New results on permutation polynomials over finite fields

In this paper, we get several new results on permutation polynomials over finite fields. First, by using the linear translator, we construct permutation polynomials of the forms $L(x)+\sum_{j=1}^k \gamma_jh_j(f_j(x))$ and $x+\sum_{j=1}^k\gamma_jf_j(x)$. These generalize the results obtained by Kyureghyan in 2011. Consequently, we characterize permutation polynomials of the form $L(x)+\sum_{i=1} ^l\gamma_i {\rm Tr}_{{\bf F}_{q^m}/{\bf F}_{q}}(h_i(x))$, which extends a theorem of Charpin and Kyureghyan obtained in 2009.Comment: 11 pages. To appear in International Journal of Number Theor

Some new results on permutation polynomials over finite fields

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, four classes of monomial complete permutation polynomials and one class of trinomial complete permutation polynomials are presented, one of which confirms a conjecture proposed by Wu et al. (Sci. China Math., to appear. Doi: 10.1007/s11425-014-4964-2). Furthermore, we give two classes of trinomial permutation polynomials, and make some progress on a conjecture about the differential uniformity of power permutation polynomials proposed by Blondeau et al. (Int. J. Inf. Coding Theory, 2010, 1, pp. 149-170).Comment: 21 pages. We have changed the title of our pape

A conjecture on permutation trinomials over finite fields of characteristic two

In this paper, by analyzing the quadratic factors of an $11$-th degree polynomial over the finite field \ftwon, a conjecture on permutation trinomials over \ftwon[x] proposed very recently by Deng and Zheng is settled, where $n=2m$ and $m$ is a positive integer with $\gcd(m,5)=1$

Further results on complete permutation monomials over finite fields

In this paper, we construct some new classes of complete permutation monomials with exponent $d=\frac{q^n-1}{q-1}$ using AGW criterion (a special case). This proves two recent conjectures in [Wuetal2] and extends some of these recent results to more general $n$'s

More new classes of permutation trinomials over F2n\mathbb{F}_{2^n}

Permutation polynomials over finite fields have wide applications in many areas of science and engineering. In this paper, we present six new classes of permutation trinomials over $\mathbb{F}_{2^n}$ which have explicit forms by determining the solutions of some equations.Comment: 17 page

Permutation trinomials over Fq3\mathbb{F}_{q^3}

We consider four classes of polynomials over the fields $\mathbb{F}_{q^3}$, $q=p^h$, $p>3$, $f_1(x)=x^{q^2+q-1}+Ax^{q^2-q+1}+Bx$, $f_2(x)=x^{q^2+q-1}+Ax^{q^3-q^2+q}+Bx$, $f_3(x)=x^{q^2+q-1}+Ax^{q^2}-Bx$, $f_4(x)=x^{q^2+q-1}+Ax^{q}-Bx$, where $A,B \in \mathbb{F}_q$. We determine conditions on the pairs $(A,B)$ and we give lower bounds on the number of pairs $(A,B)$ for which these polynomials permute $\mathbb{F}_{q^3}$

New Classes of Permutation Binomials and Permutation Trinomials over Finite Fields

Permutation polynomials over finite fields play important roles in finite fields theory. They also have wide applications in many areas of science and engineering such as coding theory, cryptography, combinatorial design, communication theory and so on. Permutation binomials and trinomials attract people's interest due to their simple algebraic form and additional extraordinary properties. In this paper, several new classes of permutation binomials and permutation trinomials are constructed. Some of these permutation polynomials are generalizations of known ones.Comment: 18 pages. Submitted to a journal on Aug. 15t

New Permutation Trinomials Constructed from Fractional Polynomials

Permutation trinomials over finite fields consititute an active research due to their simple algebraic form, additional extraordinary properties and their wide applications in many areas of science and engineering. In the present paper, six new classes of permutation trinomials over finite fields of even characteristic are constructed from six fractional polynomials. Further, three classes of permutation trinomials over finite fields of characteristic three are raised. Distinct from most of the known permutation trinomials which are with fixed exponents, our results are some general classes of permutation trinomials with one parameter in the exponents. Finally, we propose a few conjectures

Finding compositional inverses of permutations from the AGW criterion

Permutation polynomials and their compositional inverses have wide applications in cryptography, coding theory, and combinatorial designs. Motivated by several previous results on finding compositional inverses of permutation polynomials of different forms, we propose a general method for finding these inverses of permutation polynomials constructed by the AGW criterion. As a result, we have reduced the problem of finding the compositional inverse of such a permutation polynomial over a finite field to that of finding the inverse of a bijection over a smaller set. We demonstrate our method by interpreting several recent known results, as well as by providing new explicit results on more classes of permutation polynomials in different types. In addition, we give new criteria for these permutation polynomials being involutions. Explicit constructions are also provided for all involutory criteria.Comment: 24 pages. Revision submitte

Permutation polynomials and complete permutation polynomials over Fq3\mathbb{F}_{q^{3}}

Motivated by many recent constructions of permutation polynomials over $\mathbb{F}_{q^2}$, we study permutation polynomials over $\mathbb{F}_{q^3}$ in terms of their coefficients. Based on the multivariate method and resultant elimination, we construct several new classes of sparse permutation polynomials over $\mathbb{F}_{q^3}$, $q=p^{k}$, $p\geq3$. Some of them are complete mappings.Comment: 31 page