Quadratic Bent Functions Research Articles

Maximally nonlinear functions over finite fields

Abstract An n-place function over a field F q $ \mathbf {F}_q $ with q elements is called maximally nonlinear if it has the largest nonlinearity among all q-valued n-place functions. We show that, for even n=2, a function is maximally nonlinear if and only if its nonlinearity is q n − 1 ( q − 1 ) − q n 2 − 1 $ q^{n-1}(q - 1) - q^{\frac n2-1} $ ; for n=1, the corresponding criterion for maximal nonlinearity is q − 2. For q > 2 $ q \gt 2 $ and even n=2, we describe the set of all maximally nonlinear quadratic functions and find its cardinality. In this case, all maximally nonlinear quadratic functions are quadratic bent functions and their number is smaller than the halved number of the bent functions.

Quadratic bent functions and their duals

Quadratic bent functions and their duals

Vectorial bent functions in odd characteristic and their components

Bent functions in odd characteristic can be either (weakly) regular or non-weakly regular. Furthermore one can distinguish between dual-bent functions, which are bent functions for which the dual is bent as well, and non-dual bent functions. Whereas a weakly regular bent function always has a bent dual, a non-weakly regular bent function can be either dual-bent or non-dual-bent. The classical constructions (like quadratic bent functions, Maiorana-McFarland or partial spread) yield weakly regular bent functions, but meanwhile one knows constructions of infinite classes of non-weakly regular bent functions of both types, dual-bent and non-dual-bent. In this article we focus on vectorial bent functions in odd characteristic. We first show that most p-ary bent monomials and binomials are actually vectorial constructions. In the second part we give a positive answer to the question if non-weakly regular bent functions can be components of a vectorial bent function. We present the first construction of vectorial bent functions of which the components are non-weakly regular but dual-bent, and the first construction of vectorial bent functions with non-dual-bent components.

Infinite families of five-valued Walsh spectrum Boolean functions

In this paper, we provide some sufficient conditions for a function of the form to be five-valued with the Walsh spectrum {0, ±2 n /2, ±2 n /2+1}. In particular, several classes of five-valued Walsh spectrum functions by adding the arbitrary product of linear functions to some known quadratic bent function are obtained.

On q-ary bent and plateaued functions

We obtain the following results. For any prime p the minimal Hamming distance between distinct regular p-ary bent functions of 2n variables is equal to $$p^n$$ . The number of p-ary regular bent functions at the distance $$p^n$$ from the quadratic bent function $$Q_n=x_1x_2+\cdots +x_{2n-1}x_{2n}$$ is equal to $$p^n(p^{n-1}+1)\cdots (p+1)(p-1)$$ for $$p>2$$ . The Hamming distance between distinct binary s-plateaued functions of n variables is not less than $$2^{\frac{s+n-2}{2}}$$ and the Hamming distance between distinct ternary s-plateaued functions of n variables is not less than $$3^{\frac{s+n-1}{2}}$$ . These bounds are tight. For $$p=3$$ we prove an upper bound on nonlinearity of ternary functions in terms of their correlation immunity. Moreover, functions reaching this bound are plateaued. For $$p=2$$ analogous result are well known but for large p it seems impossible. Constructions and some properties of p-ary plateaued functions are discussed.

The connection between quadratic bent–negabent functions and the Kerdock code

In this paper we prove that all bent functions in the Kerdock code, except for the coset of the symmetric quadratic bent function, are bent–negabent. In this direction, we characterize the set of quadratic bent–negabent functions and show some results connecting quadratic bent–negabent functions and the Kerdock code. Further, we note that there are bent–negabent preserving nonsingular transformations outside the well known class of orthogonal ones that might provide additional functions in the bent–negabent set. This is the first time we could identify non-orthogonal (nonsingular) linear transformations that preserve bent–negabent property for a special subset.

