Permutation trinomials over F2m

Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we determine all permutation trinomials over ${\mathbb{F}}_{2^m}$ in Zieve's paper [30]. We prove a conjecture proposed by Gupta and Sharma in [8] and obtain some new permutation trinomials over ${\mathbb{F}}_{2^m}$. Finally, we show that some classes of permutation trinomials with parameters are QM equivalent to some known permutation trinomials.     

Constructing new differentially 4-uniform permutations from the inverse function

Two new families of differentially 4-uniform permutations over ${\mathbb{F}}_{2^{2m}}$ are constructed by modifying the values of the inverse function on some subfield of ${\mathbb{F}}_{2^{2m}}$ and by applying affine transformations on the function. The resulted 4-uniform permutations have high nonlinearity and algebraic degree. A family of differentially 6-uniform permutations with high nonlinearity and algebraic degree is also constructed by making the modification on an affine subspace of ${\mathbb{F}}_{2^{2m}}$.     

On two upper bounds for hypersurfaces involving a Thas’ invariant

Let $X^n$ be a hypersurface in ${\mathbb{P}}^{n+1}$ with $n\ge 1$ defined over a finite field ${\mathbb{F}}_q$ of $q$ elements. In this note, we classify, up to projective equivalence, hypersurfaces $X^n$ as above which reach two elementary upper bounds for the number of ${\mathbb{F}}_q$-points on $X^n$ which involve a Thas’ invariant.