Cubic monomial bent functions: a subclass of \(\mathcal{M}\).

*(English)*Zbl 1171.11062Let \(\mathbb{F}_{2^n}\) be the Galois field of order \(n\) and characteristic 2. If \(m| n\), the trace map is \(T_m^n:\mathbb{F}_{2^n}\to\mathbb{F}_{2^m}\), \(x\mapsto \sum_{k=0}^{\frac{n}{m}-1} x^{2^{mk}}\), and it is \(\mathbb{F}_{2^m}\)-linear. A monomial map has the form \(f_{rd}:x\mapsto T_1^n(r\,x^d)\), where \(d\) is called the exponent. For even \(n\), a Boolean map \(f:\mathbb{F}_{2^n}\to\mathbb{F}_2\) is bent if for each \(r\), \(\sum_x (-1)^{f(x)+T_1^n(r\,x)}\) has absolute value \(\sqrt{2^n}\). The paper introduces a family of monomial bent functions with exponents of the form \(d(i,j) = 2^i+2^j+1\), with \(0<j<i\) and \(j<n/2\). Indeed, the introduced family consists of Maiorana-McFarland maps: each such function can be written as the concatenation of \(2^{m}\) pairwise-different affine maps defined on \(\mathbb{F}_{2^{m}}\), with \(m=\frac{n}{2}\).

Theorem 3.3 in the paper gives several families of triplets \((n,i,j)\) for which the monomials \(f_{rd}\), with \(r\) being a root of the polynomial \(X+X^{2^j}+X^{2^i}\) are expressed as concatenations of affine maps on \(\mathbb{F}_{2^{m}}\). One of these families gives the collection of bent functions reported by the same authors in a former paper.

Another family results for the case in which \(i=m\), thus the exponents are of the form \(2^m+2^j+1\). The proposition 4.2 gives conditions and manners to express the monomial functions as concatenations of affine maps on \(\mathbb{F}_{2^{m}}\). In order to have pairwise-different affine maps, the theorem 5.1 in the paper gives necessary and sufficient conditions in order that a polynomial \(\rho(X)=X^{2^k+2} + s X\in\mathbb{F}_{2^m}[X]\) defines a permutation \(\mathbb{F}_{2^m}\to\mathbb{F}_{2^m}\).

In the introduction of the paper the authors list the known families of monomial bent functions, including the families obtained in the paper and they pose as a final open problem to decide whether there are further bent monomial functions with exponents of the form \(d(i,j) = 2^i+2^j+1\).

Theorem 3.3 in the paper gives several families of triplets \((n,i,j)\) for which the monomials \(f_{rd}\), with \(r\) being a root of the polynomial \(X+X^{2^j}+X^{2^i}\) are expressed as concatenations of affine maps on \(\mathbb{F}_{2^{m}}\). One of these families gives the collection of bent functions reported by the same authors in a former paper.

Another family results for the case in which \(i=m\), thus the exponents are of the form \(2^m+2^j+1\). The proposition 4.2 gives conditions and manners to express the monomial functions as concatenations of affine maps on \(\mathbb{F}_{2^{m}}\). In order to have pairwise-different affine maps, the theorem 5.1 in the paper gives necessary and sufficient conditions in order that a polynomial \(\rho(X)=X^{2^k+2} + s X\in\mathbb{F}_{2^m}[X]\) defines a permutation \(\mathbb{F}_{2^m}\to\mathbb{F}_{2^m}\).

In the introduction of the paper the authors list the known families of monomial bent functions, including the families obtained in the paper and they pose as a final open problem to decide whether there are further bent monomial functions with exponents of the form \(d(i,j) = 2^i+2^j+1\).

Reviewer: Guillermo Morales-Luna (Mexico)