椭圆曲线上与密码算法相关的计算问题

项目名称： 椭圆曲线上与密码算法相关的计算问题

项目编号： No.61272035

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 自动化技术、计算机技术

项目作者： 王明强

作者单位： 山东大学

项目金额： 61万元

中文摘要： 本课题主要研究在椭圆曲线密码算法中广泛应用的椭圆曲线上的离散对数、点乘和双线性对的计算问题。本课题拟利用椭圆曲线函数域的扩域的性质,研究曲线之间映射的特殊性质,找到合适的曲线提高某些特殊类型椭圆曲线上离散对数问题的计算速度；利用椭圆曲线点群的子集的性质,研究构造较少个数的因子基的方法,提高椭圆曲线上离散对数问题的计算速度；利用曲线上点乘运算与曲线方程的某些参数无关,研究基于超椭圆曲线的离散对数困难问题的密码算法的植入错误攻击；利用特殊类型的椭圆曲线上的自同态映射，将窗口算法应用到特殊类型的椭圆曲线点乘的计算中；利用椭圆曲线上不同坐标表示的点加公式，研究加法公式在窗口算法预计算中的应用,提高一次点加计算的效率；利用构造范数较小的椭圆曲线方法, 研究椭圆曲线上双线性对的构造方法，构造需要Miller迭代次数尽可能少的双线性对。

中文关键词： 椭圆曲线；离散对数；密码分析；格算法；分组密码

英文摘要： This subject focus on the computation problems of discrete logarithm, point multiplication and bilinear pairing which are main parts of elliptic curve and widely applied in cryptography. For some special elliptic curves, we utilize the properties of the mapping between curves and the extention of function fields to find out some proper elliptic curves on which the discrete logarithm problems can be efficiently solved, thereby accelerate the algorithm complexity of discrete logarithm problems. We research on the method of constructing smaller factor basis utilizing the properties of the subset of the point group of elliptic curves, and thereby improve the computation complexity of the elliptic curve discrete logarithm problem. We research on the fault attack on the discrete logarithm problem of hyperelliptic curves utilizing the property that the point multiplication is independent of some parameters. Using the endomorhpism over some special elliptic curves, We try to apply the window method to the computation of point multiplication on these curves. We research on the different addition formulas with different representation of coordinate and the application of these formulas in the pre-computation of window method to improve the efficiency of point adding calculation. We research on the method of constructing t

英文关键词： Hyperelliptic Curve；Discrete Logarithm；Cryptanalysis；lattice Algorithm；Block-cipher