【摘要】：最佳非線性函數(shù)即Bent函數(shù)和完全非線性函數(shù)分別是抵抗線性密碼攻擊和差分密碼攻擊能力最強(qiáng)的密碼函數(shù),故其在密碼學(xué)中扮演著非常重要的角色。而且,最佳非線性函數(shù)在編碼理論、序列設(shè)計和組合理論等領(lǐng)域中亦有重要的應(yīng)用。 本論文的第一個主要研究內(nèi)容是Bent函數(shù)的構(gòu)造�；诃h(huán)上的二次型理論和線性化方程途徑,本文首先構(gòu)造出幾類新的二次廣義布爾Bent函數(shù)。結(jié)合布爾Bent函數(shù)與廣義布爾Bent函數(shù)之間的關(guān)系并將構(gòu)造廣義布爾Bent函數(shù)的方法應(yīng)用于奇特征域中,本文相繼得到新的二次布爾Bent函數(shù)和二次p-元Bent函數(shù),其中p是-奇素數(shù)。而對于高次Bent函數(shù),本文著重研究了具有最佳代數(shù)次數(shù)的Dillon型Bent函數(shù)和Niho型Bent函數(shù)。通過對有限域中某些部分指數(shù)和的討論,本文成功刻畫出幾類新的Dillon型布爾Bent函數(shù)和Dillon型p-元Bent函數(shù),并推廣了部分已知結(jié)果。將研究Dillon型Bent函數(shù)的方法運(yùn)用在Niho型函數(shù)上,本文推廣了偶特征域中Leander-Kholosha類Niho型Bent函數(shù)的結(jié)論,并給出了其Bent性的一個簡潔的證明。同時,本文證明了所考察的Niho型函數(shù)在奇特征域中具有四值Walsh譜且確定了其譜值分布。 本論文的第二個主要研究內(nèi)容是利用完全非線性函數(shù)和幾乎完全非線性函數(shù)構(gòu)造最佳循環(huán)碼。通過利用有限域上低次多項式的因式分解以及不可約多項式次數(shù)與其對應(yīng)方程解之間的關(guān)系,本論文成功解決了由Ding和Helleseth提出的一個關(guān)于最佳三元單糾錯循環(huán)碼的公開問題。借助于有限域上的二次特征,運(yùn)用同樣的方法,對于正整數(shù)m,本論文得到了四類新的參數(shù)為[3m-1,3m-2m-1,4]的最佳三元單糾錯循環(huán)碼。更進(jìn)一步地,通過利用完全非線性函數(shù)的性質(zhì),本論文亦構(gòu)造出兩類新的參數(shù)為[3m-1,3m-2m-2,5]的最佳三元雙糾錯循環(huán)碼。而且,本論文亦考慮了上述所得最佳循環(huán)碼的覆蓋半徑及其對偶碼的重量分布。然而,本論文僅得到部分相關(guān)結(jié)果,目前仍有較多問題尚未解決。 本論文的第三個主要研究內(nèi)容是利用廣義布爾Bent函數(shù)和高非線性Gold函數(shù)研究最佳或幾乎最佳四元序列集。借助于環(huán)上的二次型理論和廣義布爾Bent函數(shù)的性質(zhì),本論文考察了環(huán)上一類指數(shù)和的性質(zhì)進(jìn)而確定了兩類最佳序列集的精確相關(guān)分布。而且,基于環(huán)上二次型理論,本文利用統(tǒng)一的方法得到了一類已知的最佳四元序列集和一類新的低相關(guān)四元序列集。另一方面,通過對伽羅華環(huán)上Gold函數(shù)性質(zhì)的考察,本論文確定了四元Gold序列集的精確相關(guān)分布。而且,依據(jù)四元序列與二元序列之間的關(guān)系,本文確定了四元Gold序列集的MSB序列的最大非平凡相關(guān)值以及四元Gold序列集的Gray序列的精確相關(guān)分布。
[Abstract]:The best nonlinear function, that is, Bent function and complete nonlinear function, are the most powerful cryptographic functions to resist linear cipher attack and differential cryptosystem attack, respectively, so they play a very important role in cryptography. Moreover, the optimal nonlinear function has important applications in the fields of coding theory, sequence design and combination theory. The first part of this thesis is the construction of Bent function. Based on the theory of quadratic form over rings and the path of linearization equations, several new classes of quadratic generalized Boolean Bent functions are constructed in this paper. Combining the relationship between Boolean Bent function and generalized Boolean Bent function, and applying the method of constructing generalized Boolean Bent function to odd characteristic domain, we obtain new quadratic Boolean Bent function and quadratic p- element Bent function, where p is an odd prime number. For higher order Bent functions, this paper focuses on the study of Dillon type Bent functions and Niho type Bent functions with the best algebraic degree. Through the discussion of some partial exponential sums in finite fields, several new classes of Dillon type Boolean Bent functions and Dillon type p-element Bent functions are successfully characterized in this paper, and some known results are generalized. In this paper, the method of studying Dillon type Bent functions is applied to Niho type functions. In this paper, we generalize the conclusion of Leander-Kholosha type Niho type Bent functions in the even characteristic domain, and give a concise proof of its Bent property. At the same time, it is proved that the Niho type function has four-valued Walsh spectrum and its spectral value distribution is determined in the odd characteristic domain. The second main content of this thesis is to construct the best cyclic codes by using completely nonlinear functions and almost completely nonlinear functions. By using the factorization of low order polynomials over finite fields and the relationship between irreducible polynomial degrees and the solutions of their corresponding equations, this paper successfully solves an open problem of optimal single error correcting cyclic codes for ternary systems proposed by Ding and Helleseth. By using the same method and by means of the quadratic characteristics over finite fields, four new classes of triple single error-correcting cyclic codes with [3m-1n 3m-2m-1m-1] are obtained for positive integers m in this paper. Furthermore, by using the properties of completely nonlinear functions, this paper also constructs two kinds of optimal binary error correcting cyclic codes with two new parameters [3m-1n 3m-2m-2m-2]. Furthermore, the coverage radius of the optimal cyclic codes and the weight distribution of the dual codes are also considered in this paper. However, only some results have been obtained in this paper, and there are still many problems to be solved. The third main content of this thesis is to study the best or almost optimal quaternion sequence set by using the generalized Boolean Bent function and the high nonlinear Gold function. With the help of the theory of quadratic form over rings and the properties of generalized Boolean Bent functions, this paper investigates the properties of a class of exponential sums over rings and determines the exact correlation distributions of two kinds of optimal sequence sets. Furthermore, based on the theory of quadratic form over rings, a class of known optimal quaternion sequences and a new low correlation quaternion sequence set are obtained by using the unified method. On the other hand, the exact correlation distribution of quaternion Gold sequence sets is determined by investigating the properties of Gold functions over Galois rings. Furthermore, according to the relation between quaternion and binary sequence, the maximum nontrivial correlation value of MSB sequence of quaternion Gold sequence set and the exact correlation distribution of Gray sequence of quaternion Gold sequence set are determined.
【學(xué)位授予單位】：西南交通大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2014
【分類號】：TN918.3
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本文編號：2161361