【摘要】：密碼函數(shù)是多種密碼系統(tǒng)的重要組成部分.要使設(shè)計的密碼系統(tǒng)能夠抵抗各種已有的攻擊,要求該系統(tǒng)所選用的密碼函數(shù)必須滿足一些相應(yīng)的密碼學性質(zhì),如平衡性、相關(guān)免疫性、彈性、高代數(shù)次數(shù)、高非線性度、高代數(shù)免疫度、低差分均勻度等.因此研究和構(gòu)造具有優(yōu)良密碼學性質(zhì)的密碼函數(shù)在理論和實際應(yīng)用上都具有重要意義.本文主要研究密碼函數(shù)幾個關(guān)鍵密碼學性質(zhì)的分析和構(gòu)造問題,得到了如下研究成果:針對非線性度、代數(shù)免疫度及差分均勻度這三類關(guān)鍵密碼學安全性指標,本文首先運用組合數(shù)學中的重要工具一 Schur·函數(shù),給出了最優(yōu)代數(shù)免疫平衡布爾函數(shù)的一種新刻畫.利用此刻畫給出了 Carlet-Feng函數(shù)是最優(yōu)代數(shù)免疫函數(shù)的新證明.同時,構(gòu)造了三類平衡的最優(yōu)代數(shù)免疫布爾函數(shù).發(fā)現(xiàn)所構(gòu)造的三類函數(shù)中存在高非線性度、高代數(shù)次數(shù)等其它優(yōu)良密碼學性質(zhì)的例子.其次,采用將函數(shù)的定義域分為兩個子集,且在這兩個子集上定義不同置換的方法,得到了一類4-差分置換.研究了其代數(shù)次數(shù)、非線性度等密碼學性質(zhì).還討論了該類函數(shù)與12類4-差分置換的CCZ不等價性.最后,構(gòu)造了五類二次Semi-bent函數(shù)及兩類Plateaued函數(shù),并與已知構(gòu)造進行了比較.本文還對Budaghyan-Carlet多項式及Dembowski-Ostrom型函數(shù)的重要密碼學性質(zhì)進行了分析.討論了一個與Budaghyan-Carlet多項式有關(guān)的集合所含元素的性質(zhì)和個數(shù).通過研究Budaghyan-Carlet多項式的分量函數(shù),得到了一類Bent函數(shù),回答了 Budaghyan-Carelt多項式是否能通過加上線性化多項式成為置換多項式這一問題.另外,證明了若Dembowski-Ostrom型多輸出布爾函數(shù)有唯一零根且其導(dǎo)函數(shù)有一個或者四個根,則該布爾函數(shù)具有經(jīng)典Walsh譜,且其Walsh譜分布可以明確給出.由此進一步得到了四類Dembowski-Ostrom型APN函數(shù)的Walsh譜分布.
[Abstract]:Cryptographic function is an important part of many cryptographic systems. In order to make the designed cryptosystem resist all kinds of existing attacks, the cryptosystem must satisfy some corresponding cryptographic properties, such as balance, correlation immunity, elasticity, high algebraic times, and high nonlinearity, and the cryptology function chosen by the system must satisfy some corresponding cryptographic properties, such as balance, correlation immunity, elasticity, high algebraic times, and high nonlinearity. High algebraic immunity, low differential uniformity, etc. Therefore, the research and construction of cryptographic functions with excellent cryptographic properties are of great significance both in theory and in practice. This paper mainly studies the analysis and construction of several key cryptology properties of cryptographic function, and obtains the following research results: aiming at the three key cryptology security indexes: nonlinearity, algebraic immunity and differential uniformity, In this paper, a new characterization of the optimal algebraic immune equilibrium Boolean function is given by using the Schur function, an important tool in combinatorial mathematics. In this paper, a new proof that Carlet-Feng function is an optimal algebraic immune function is given. At the same time, three kinds of optimal algebraic immune Boolean functions are constructed. It is found that there are some examples of other excellent cryptographic properties, such as high nonlinearity, high algebraic times, and so on, among the three classes of functions constructed in this paper. Secondly, a class of 4-difference permutations is obtained by dividing the defined domain of a function into two subsets and defining different permutations on the two subsets. The cryptology properties such as algebraic number, nonlinearity and so on are studied. The CCZ inequality of this class of functions with 12 kinds of 4-difference permutations is also discussed. Finally, five classes of quadratic Semi-bent functions and two classes of Plateaued functions are constructed and compared with the known constructions. In this paper, the important cryptographic properties of Budaghyan-Carlet polynomials and Dembowski- Ostrom-type functions are also analyzed. The properties and number of elements in a set related to Budaghyan-Carlet polynomials are discussed. By studying the component functions of Budaghyan-Carlet polynomials, we obtain a class of Bent functions and answer the question whether Budaghyan-Carelt polynomials can be permutation polynomials by adding linearized polynomials. In addition, it is proved that if Dembowski- Ostrom type multi-output Boolean function has unique zero root and its derivative function has one or four roots, then the Boolean function has classical Walsh spectrum, and its Walsh spectrum distribution can be clearly given. The Walsh spectral distributions of four types of Dembowski- Ostrom-type APN functions are obtained.
【學位授予單位】：湖北大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：TN918.1;O174

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