GF(2m)域高效椭圆曲线标量乘结构的研究

为提高椭圆曲线加密运算的速度,提出了一种多项式基表示的GF(2m)域高效标量乘加速器结构.该结构对面积和性能进行了合理的权衡,采用点加、倍点模块并行运算以提高速度;为了减少面积采用并行和串行相结合的方法对点加和倍点模块进行优化,初始化和最后的坐标变换求逆模块通过优化分解成一系列乘和加运算,合并在一个模块中用串行结构实现.采用Xilinx公司的VirtexE XCV2600 FPGA硬件实现结果表明,完成有限域GF(2163)上任意椭圆曲线上的一次标量乘的全部运算时间消耗为36.5μs,适合高性能椭圆曲线加密应用的要求.     

GF(2m)域上椭圆曲线点积算法的一种改进

提高椭圆曲线点积运算的效率是椭圆曲线研究的一个核心问题。文章对有限域GF(2m)上的椭圆曲线的点积运算作了较为深入的研究,并利用正则的二进制冗余序列构造了一种新的窗口算法,从算法的效率比较来看,本算法有一定的提高。     

Elliptic Curve Scalar Multiplication with x-Coordinate

This paper gives a comprehensive method to do Elliptic Curve Scalar Multiplication with only x-coordinate. Explicit point operation formulae for all types of defining equations of the curves are derived. For each type of curve, the performance is analyzed. The formulae are applied in Montgomery Ladder to get scalar multiplication algorithm operated with only x-coordinate. The new scalar multiplication has the same security level and computation amount with protected binary scalar multiplication （PBSM） against side channel attack, and has the advantages of higher security and little memory needed.     

Addition Sequence Method of Scalar Multiplication of Elliptic Curve over OEF

A new elliptic curve scalar multiplication algorithm is proposed. The algorithm uses the Frobenius map on optimal extension field (OEF) and addition sequence. We introduce a new algorithm on generating addition sequence efficiently and also give some analysis about it. Based on this algorithm, a new method of computing scalar multiplication of elliptic curve over an ()EF is presented. The new method is more efficient than the traditional scalar multiplication algorithms of elliptic curve over OEF. The comparisons of traditional method and the new method are also given.     

有限域GF(pm)上圆锥曲线标量乘快速算法

圆锥曲线密码体制是一种新型的公钥密码体制,得到了广泛关注,在已有研究成果中,圆锥曲线都是定义在大的素域GF(p)上、特征为2的有限域GF(2m)上、或剩余类环Z/nZ上,其中n=pq是两个奇素数的乘积.为实现更高效的圆锥曲线密码体制,本文讨论有限域GF(pm)上的圆锥曲线,并定义了其上的Frobenius映射,基于此设计了新的快速标量乘算法.理论分析和数值结果都表明,在使用同等的预存储空间的前提下,新算法的时间复杂度较传统算法有很大程度的降低.     

一种改进的2w-ary快速标量乘算法

标量乘是椭圆曲线密码体制(ECC)的基本运算,也是最耗时和极易受到攻击的运算之一。针对利用重编码和R-L技术实现2w-ary快速标量乘算法的不足,对该算法进行了改进。在保持抗SPA的前提下,改进算法使用绝对值,避免了负数做数组下标的使用,减少了50%的存储量,节约了赋值后的消除操作。理论分析表明,改进算法优于原算法,算法的运算量降低2w次。数字验证表明,改进算法比原算法快约17%。     

基于双基数子集表示的快速标量乘算法

为了减少双基数表示时间,提出双基数子集的概念,将存储空间和预计算时间从n2 降低到3n-2,同时给出求子集表示的贪婪算法.给出基于双基数子集的标量乘算法,由于搜索空间的降低,整数的双基数子集表示个数稍微多于双基数表示个数,新算法使用混合坐标进行优化.从运算量、存储空间上,将新算法分别与传统固定基点的窗口算法和Dimitrov的基于双基数链的算法进行比较,结果证明新算法显著优于前人算法.     

Steinitz class of elliptic curves with complex multilication

Let E be any elliptic curve having complex multiplication by the ring CK of integers of the quadratic number field K= Q(- D). Let H be the Hilbert class field of K. The Mordell-Weil group E(H) of H-rational points is a module over the Dedekind domain CK, its structure depends on its Steinitz class. Here the Steinits class is determined when D is any prime number. This result advances the result for the specific elliptic curves when D=10.A general theorem on structure of modules over Dedekind domain is also proposed.     

