigenvalues.

我们从安全性和效率两个方面设计了超椭圆曲线密码体制，并在软件和硬件上实现了所设计的超椭圆曲线密码体制，研究了GF（p）上的超椭圆曲线密码体制与GF（2 ^n）上的超椭圆曲线密码体制的性能，分析了Jacobian群律的复杂性，并比较了GF（p）上的超椭圆曲线密码体制与GF（2 ^n）上的超椭圆曲线密码体制的性能。相对于GF（2^n）。考虑到单词大小的有效性，（32位或64位）的应用CPU（Alpha和Pentium）在定义域上的算法，我们还发展了有限域GF（p^n）上由方程y ^2 = x^p-x+1$定义的超椭圆曲线的雅可比矩阵的有效算法我们首先确定zeta函数的曲线产生的顺序雅可比。我们还研究了Frobenius算子，并利用它证明了对于n次与p互素的域扩张GF（p^n），雅可比矩阵具有循环群结构。我们还提出了一种通过使用除Frobenius之外的具有较小特征值的高效运算符来实现雅可比矩阵中更快的标量乘法的方法。

项目成果

酒井康行,櫻井幸一: "有限体F_<2^n>上の超楕円曲線暗号のソフトウエア実装"電子情報通信学会論文誌. J82-A,No.8. 1305-1306 (1999)

Yasuyuki Sakai、Koichi Sakurai：“有限域 F_<2^n> 上超椭圆曲线密码学的软件实现”，电子、信息和通信工程师学会汇刊 J82-A，第 8 期。1305-1306 (1999)。

Okeya, K., Sakurai, K.: "Efficient elliptic curve cryptosystems from a scalar multiplication algorithm with recovery of the y-coordinate on a Montgomery-form elliptic curve"Proc. Workshop on Cryptographic Hardware and Embedded Systems 2001 (May 2001, Pari

Okeya, K.、Sakurai, K.：“基于标量乘法算法的高效椭圆曲线密码系统，可恢复蒙哥马利型椭圆曲线上的 y 坐标”Proc。

Masato YAMAMICHI, Masahiro MAMBO, Hiroki SHIZUYA: "On the Complexity of Constructing an Elliptic Curve of a Given Order"Ieice Trans.. E84-A・No.1. 140-145 (2001)

Masato YAMAMICHI、Masahiro MAMBO、Hiroki SHIZUYA：“论构造给定阶数的椭圆曲线的复杂性”Ieice Trans.. E84-A·No.1 (2001)。

CHIDA,OHMORI,and SHIZUYA: "A Way of Making Trapdoor One-Way Functions Trapdoor No-Way"IEICE Trans.Fundamentals. E84-A,1. 151-156 (2001)

CHIDA、OHMORI 和 SHIZUYA：“一种制作活板门单向函数活板门无路的方法”IEICE Trans.Fundamentals。

Y. Sakai and K. Sakurai: "Over F_p vs.F_<2^n> over and on Pentium vs. Alpha in Software Implementation of Hyperelliptic Curve"PreProc, 1999 International Conference on Information Security and Cryptology December 9-10,1999 Korea University,Seoul,Korea. 67

Y. Sakai 和 K. Sakurai：“Over F_p vs.F_<2^n> over and on Pentium v​​s. Alpha in Software Implementing of Hyperelliptic Curve”PreProc，1999 年信息安全与密码学国际会议，1999 年 12 月 9-10 日，韩国