Monomial Bent Functions Research Articles

An Open Problem About Monomial Bent Functions

An Open Problem About Monomial Bent Functions

Several new infinite families of bent functions via second order derivatives

Inspired by a recent work of Tang et al. on constructing bent functions [14, IEEE TIT, 63(1): 6149-6157, 2017], we introduce a property (Pτ) of any Boolean function that its second order derivatives vanish at any direction (ui,uj) for some τ-subset {u1,…,uτ} of $\mathbb {F}_{2^{n}}$ , and then establish a link between this property and the construction of Tang et al. (IEEE Trans. Inf. Theory 63(10), 6149–6157 2017). It enables us to find more bent functions efficiently. We construct (at least) five new infinite families of bent functions from some known functions: the Gold’s bent functions and some quadratic non-monomial bent functions, Leander’s monomial bent functions, Canteaut-Charpin-Kyureghyan’s monomial bent functions, and the Maiorana-McFarland class of bent functions, respectively. Our result generalizes some recent works on bent functions. We also provide the corresponding dual functions in all our constructions except the quadratic non-monomial one. It also turns out that we can get new bent functions outside the Maiorana-McFarland completed class.

A note on vectorial bent functions

In this paper we give three methods on the constructions of vectorial bent functions from F2n to F2n2, where n is a positive even integer. The first two kinds of functions are based on monomial bent functions. The third kind of functions are based on partial spreads bent functions, and those vectorial bent functions can achieve the largest algebraic degree.

Comments on "Monomial Bent Functions" [Feb 2006 738-743

In this paper we correct the wrong proof of Lemma 7 of the above titled paper (ibid., vol. 52, no. 2, pp. 738-743, Feb. 06), which is very important to the proof of Theorem 8 of the same paper.

Monomial and quadratic bent functions over the finite fields of odd characteristic

Considered are p-ary bent functions having the form f(x)=Tr/sub n/(/spl sigma//sub i=0//sup s/a/sub i/x/sup di/). A new class of ternary monomial regular bent function with the Dillon exponent is discovered. The existence of Dillon bent functions in the general case is an open problem of deciding whether a certain Kloosterman sum can take on the value -1. Also described is the general Gold-like form of a bent function that covers all the previously known monomial quadratic cases. The (weak) regularity of the new as well as of known monomial bent functions is discussed and the first example of a not weakly regular bent function is given. Finally, some criteria for an arbitrary quadratic function to be bent are proven.