3特征域椭圆曲线群点的快速计算算法

GF(3m)作为小素数扩域GF(Pm)更加特殊的一种类型,定义于其上的椭圆曲线密码算法更加优越.基于有限域GF(3m)特殊性质,详细研究了GF(3m)上椭圆曲线群基本点算术运算,给出并证明了GF(3m)上超奇异和非超奇异椭圆曲线在仿射、射影、雅可比、Chudnovsky雅可比和López Dahab坐标系下普通点加、混...     

GF（2^m）域上椭圆曲线点乘算法的改进

介绍了在GF(2m)域上实现非超异椭圆曲线的点乘的算法,它是Montgomery 算法的改进.该算法无需乘法预处理,运算速度快于IEEE P1363草案标准上"加-减"算法,而且占用的内存资源少,易于软、硬件方式的实现.因此,该算法更利于在那些资源有限的环境中实现椭圆曲线加密体制.     

CM elliptic curves and p-adic valuations of their L-series at s=1

For rational integers γ and λ, consider two families of (-1) and (-3) respectively. General formulae expressed by Weierstrass -functions are given for special values of Hecke L-series attached to such elliptic curves. The uniform lower bounds of 2-adic and 3-adic valuations of these values of Hecke L-series as well as global criteria for reaching these bounds are obtained. Moreover, when γ=2 and λ=2, 4, further results of 2-adic and 3-adic valuations are obtained for the corresponding curves in more general case of D1 and some restricted D2 respectively. These results are consistent with the predictions of the conjecture of Birch and Swinnerton-Dyer, and greatly develop and generalize some results in recent literature for more special cases.     

Formula and 2-adic valuation of L(1) of elliptic curves with CM by -3

For the rational integers λ≡1, 3, or 5 (mod 6), considering elliptic curves y2＝x3-2433Dλ over the field (-3), the formula for the value at s=1 of Hecke L-series attached to such elliptic curves, expressed as a finite sum of values of Weierstrass -functions, is obtained. Moreover, when λ≡3 (mod 6), the lower bounds of 2-adic valuations of these values are also obtained. These results are consistent with the predictions of the conjecture of Birch and Swinnerton-Dyer in a sense, and have generalized and advanced some results in recent literature.     

一种素数域上的非超奇椭圆曲线构造方案

对基域为素数域的非超奇椭圆曲线进行了研究，给出了构造素域上具有特定素数阶的非超奇椭圆曲线的一种算法，分析了算法时间复杂度．     

一种适用于无线网络的椭圆曲线标量乘算法研究

单标量乘和多点标量乘是椭圆曲线密码体制中最主要的运算,在各种密码协议中起到核心作用。提出一种基于Shamir方法优化预计算的多标量乘算法,算法对标量从左到右进行编码,并将编码阶段与多标量乘的主计算阶段合并进行,节省了存储标量新编码的内存空间,更适合于内存受限的无线环境中应用。分析表明新算法在一般情况下效率可提高6％～50％。     

有限域F2n上安全椭圆曲线的构造

详细阐述了椭圆曲线密码系统的安全性及其理论,讨论了在有限域F2n上寻找安全椭圆曲线的基本思想,并利用l-adic的基本思想给出了在特征为2的有限域F2n上构造安全椭圆曲线的有效算法,设计实现了该算法并获得了实验结果.所构造的椭圆曲线可提高应用系统的安全性.     

椭圆曲线密码的一种合适的对算法

在椭圆曲线密码的应用中,有些密码体制需要进行标量乘法对计算,通常在标量乘法计算时,任一整数对采用的是三元联合稀疏形式表示.在三元稀疏形式的基础上提出任一整数对的五元联合稀疏形式表示,并把这种形式用于快速Shamir算法计算标量乘法对,并证明五元联合稀疏形式比三元联合形式更有效.     

A complete set of addition laws for twisted jacobi intersection curves

A complete system of addition laws on an elliptic curve E is a collection of addition laws with the property that for any pair of points P1,P2 on E at least one of the addition laws in the collection can be used to compute P1+P2.This paper proposes a complete set of the addition laws for arbitrary twisted Jacobi intersection curve.     

基于椭圆曲线密码体制的无线传感器网络对偶密钥管理的研究

提出了一种适用于无线传感器网络的基于椭圆曲线密码体制的对偶密钥管理方案,引入了证书服务,可以实现密钥的分配、更新、撤销以及新加入节点时的密钥生成．分析结果表明,与现有的密钥更新方案比较,基于椭圆曲线密码体制的无线传感器密钥更新方案有效地减少了密钥的存储空间;在密钥更新阶段中加入保密和认证功能,增加了网络攻击的难度．