Research On Low Differential Uniformity Functions

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.