一类椭圆曲线的特征多项式计算
发布时间：2018-08-05 15:15
【摘要】：椭圆曲线密码体制（ECC），可以看做是基于有限域上离散对数问题（ECDLP）的公钥密码体制在椭圆曲线上的推广。目前，由于其它公钥密码体制的有效攻击算法是压指数时间的，而ECDLP是完全指数时间的，这就意味着，，在相同安全条件下，ECC密钥长度比其它密码体制的密钥长度更短。这带来的优势是，椭圆曲线能够用较小的开销(如宽带、计算量、软硬件实现规模、存储量等)和时延（如加密和签名速度等）来实现较高的安全性。因此，ECC特别适用于集成电路、宽带和计算机能力受限的情况如Smart卡、无线通信和某些计算机网络等。 椭圆曲线特征多项式的计算，对加快Jacobian群上的除子标量乘和提高ECC实现速度有着重要意义。同时，对构造安全的双线性对密码体制的加密、签名和密钥协商方案，也有实际意义。 在研究椭圆曲线特性时，一般从同构曲线入手，因为同构的曲线具有相同的特征多项式和群结构。本文主要研究了一类Jacobian四次曲线E20: y=x4+ax2+b,其中， a,b∈F和素数域Fq上的超奇异的椭圆曲线，并分别计算了其特征多项式。主要工作包括以下几个方面： （1）第一章首先介绍了ECC的研究现状以及一些亟待解决的关键问题，然后重点归纳了现有的、求解椭圆曲线特征多项式方法，主要包括：ECC求阶算法，经典曲线提升法，Selberg迹公式和指数方法研究有理点分布，曲线同构类计算，特殊曲线的特征多项式计算。 （2）第三章主要介绍了有限域上两类经典的求阶算法：Schoof算法和SEA算法，并提出了袋鼠加速、大步小步（BSGS）改进策略，改进算法在原算法的基础上提高了30%和6%左右。 （3）第四章讨论了一类Jacobia四次曲线E20: y=x4+ax2+b,并根据其二次特征的性质，分三类情况探讨了该Jacobia四次曲线的有理点个数和特征多项式。 （4）第五章讨论了素数域Fq上超奇异Weistrass曲线E21: y+a1xy+a3y=x3+a2x2+a4x+a6的特征多项式，其ai∈Fq, q=pm， m为任意正整数。我们首先介绍了E1曲线的同构类，然后分别讨论各个同构类的特征多项式。
[Abstract]:The elliptic curve cryptosystem (ECC),) can be regarded as a generalization of the public key cryptosystem based on the discrete logarithm problem (ECDLP) on the elliptic curve. At present, because the effective attack algorithms of other public-key cryptosystems are exponential time, and ECDLP is completely exponential time, this means that the length of ECDLP keys is shorter than that of other cryptosystems under the same security conditions. The advantage of this is that elliptic curves can achieve higher security with lower overhead (such as broadband, computation, hardware and software implementation scale, storage capacity, etc.) and delay (such as encryption and signature speed). Therefore, ECC is especially suitable for integrated circuits, broadband and limited computer capabilities such as Smart cards, wireless communications and some computer networks. The calculation of characteristic polynomials of elliptic curves is of great significance to accelerate the multiplication of divider scalars on Jacobian groups and to improve the speed of ECC realization. At the same time, it is of practical significance to construct a secure bilinear cryptosystem encryption, signature and key agreement scheme. When we study the characteristics of elliptic curves, we usually start with isomorphism curves, because the isomorphic curves have the same characteristic polynomial and group structure. In this paper, we mainly study a class of Jacobian quartic curves E20: y=x4 ax2 b, where a b 鈭
本文编号：2166185
[Abstract]:The elliptic curve cryptosystem (ECC),) can be regarded as a generalization of the public key cryptosystem based on the discrete logarithm problem (ECDLP) on the elliptic curve. At present, because the effective attack algorithms of other public-key cryptosystems are exponential time, and ECDLP is completely exponential time, this means that the length of ECDLP keys is shorter than that of other cryptosystems under the same security conditions. The advantage of this is that elliptic curves can achieve higher security with lower overhead (such as broadband, computation, hardware and software implementation scale, storage capacity, etc.) and delay (such as encryption and signature speed). Therefore, ECC is especially suitable for integrated circuits, broadband and limited computer capabilities such as Smart cards, wireless communications and some computer networks. The calculation of characteristic polynomials of elliptic curves is of great significance to accelerate the multiplication of divider scalars on Jacobian groups and to improve the speed of ECC realization. At the same time, it is of practical significance to construct a secure bilinear cryptosystem encryption, signature and key agreement scheme. When we study the characteristics of elliptic curves, we usually start with isomorphism curves, because the isomorphic curves have the same characteristic polynomial and group structure. In this paper, we mainly study a class of Jacobian quartic curves E20: y=x4 ax2 b, where a b 鈭
本文编号：2166185
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