【作者】 查正邦；

【导师】 王学理；

【作者基本信息】 湖南大学 ， 应用数学， 2008， 博士

【摘要】 低差分一致性函数可分为几乎完全非线性(almost perfect nonlinear, APN)函数、完全非线性(perfect nonlinear, PN)函数和其它低差分一致性函数.它们在密码学和代数学中都有广泛的应用.本文构造和分析了一批低差分一致性函数,具体成果如下:在第二章中,由已知的APN幂函数特例归纳出奇特征域中的两类APN幂函数,并以二次特征和Dickson多项式为工具给出证明.新的APN函数可用来解释Helle-seth提出的两种公开情况,并且可用来证明Dobbertin猜想.最后,将前面提出的两类APN函数加以推广,得出特定形式下幂函数的差分一致属性.在证明过程中创造性地引入Dickson多项式来求解方程,使得方程的求解变得简单可行.在第三章中,首先讨论了特征为2的域中已知的几类APN多项式函数的等价性质,接着给出了一类新的APN多项式函数,并进一步分析了它的bent属性.该APN函数不Carlet-Charpin-Zinoviev (CCZ)等价于Dobbertin函数,且在特定条件下不等价于已知的函数.其次,将特征为3的域中一类已知的APN函数推广到奇特征域中,新的APN函数包含已知的APN函数为特例.最后,利用线性多项式和迹函数,通过引入中间变量的方法计算了一大类APN函数的Walsh谱.计算结果表明该类函数的Walsh谱和Gold函数的Walsh谱相同,从而确定了它们的非线性度,而函数的非线性度可以衡量其抗线性分析能力.在第四章中,将已知的APN多项式函数推广到奇特征域中,加上特定的条件而得出几类PN多项式函数,且证明了其中两类PN函数不CCZ等价于已知的PN函数,由此给出了两类半域.在特定的条件下,所构造的半域也与已知的半域不合痕.我们在证明过程中给出了判断CCZ等价和扩张仿射(extended a?ne, EA)等价的一种方法.最后,讨论了特定条件下Dillon多项式的差分一致性.在第五章中,结合前面的工具和方法,构造了几类低差分一致性函数.我们注意到除了Edel和Pott最近发现的APN函数外,其它已知的APN多项式函数都是二次的,而且在F22n中还没有发现APN置换函数.我们构造的低差分一致性函数不限于二次,而且具有置换特性,从而给设计和构造S盒提供了更多的方法.在第3节中,引入了几乎低差分一致性概念,并构造了几类几乎低差分一致性函数.更多还原

【Abstract】 Low di?erential uniformity functions can be divided into three classes: almostperfect nonlinear (APN) functions, perfect nonlinear (PN) functions and other lowdi?erential uniformity functions. They play an important role in cryptography andalgebra. In this dissertation, we construct and analyze a series of low di?erentialuniformity functions. The main contributions are as follows.In chapter 2, two families of APN power functions on odd prime field arededuced from the known results of APN power functions. Their APN property canbe proved by applying quadratic character and Dickson polynomial. Based on thenew APN functions, we can explain the two open cases introduced by Helleseth andthen prove Dobbertin’s conjecture. Finally, we generalize the new APN functionsand get the di?erential uniformity of power functions with a certain form. Weinnovativly introduce Dickson polynomial to solve the equation, which makes theprocess easy and feasible.In chapter 3, we firstly discuss the equivalence of the known APN polynomialfunctions on binary field. Moreover, we present a new family of APN polynomialfunctions and analyze their bent property. These new APN polynomial functionsare not Carlet-Charpin-Zinoviev (CCZ) equivalent to Dobbertin functions, and arenot equivalent to the known functions under certain conditions. Secondly, fromthe known result on the field of character 3, we similarly construct a family ofAPN functions on odd prime field. The known APN functions can be seen as thespecial cases of our new results. Finally, we obtain the Walsh spectrum of a familyof APN functions based on linear polynomial, trace mapping and intermediatevariable. We find that the Walsh spectrum of these functions is same as that ofGold functions. Our results determinate the nonlinearity of the functions whichmeasures their resistance to linear cryptanalysis.In chapter 4, we further study the known APN polynomial functions on oddprime field and then get several families of PN polynomial functions under certainconditions. We prove that two families of them are not CCZ equivalent to theknown PN functions. Then we get two families of semifields. Under certain con-ditions, the new semifields are not isotopic to any other known one. We present amethod to determine CCZ-equivalence and EA (extended a?ne) equivalence in our proof. Finally, we discuss the di?erential uniformity of Dillon’s polynomial undercertain conditions.In chapter 5, we give some low di?erential uniformity functions by the waysintroduced before. We note that the known APN polynomial functions are allquadratic except the APN functions recently found by Edel and Pott. In F22n, theAPN permutation has not yet been found. We construct low di?erential uniformityfunctions which are not quadratic. Especially, some of them are permutations. Weprovide more methods to design the S-box by using them. In section 3, we intro-duce the concept of almost low di?erential uniformity and construct several familiesof almost low di?erential uniformity functions.更多还原