New classes of permutation polynomials with coefficients 1 over finite fields

Permutation polynomials with coefficients 1 over finite fields attract researchers' interests due to their simple algebraic form. In this paper, we first construct four classes of fractional permutation polynomials over the cyclic subgroup of $\mathbb{F}_{2^{2m}}$. From these permutation polynomials, three new classes of permutation polynomials with coefficients 1 over $\mathbb{F}_{2^{2m}}$ are constructed, and three more general new classes of permutation polynomials with coefficients 1 over $\mathbb{F}_{2^{2m}}$ are constructed using a new method we presented recently. Some known permutation polynomials are the special cases of our new permutation polynomials. Furthermore, we prove that, in all new permutation polynomials, there exists a permutation polynomial which is EA-inequivalent to known permutation polynomials for all even positive integer $m$. This proof shows that EA-inequivalent permutation polynomials over $\mathbb{F}_{q}$ can be constructed from EA-equivalent permutation polynomials over the cyclic subgroup of $\mathbb{F}_{q}$. From this proof, it is obvious that, in all new permutation polynomials, there exists a permutation polynomial of which algebraic degree is the maximum algebraic degree of permutation polynomials over $\mathbb{F}_{2^{2m}}$.

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