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  本文選題：橢圓曲線加密 + 有限域��； 參考：《哈爾濱工業(yè)大學》2017年碩士論文

【摘要】：橢圓曲線加密算法在現(xiàn)實生活中的應用是非常廣泛的,其加密效果是經(jīng)過實踐檢驗的。橢圓曲線加密算法的加解密過程會涉及到有限域上的基本的算術運算。而且本文所涉及的算術運算都是在有限域GF(2~m)中進行的。完成這些算術運算需要用到高效的乘法器。在有限域GF(2~m)中實現(xiàn)一個乘法器,那么這個乘法器的時間復雜度和空間復雜度跟它所在有限域中元素所使用的基有很大的關聯(lián)。換句話說就是基決定效率。正規(guī)基,多項式基和對偶基是有限域中三種常用的基形式。每一種基的表示形式都有其獨有的特性。正規(guī)基的最大優(yōu)點就是在平方操作的時候只需要對元素進行循環(huán)移位操作就可以了。偶型高斯正規(guī)基屬于正規(guī)基。而且偶型高斯正規(guī)基在探索乘法器效率的方面已經(jīng)有了很廣泛的應用。基于對空間復雜度的考慮,本文選取了偶型高斯正規(guī)基。本文的目的在于設計一種保證時間復雜度的前提下,盡可能使空間復雜度小的乘法器,提高橢圓曲線加密算法的效率。本文提出了三種乘法器結(jié)構(gòu)并應用到偶型高斯正規(guī)基中。第一種是基于對稱矩陣和向量相乘的乘法器結(jié)構(gòu);第二種是基于分塊對稱矩陣和向量相乘的乘法器結(jié)構(gòu);第三種是基于陣列式的乘法器結(jié)構(gòu)。通過對三種乘法器的復雜度分析,三種乘法器結(jié)構(gòu)在降低空間復雜度上都有很好的效果。三種乘法器結(jié)構(gòu)都可以一定程度上提高橢圓曲線加密算法的效率。除了在空間復雜度上的優(yōu)勢以外,我們提出的三種乘法器共同的優(yōu)點還在于,都能夠統(tǒng)一乘法器的結(jié)構(gòu)。對于高斯正規(guī)基中的乘積運算,本文提出的乘法器結(jié)構(gòu)只需要一個乘法器就能解決,不需要多個乘法器并行。本文提出的乘法器結(jié)構(gòu)比較適合應用到對空間復雜度要求比較嚴格的場景。
[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.
【學位授予單位】：哈爾濱工業(yè)大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：TP332.22

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本文編號：1886175