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We introduce a class of permutation polynomial over F q n that can be written in the form L (x) x q + 1 or L (x q + 1) x for some q -linear polynomial L over F q n . Specifically, we present those permutation polynomials explicitly as well as their inverses. In addition, more permutation polynomials can be derived in a more general form. [ABSTRACT FROM AUTHOR]

Planar functions, introduced by Dembowski and Ostrom, have attracted much attention in the last decade. As shown in this paper, we present a new class of planar functions of the form Tr (a x q + 1) + ℓ (x 2) on an extension of the finite field F q n / F q. Specifically, we investigate those functions on F q 2 / F q and construct several typical kinds of planar functions. We also completely characterize them on F q 3 / F q. When the degree of extension is higher, it will be proved that such planar functions do not exist given certain conditions. [ABSTRACT FROM AUTHOR]

Permutation rational functions over finite fields have attracted much attention in recent years. In this paper, we introduce a class of permutation rational functions over F q 2 , whose numerators are so-called q -quadratic polynomials. To this end, we will first determine the exact number of zeros of a special q -quadratic polynomial in F q 2 , by calculating character sums related to quadratic forms of F q 2 / F q. Then given some rational function, we can demonstrate whether it induces a permutation of F q 2 . [ABSTRACT FROM AUTHOR]

In the present article, we provide a complete characterization of permutations on the finite field F 4 m of shape f ϵ _ (X) : = ϵ 1 X ‾ q + 1 + ϵ 2 X ‾ q X + ϵ 3 X ‾ X q + ϵ 4 X q + 1 , definitively (where q = 2 k , Q = 2 m , m is odd, (m , k) = 1 , X ‾ = X Q). Our achievement firstly extends the very recent literature (a long series of interesting recent (2019-2022) results derived in (at least) seven articles) about the bijectivity on F Q 2 of f ϵ _ . It also gives a complete proof of the conjecture (Conjecture 19) posed in the recent literature by Li et al. (2021) [24] about the characterization of permutations f ϵ _ with 4-uniform BCT (which can be viewed, thanks to their nice cryptographic properties, as promising candidates as S-boxes to be used in designing secure block ciphers in symmetric cryptography). To derive our results about the quadrinomials f ϵ _ , we shall keep the essence of our successful approach initiated by Kim et al. in [23] by relating the problem of characterization of the bijectivity of f ϵ _ over F Q 2 to the equation X q + 1 + X + a = 0 but also by performing algebraic developments with new nature by considering some rational functions over finite fields. Besides, we shall employ novel algebraic techniques and derive new auxiliary results (some of them have their independent interest) going beyond the recent results given in [23]. Notably, we present a complete proof of the bijectivity of f ϵ _ over F Q 2 without any restriction. Specifically, our results are valid when the coefficients ϵ i (i ∈ { 1 , 2 , 3 , 4 }) lie in F Q 2 and for any value of k ≥ 1 (at the opposite of the results in [23] valid with the condition that the coefficients ϵ i (i ∈ { 1 , 2 , 3 , 4 }) lie in the finite field F Q while covering all values of k ≥ 1). Finally, we emphasize that very recently, Göloğlu has presented in (Göloğlu, 2022 [13]), an alternative novel approach providing a complete description of permutations of the form mentioned above using biprojective polynomials. [ABSTRACT FROM AUTHOR]

Abstract: One of the classes of bent Boolean functions introduced by John Dillon in his thesis is family H. While this class corresponds to a nice original construction of bent functions in bivariate form, Dillon could exhibit in it only functions which already belonged to the well-known Maiorana–McFarland class. We first notice that H can be extended to a slightly larger class that we denote by . We observe that the bent functions constructed via Niho power functions, for which four examples are known due to Dobbertin et al. and to Leander and Kholosha, are the univariate form of the functions of class . Their restrictions to the vector spaces , , are linear. We also characterize the bent functions whose restrictions to the ʼs are affine. We answer the open question raised by Dobbertin et al. (2006) in on whether the duals of the Niho bent functions introduced in the paper are affinely equivalent to them, by explicitly calculating the dual of one of these functions. We observe that this Niho function also belongs to the Maiorana–McFarland class, which brings us back to the problem of knowing whether H (or ) is a subclass of the Maiorana–McFarland completed class. We then show that the condition for a function in bivariate form to belong to class is equivalent to the fact that a polynomial directly related to its definition is an o-polynomial (also called oval polynomial, a notion from finite geometry). Thanks to the existence in the literature of 8 classes of nonlinear o-polynomials, we deduce a large number of new cases of bent functions in , which are potentially affinely inequivalent to known bent functions (in particular, to Maiorana–McFarlandʼs functions). [Copyright &y& Elsevier]