New quadratic bent functions in polynomial forms with coefficients in extension fields

In this paper, we first discuss the bentness of a large class of quadratic Boolean functions in polynomial form $$f(x)=\sum _{i=1}^{{n}/{2}-1}\mathrm {Tr}^n_1(c_ix^{1+2^i})+ \mathrm {Tr}_1^{n/2}(c_{n/2}x^{1+2^{n/2}})$$ , where n is even, $$c_i\in \mathrm {GF}(2^n)$$ for $$1\le i \le {n}/{2}-1$$ and $$c_{n/2}\in \mathrm {GF}(2^{n/2})$$ . The bentness of these functions can be connected with linearized permutation polynomials. Hence, methods for constructing quadratic bent functions are given. Further, we consider a subclass of quadratic Boolean functions of the form $$f(x)=\sum _{i=1}^{{m}/{2}-1}\mathrm {Tr}^n_1(c_ix^{1+2^{ei}})+ \mathrm {Tr}_1^{n/2}(c_{m/2}x^{1+2^{n/2}})$$ , where $$n=em$$ , m is even, and $$c_i\in \mathrm {GF}(2^e)$$ . The bentness of these functions is characterized and some methods for deriving new quadratic bent functions are given. Finally, when m and e satisfy some conditions, we determine the number of these quadratic bent functions.

The graph of minimal distances of bent functions and its properties

A notion of the graph of minimal distances of bent functions is introduced. It is an undirected graph (V, E) where V is the set of all bent functions in 2k variables and $$(f, g) \in E$$ if the Hamming distance between f and g is equal to $$2^k$$ . It is shown that the maximum degree of the graph is equal to $$2^k (2^1 + 1) (2^2 + 1) \cdots (2^k + 1)$$ and all its vertices of maximum degree are quadratic bent functions. It is obtained that the degree of a vertex from Maiorana—McFarland class is not less than $$2^{2k + 1} - 2^k$$ . It is proven that the graph is connected for $$2k = 2, 4, 6$$ , disconnected for $$2k \ge 10$$ and its subgraph induced by all functions EA-equivalent to Maiorana—McFarland bent functions is connected.

A construction of codes with linearity from two linear codes

Let źźp$\mathbb {F}_{p}$ be a finite field, where p is a prime. This paper presents a new construction of a code C$\mathcal C$ over źźp2$\mathbb {F}_{p^{2}}$ from two linear codes over źźp$\mathbb {F}_{p}$, which has źźp$\mathbb {F}_{p}$-linearity and its minimum distance is equal to its minimum nonzero weight. Based on our construction, we shall determine the weight enumerators of some codes defined by quadratic bent functions and simplex codes. It will be shown that these codes have a few nonzero weights.

Complete weight enumerators of some linear codes from quadratic forms

Linear codes with few weights have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, a construction of q-ary linear codes with few weights employing general quadratic forms over the finite field Fq${\mathbb {F}}_{q}$ is proposed, where q is an odd prime power. This generalizes some earlier constructions of p-ary linear codes from quadratic bent functions over the prime field Fp${\mathbb {F}}_{p}$, where p is an odd prime. The complete weight enumerators of the resultant q-ary linear codes are also determined.

Linear codes with two or three weights from quadratic Bent functions

Linear codes with few weights have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, several classes of $p$-ary linear codes with two or three weights are constructed from quadratic Bent functions over the finite field $\gf_p$, where $p$ is an odd prime. They include some earlier linear codes as special cases. The weight distributions of these linear codes are also determined.

An approach to the classification of Boolean bent functions of the nonlinearity degree 3

AbstractWe consider an approach to the classification of n-variable Boolean bent functions of the nonlinearity degree 3. We utilize the apparatus of bent rectangles introduced by S. V. Agievich. This apparatus was used for the classification of 8-variable Boolean cubic bent functions. The results of our research allow to construct cubic bent functions that depend on an arbitrary even number of variables; the construction is based on well studied quadratic bent functions.

Characterisation and enumeration of a class of semi-bent quadratic Boolean functions

In this paper, we consider semi-bentness of quadratic Boolean functions defined for even n and give the characterisation of these functions. Up to our knowledge, semi-bentness of this class has not been investigated before and we proved that semi-bent functions of this form exist only for 6|n. Furthermore, we present a method for enumeration of semi-bent and bent functions in certain classes. Using this method we find the exact number of semi-bent functions of this form. Moreover, we complete some previous partial and incomplete enumeration results for three other classes of semi-bent/bent functions in the literature using this method. We also correct some results on quadratic bent functions stated in Ma et al. 2005.

