Construction And Analysis Of Functions With Low Differential Uniformity Over Finite Fields

As an important component of a cipher algorithm,cryptographic function has crucial effect on the security of the algorithm.In the differential attacks,the lower the differential uniformity of a cryptographic function is,the more uniform of its differential distribution will be,which leads to a stronger resistibility against the differential attacks.Therefore,constructing the functions with low differential uniformity over finite fields has an signif-icant importance in both theory and practice.In this thesis,we make full use of the construction idea of the“Switching Method”and construct some classes of functions with low differential uniformity over both even characteristic finite fields and odd characteristic finite fields.The main results obtained are as follows:1.When the characteristic of the finite field is even,we focus on the construction of differentially 4-uniform permutations.First,based on the concept of the preferred boolean function and its correlation with the construction of differentially 4-uniform permutations,some new differentially 4-uniform permutations are discovered by constructing some bi-nomial and trinomial preferred boolean functions.Then,we change the boolean functions used in the former switching method to a mapping from F₂nto F₄.After constructing the functions with special restrictions,we also get some new differentially 4-uniform permu-tations.Besides,some computational results on the nonlinearity of the newly discovered differentially 4-uniform permutations are given.At last,a theorem of verifying the reality quantitatively that more differentially 4-uniform functions can be constructed in case of the permutation is also given.2.When the characteristic of the finite field is odd,we mainly study the construction of APN function and PN function.Under the guidance of the“Switching Method”,a series of transformation are carried on the known PN functions.First of all,the concept of“circle”is reviewed.By constructing some special form of circles and combining them with the known PN function,some new APN functions are obtained.Then,with the idea of piecewise function,some new APN functions are constructed after modifying the known PN functions in the subfield slightly.Furthermore,via jointing the two known PN functions together,some APN functions are also discovered after giving proper condition-s.The computational results from Magma procedure and analysis about these newfound APN functions are given at last.