Study On Nonlinear Permutation Polynomials Over Finite Fields

Permutation polynomials over finite fields are important objects in the theory of finite fields,and they have been widely used in cryptography,coding theory and combi-natorial design theory.For example,the inverse function used in the S-box(the unique nonlinear component in symmetric cryptographic algorithm)of the Advanced Encryp-tion Standard(AES)algorithm is a nonlinear permutation over the finite field F28.Per-mutation polynomials in the realistic applications of mathematics and cryptography are always required to either have simple algebraic forms or possess good cryptographic properties,such as low differential uniformity,high algebraic degree,high nonlinearity and so on.So it has great theoretical and realistic significance to construct permutation polynomials with good properties over finite fields.Based on the previous results,we propose abundant permutation polynomials over finite fields in this thesis.Our result-s not only enrich the known permutation polynomials,but also provide methods and ideas for the construction of new permutation polynomials.To ensure the algebraic de-gree of the permutation polynomials as high as possible,we restrict these polynomials to be nonlinear.The main results of this thesis are listed as follows:(1)Permutation polynomials based on linearized polynomials are easy to be im-plemented in engineering,and the polynomials from Kloosterman sum identities are closely related to linearized polynomials.According to the AGW criterion,we system-atically study the permutation properties of polynomials from Kloosterman sum iden-tities with the form(xpm-x+δ)s1+(xpm-x+δ)s2+x,and obtain two classes of such permutation polynomials.Our results enrich the known permutation polynomials,as well as generalize some previous works.(2)The complete permutation polynomials form a special class of permutation polynomials.Compared to the construction of permutation polynomials,the method of constructing complete permutation polynomials seems more limited,and the known classes of complete permutation polynomials are fewer.We propose two classes of complete permutation polynomials over finite fields based on linearized polynomials,i.e.,complete permutation polynomials over finite fields from trace functions and com-plete permutation polynomials of the form axpm+bx+h(xpm±x).Utilizing the AGW criterion,we establish a link between these two classes of complete permutation poly-nomials over finite fields with permutation polynomials on their subsets,and obtain a large number of complete permutation polynomials through investigating permutation polynomials on their subsets.(3)Niho exponents have been shown as important parameters in constructing pe-mutation polynomials over finite fields with even dimension,so the construction of permutation polynomials with Niho exponents has attracted much attention.We inves-tigate complete permutation polynomials with Niho exponents in this thesis.Using the AGW criterion,we construct complete trinomials,pentanomials and complete permu-tation polynomials with fractional and piecewise forms over finite fields.Further,with the help of the results related to some known complete permutation trinomials,we give the necessary and sufficient conditions which yield complete permutation trinomials.(4)Dickson polynomials often contain abundant classes of permutation polyno-mials,therefore,their permutation properties are widely concerned by scholars.As for the second kind of Dickson polynomials,there is a famous conjecture about the neces-sary and sufficient condition that yields it a permutation over the finite field Fpm with odd characteristic.So far,only the cases m=1 and 2 have been proved.Since the second kind of Dickson polynomials can be written as the product of two generalized Lucas polynomials,we investigate the permutation properties of these two classes of generalized Lucas polynomials over the finite field with odd characteristic,and hope to find methods to verify the conjecture.Ultimately,we obtain some necessary conditions such that these two classes of polynomials are permutations over the finite field with odd characteristic,and completely characterize their permutation behaviours over the prime field with odd characteristic.