基于偶型高斯正规基乘法器设计

发布时间：2018-05-14 03:34

本文选题：椭圆曲线加密 + 有限域　；参考：《哈尔滨工业大学》2017年硕士论文

【摘要】：椭圆曲线加密算法在现实生活中的应用是非常广泛的,其加密效果是经过实践检验的。椭圆曲线加密算法的加解密过程会涉及到有限域上的基本的算术运算。而且本文所涉及的算术运算都是在有限域GF(2~m)中进行的。完成这些算术运算需要用到高效的乘法器。在有限域GF(2~m)中实现一个乘法器,那么这个乘法器的时间复杂度和空间复杂度跟它所在有限域中元素所使用的基有很大的关联。换句话说就是基决定效率。正规基,多项式基和对偶基是有限域中三种常用的基形式。每一种基的表示形式都有其独有的特性。正规基的最大优点就是在平方操作的时候只需要对元素进行循环移位操作就可以了。偶型高斯正规基属于正规基。而且偶型高斯正规基在探索乘法器效率的方面已经有了很广泛的应用。基于对空间复杂度的考虑,本文选取了偶型高斯正规基。本文的目的在于设计一种保证时间复杂度的前提下,尽可能使空间复杂度小的乘法器,提高椭圆曲线加密算法的效率。本文提出了三种乘法器结构并应用到偶型高斯正规基中。第一种是基于对称矩阵和向量相乘的乘法器结构;第二种是基于分块对称矩阵和向量相乘的乘法器结构;第三种是基于阵列式的乘法器结构。通过对三种乘法器的复杂度分析,三种乘法器结构在降低空间复杂度上都有很好的效果。三种乘法器结构都可以一定程度上提高椭圆曲线加密算法的效率。除了在空间复杂度上的优势以外,我们提出的三种乘法器共同的优点还在于,都能够统一乘法器的结构。对于高斯正规基中的乘积运算,本文提出的乘法器结构只需要一个乘法器就能解决,不需要多个乘法器并行。本文提出的乘法器结构比较适合应用到对空间复杂度要求比较严格的场景。
[Abstract]:Elliptic curve encryption algorithm is widely used in real life, and its encryption effect is verified by practice. The encryption and decryption process of elliptic curve encryption algorithm involves basic arithmetic operations on finite fields. Moreover, the arithmetic operations in this paper are all carried out in the finite field GF ~ (2 +). Efficient multipliers are needed to complete these arithmetic operations. The time and space complexity of the multiplier is related to the basis used by the elements in the finite domain. In other words, basic decision efficiency. Normal basis, polynomial basis and dual basis are three common basis forms in finite domain. Each representation of a base has its own unique characteristics. The greatest advantage of normal bases is that they only need to be rotated through the square operation. The Gao Si normal basis of even type belongs to the normal basis. And even Gao Si normal basis has been widely used in exploring multiplier efficiency. Considering the space complexity, we select the Gao Si normal basis of even type. The purpose of this paper is to design a multiplier with low space complexity to improve the efficiency of the elliptic curve encryption algorithm on the premise of ensuring the time complexity. In this paper, three multiplier structures are proposed and applied to even Gao Si normal bases. The first is a multiplier structure based on symmetric matrix and vector multiplication; the second is a multiplier structure based on block symmetric matrix and vector multiplication; the third is an array based multiplier structure. By analyzing the complexity of the three multipliers, the results show that the three multipliers have good performance in reducing the space complexity. All three multiplier structures can improve the efficiency of elliptic curve encryption algorithm to some extent. In addition to the advantages in space complexity, the common advantage of the three multipliers is that they can unify the structure of multipliers. For the product operation in Gao Si normal basis, the multiplier structure proposed in this paper needs only one multiplier to solve, and does not need multiple multipliers in parallel. The multiplier structure proposed in this paper is more suitable for scenarios with strict spatial complexity requirements.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：TP332.22

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