A Class Of Permutation Trinomial Over Finite Fields Of Characteristic 3

Permutation polynomials have a research history of nearly one hundred and six-ty years.Since the last century,permutation polynomials with specific forms or fewer terms have been favored by experts and scholars.Therefore,some permutation trinomi-als have received wide attention in the design of cryptographic algorithms due to their simple algebraic form and excellent cryptographic properties.Last few years,a lot of great progress has been made in the research of permutation trinomials,but the rapid development of cryptography,coding,combinatorics and other fields has raised many new research questions for permutation polynomials,and the construction of new per-mutation trinomials is still a very difficult task.This paper will study the construction of a class of sparse permutation polynomials.The main research results are as follows:This paper mainly studies the permutation trinomials with the form f(x)=x-Axq(q2-q+1)-Bxq2-q+1 over finite fields Fq3 of characteristic 3,where q=3m,A,B∈Fq3.By using the multivariate method with the resultant elimination,we first characterize the conditions that f(x)=0 has only zero solution and f(x)=a,a∈Fq3*,has a unique nonzero solution.Further we present four classes of permuta-tion trinomials over Fq3 with the above form when A and B satisfy some conditions.Then combining with computer data and using similar methods,we extend two of them to finite fields with arbitrary odd characteristic.Finally,we theoretically analyzed the QM equivalence of the permutation trinomials proposed in this paper and the known permutation polynomials,and obtained the conclusion that three of them are QM in-equivalent.