A Permutation Polynomial Over A Finite Field

Let p be a prime, m a positive integer, and Fpm the finite field with pm elements.A polynomial f(x) in Fpm[x] is said to be a permutation polynomial over Fpm if it induces a permutation from Fpm to Fpm. This paper dedicated to the permutation polynomial with the form (xpk - x + ?)s+ L(x) over finite field Fpm. We obtain several kinds of permutation polynomials as mentioned above over finite fields F2m.In particular:Theorem 3.1 Let ? ? F2m and Tr(?) = 1, where m?0 (mod 4), Then for each positive integer number k with (2m/2 - 2)k ? 2m/2 - 1(mod 2m - 1), we obtain that f (x) = (x2 + x + ?)k + x is a permutation polynomial over F2m.Theorem 3.2 Let ? ? F2m and Tr(?) = 1, k |m,m/k odd. Then for each positive integer k' and a ? F2k with (2k + 1)k' ? 1(mod 2m - 1), we obtain that f(x) = (ax2k +ax + ?)k'+ x is a permutation polynomial over F2m.Theorem 3.3 Let ? ? F2m and Tr(?) = 1. If m, k is positive number and m/gcd(k,m) is odd, then for each positive integer k' and a ? F2m/2 with (2k + 1)k'?2m/2-1(mod 2m - 1) we obtain that f(x)=(1/ax2k+ax+?)k'+x is a permutation polynomial over F2m.Theorem 3.4 Let ?,a ? F2m, m, k, s positive integers such that gcd(m ,k) > 1 and s(2k - 1) ? 0(mod 2m - 1), we obtain that f(x) = (ax2k + ax + ?)s + x is a permutation polynomial over F2m.Theorem 3.5 Let ? ? F2m,where m is even, 0?a ? F2m/2 and Tr(?/a2) = 1, we obtain that f(x) = (x2 + ax + ?)2m/2-1 + x2m/2+1 is a permutation polynomial over F2m.