Bent and Permutational Properties of Budaghyan-Carlet Hexanomials

The set related to the existence of Budaghyan-Carlet hexanomials is characterized. By investigating the component functions, it is also proved that none of Budaghyan-Carlet hexanomials cannot be turned into a permutation by adding any linearized polynomial. As a byproduct, a class of quadratic bent functions is obtained.

New Constructions of Quadratic Bent Functions in Polynomial Form

New quadratic bent functions in polynomial form are constructed in this paper. The constructions give new Boolean bent, generalized Boolean bent and p-ary bent functions. Based on Z 4 -valued quadratic forms, a simple method provides several new constructions of generalized Boolean bent functions. From these generalized Boolean bent functions a method is presented to transform them into Boolean bent and semi-bent functions. Moreover, many new p-ary bent functions can also be obtained by applying similar methods.

Classification of p-ary self dual quadratic bent functions, p odd

Let p be an odd prime. We classify all self dual quadratic bent functions from Fpn to Fp under the action of the orthogonal group O(n,Fp). The sizes of the O(n,Fp)-orbits of such self dual bent functions are explicitly determined. These results are obtained through the following steps: 1. n×n symmetric matrices A over Fp satisfying A2=cI (c∈Fp⁎) are classified under the conjugation by O(n,Fp). 2. For each representative A in step 1, the orbits of ker(A−I) under the action of centO(n,Fp)(A) are determined. 3. The sizes of the orbits in steps 1 and 2 are computed.

Enumeration of the bent functions of least deviation from a quadratic bent function

We study a construction of the bent functions of least deviation from a quadratic bent function, describe all these bent functions of 2k variables, and show that the quantity of them is 2k(21 + 1) ... (2k + 1). We find some lower bound on the number of the bent functions of least deviation from a bent function of the Maiorana-McFarland class.

Classification of self dual quadratic bent functions

We classify all self dual and anti self dual quadratic bent functions in 2n variables under the action of the orthogonal group $${{O}(2n,\mathbb F_2)}$$ . This is done through a classification of all 2n × 2n involutory alternating matrices over $${\mathbb F_2}$$ under the action of the orthogonal group. The sizes of the $${{O}(2n,\mathbb F_2)}$$ -orbits of self dual and anti self dual quadratic bent functions are determined explicitly.

Strongly regular graphs associated with ternary bent functions

We prove a new characterization of weakly regular ternary bent functions via partial difference sets. Partial difference sets are combinatorial objects corresponding to strongly regular graphs. Using known families of bent functions, we obtain in this way new families of strongly regular graphs, some of which were previously unknown. One of the families includes an example in [N. Hamada, T. Helleseth, A characterization of some { 3 v 2 + v 3 , 3 v 1 + v 2 , 3 , 3 } -minihypers and some [ 15 , 4 , 9 ; 3 ] -codes with B 2 = 0 , J. Statist. Plann. Inference 56 (1996) 129–146], which was considered to be sporadic; using our results, this strongly regular graph is now a member of an infinite family. Moreover, this paper contains a new proof that the Coulter–Matthews and ternary quadratic bent functions are weakly regular.

Constructions of p-ary Quadratic Bent Functions

Bent functions have many applications in the fields of coding theory, communications and cryptography. This paper studies the constructions of bent functions having the form $\sum_{i=1}^{(n-1)/2}c_{i}{\mathrm{tr}}_{1}^{n}(x^{p^{i}+1})$ for odd n and $\sum_{i=1}^{n/2-1}c_{i}{\mathrm{tr}}_{1}^{n}(x^{p^{i}+1})+c_{n/2}{\mathrm{tr}}_{1}^{n/2}(x^{p^{n/2}+1})$ for even n, over the finite field $\mathbb{F}_{p^{n}}$ of odd characteristic p, where $c_{i}\in \mathbb{F}_{p}$ . Based on the irreducibility of some polynomials on $\mathbb{F}_{p}$ , we focus on characterizing the bent functions for n=p v q r and n=2p v q r , where $v\geq0,\;r\geq1,\;q$ is an odd prime and p a primitive root modulo q 2. Moreover, the enumerations of those functions are also considered.