Research On Constructions Of Permutation Polynomials Over Finite Fields

Permutation polynomials have important applications in a wide range of areas such as algebra,number theory,combinatorial designs,cryptography and coding theory.For instance,the S-box,as a core component in symmetric ciphers,has been designed by using permutation polynomials over finite fields with even characteristic.Thus,con-structing permutation polynomials over finite fields is a challenging and hot problem.Recently,new results and methods have sprung up.The AGW criterion,piece by piece construction,switching construction and exchanging construction have been the major methods.Permutation polynomials of short cycles and permutation polynomials based on linearized polynomials have received a lot of attention for their effective implemen-tation of encryption and decryption of data in cryptographic systems.And permutation polynomials with fewer terms attract researchers' interest due to their wide applications in cryptography,coding theory and finite geometry.As we know,the well known Welch conjecture and Niho conjecture have been proved by using them.In this paper,we s-tudy 3-cycle permutations,permutation polynomials based on linearized polynomials and permutation quadrinomials.Permutation polynomials of short cycles are important in cryptography.Involu-tions have been studied deeply and used widely.Based on cycle structure of permuta-tion polynomials and the numbers of solutions of congruence equations,we obtain the necessary and sufficient conditions which make the monomials,Dickson polynomials and linearized polynomials be 3-cycle permutations.Then the numbers of such 3-cycle permutations are determined accurately.Furthermore,we use the known 3-cycle per-mutation polynomials to construct new ones by the switching method,and extend the 3-cycle permutation polynomials as candidates of cryptographic functions.Permutation polynomials based on linearized polynomials are closely related to Kloosterman sum and have important theoretical and practical significance.Permuta-tion polynomials of the form(xpm-x+?)s+L(x)over Fpn have received a lot of attention and have been deeply studied.In this paper,we construct eight new classes of permutation polynomials of the form(x2m+x+?)s1+(x2m+x+?)s2+x over F2n for the first time.The permutation behavior is investigated by the polar coordinate representation of the affine polynomial x2m+x+?,transforming the problem to a related cubic equation and the AGW criterion.The complete characterization of the permutation polynomials with fewer terms is a challenging task.Seldom classes of permutation binomials and trinomials have been done perfectly.Recently,based on the additive character criterion,a class of permu-tation quadrinomials of the form f(x)=x3(x3(q-1)+a1x2(q-1)+a2xq-1+a3)have been found,the permutation problem was transformed into the determination of solu-tions of some cubic equations in the unit circle.However,the characterized coefficients are incomplete and it doesn't seem to be a feasible approach to characterize all the co-efficients.In this paper,we manage to transform the fine equation f(x+a)+f(x)=0 to some lower-degree affine equations.Consequently,we obtain the sufficient condi-tions which make the quadrinomials permute.Furthermore,it seems to produce all permutation quadrinomials of this form according to our exhaustive searches on small fields.That is to say,we may have found the necessary and sufficient conditions for the permutation quadrinomials of this form.