More classes of permutation hexanomials and pentanomials over finite fields with even characteristic

Abstract

In this paper, we further push the studies on permutation polynomials over the finite fields Fq2 with q=2m. We find the coefficients of f(x)=x+a1xs1(q−1)+1+a2xs2(q−1)+1+a3xs3(q−1)+1+a4xs4(q−1)+1+a5xs5(q−1)+1 over Fq2 that lead f(x) to be a permutation for (s1,s2,s3,s4)=(14,12,1,34) in case of a5=0, and find more new (si) such that f(x) is a permutation hexanomial of Fq2 with coefficients in F4. Some well-known results are covered by our results. The numerical results suggest that the sufficient conditions we find in the first class of permutation pentanomials are also necessary.