More classes of permutation quadrinomials from Niho exponents in characteristic two

Abstract

In this paper, we characterize the coefficients of f(x)=x+a1xs1(2m−1)+1+a2xs2(2m−1)+1+a3xs3(2m−1)+1 over F22m that lead f(x) to be a permutation of F22m for the following three cases: 1) (s1,s2,s3)=(−12k−1,1,2k2k−1); 2) (s1,s2,s3)=(12k+1,1,2k2k+1), where m, k are two positive integers; 3) (s1,s2,s3)=(14,1,34). We transform the problems for the first two cases in the investigation of some (2k+1)-th degree equations with variable in the unit circle. The numerical results suggest that the sufficient conditions that we find in this paper on the coefficients are also necessary. It turned out that some well-known results can be recovered (or covered) by our results.