## Cubic monomial bent functions: a subclass of $$\mathcal{M}$$.(English)Zbl 1171.11062

Let $$\mathbb{F}_{2^n}$$ 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 $$d(i,j) = 2^i+2^j+1$$.

### MSC:

 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) 11T06 Polynomials over finite fields 94A55 Shift register sequences and sequences over finite alphabets in information and communication theory

### Keywords:

monomial maps; bent functions; Maiorana-McFarland class
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