1 Introduction

Boolean bent functions introduced by Rothaus [37] are an interesting combinatorial object with the maximum Hamming distance to the set of all affine functions. Such functions have been extensively studied because of their important applications in cryptography (stream ciphers [5]), sequences [33], graph theory [35], coding theory (Reed–Muller codes [13], two-weight and three-weight linear codes [1, 17]), and association schemes [36]. A complete classification of bent functions is still elusive. Further, not only their characterization, but also their generation are challenging problems. Many papers on bent functions are devoted to the construction of bent functions [2,3,4,5, 9, 11, 12, 15, 16, 18, 22,23,24,25,26,27,28,29,30,31,32, 42].

Rotation symmetric Boolean functions, introduced by Pieprzyk and Qu [34], are invariant under circular translation of indices. Due to less space to be stored and allowing faster computation of the Walsh transform, they are of great interest. They can be obtained from idempotents (and vice versa) [19, 20]. Characterizing and constructing rotation symmetric bent functions are difficult and have theoretical and practical interest. The dual of a rotation symmetric bent function is also a rotation symmetric bent function. In the literature, few constructions of bent idempotents have been presented, which are restricted by the number of variables and have algebraic degree no more than 4. See more rotation symmetric bent functions in [7, 8, 14, 21, 38,39,40].

Quadratic rotation symmetric bent functions have been characterized by Gao et al. [21]. They proved that the quadratic function

$$\begin{aligned} \sum _{i=1}^{m-1}c_i\left( \sum _{j=0}^{n-1}x_jx_{i+j}\right) +c_m\left( \sum _{j=0}^{m-1}x_jx_{m+j}\right) \end{aligned}$$

is rotation symmetric bent if and only if the polynomial \(\sum _{i=1}^{m-1} c_i(X^i+X^{n-i})+c_mX^m\) is coprime with \(X^n+1\), where \(c_i\in {\mathbb {F}}_2\). Stanica et al. [38] conjectured that there are no homogeneous rotation symmetric bent functions of algebraic degree greater than 2. The construction of rotation symmetric bent functions of algebraic degree greater than 2 is an interesting problem [6]. Charnes et al. [10] constructed homogeneous bent functions of algebraic degree 3 in 8, 10, and 12 variables by applying the machinery of invariant theory. Up to now, there are few known constructions of rotation symmetric bent functions. Gao et al. [21] constructed an infinite class of cubic rotation symmetric bent functions of the form

$$\begin{aligned} f_t(x_0,x_1,\ldots ,x_{n-1}) =\sum _{i=0}^{n-1}(x_ix_{i+t}x_{i+m} +x_ix_{i+t})+\sum _{i=0}^{m-1} x_ix_{i+m}, \end{aligned}$$

where \(1\le t\le m-1\) and m / gcd(m, t) is odd. Carlet et al. [7] presented n-variable cubic rotation symmetric bent functions of the form

$$\begin{aligned} f(x_0,x_1,\ldots ,x_{n-1}) =\sum _{i=0}^{n-1}x_ix_{i+r}x_{i+2r}+ \sum _{i=0}^{2r-1}x_ix_{i+2r}x_{i+4r}+ \sum _{i=0}^{m-1}x_ix_{i+m}, \end{aligned}$$

where \(n=2m=6r\). Carlet et al. [8] proposed an infinite class of quartic rotation symmetric bent functions from two known semi-bent rotation symmetric functions by the indirect sum. Su and Tang [40] gave a class of n-variable rotation symmetric bent functions of any possible algebraic degree ranging from 2 to n / 2 of the form

$$\begin{aligned} f(x)=\sum _{i=0}^{m-1} x_ix_{i+m}+ \sum _{{\updelta }\in A} \sum _{{{\upbeta }'\boxplus {\upbeta }''\in {\mathcal {O}}_m({\updelta })} }\prod _{i=0}^{m-1}x_i^{{\upbeta }'} x_{i+m}^{{\upbeta }''}, \end{aligned}$$
(1)

where

  • \({\updelta }\in {\mathbb {F}}_2^m\).

  • \({\mathcal {O}}_n({\updelta })\) is the orbit of \({\updelta }\) by cyclic shift.

  • A is a subset of the representative elements of all the orbits \({\mathcal {O}}_m({\updelta })\).

  • \({\upbeta }'=({\upbeta }_0',{\upbeta }_1', \ldots ,{\upbeta }_{m-1}')\) and \({\upbeta }''=({\upbeta }_0'',{\upbeta }_1'', \ldots ,{\upbeta }_{m-1}'')\).

  • \(\boxplus \) denotes the sum over \({\mathbb {Z}}\).

These functions contain functions by Carlet et al. [7].

Motivated by the constructions of Gao et al. [21] and Su et al. [40], this paper constructs new rotation symmetric bent functions from some known rotation symmetric bent functions. We obtain two infinite classes of rotation symmetric bent functions which are equivalent to functions in the class of Maiorana–McFarland. Let \({\upgamma }(X_0,X_1,\ldots , X_{m-1})\) be a rotation symmetric polynomial in \({\mathbb {F}}_2[X_0,X_1,\ldots , X_{m-1}]\), i.e., \({\upgamma }(X_0,X_1,\ldots , X_{m-1}) ={\upgamma }(X_1,\ldots ,X_{m-1} ,X_{0})\). We obtain two classes of rotation symmetric bent functions of the form

$$\begin{aligned} f(x)&=\sum _{i=0}^{m-1}x_ix_{i+m} +{\upgamma }(x_0+x_m,\ldots , x_{m-1}+x_m),\\ f_t(x)&=\sum _{i=0}^{n-1}(x_ix_{i+t}x_{i+m} +x_{i}x_{i+t})+ \sum _{i=0}^{m-1}x_ix_{i+m}+ {\upgamma }(x_0+x_m,\ldots , x_{m-1}+x_{2m-1}), \end{aligned}$$

where \(1\le t\le m-1\) and m / gcd(m, t) is odd. In fact, these bent functions belong to the Maiorana–McFarland class of bent functions. Moreover, the two classes of rotation symmetric bent functions are disjoint.

The rest of the paper is organized as follows. Section 2 introduces some basic notations of Boolean functions and rotation symmetric bent functions. Section 3 presents the constructed rotation symmetric bent functions. Section 4 proves main results on rotation symmetric bent functions. Section 5 makes a conclusion.

2 Preliminaries

Let \({\mathbb {F}}_2^n\) denote the n-dimensional vector space over the finite field \({\mathbb {F}}_2\). An n-variable Boolean function \(f(x_0,x_1,\ldots ,x_{n-1})\) is a mapping from \({\mathbb {F}}_2^n\) to \({\mathbb {F}}_2\). And \(f(x_0,x_1,\ldots ,x_{n-1})\) can be represented by a polynomial called its algebraic normal form (ANF):

$$\begin{aligned} f(x_{0},x_1 \ldots , x_{n-1})= \sum _{u\in {\mathbb {F}}_2^n} c_{u}\left( \prod _{i=0}^{n-1}x_{i}^{{\upbeta }_i}\right) , \end{aligned}$$
(2)

where \(u=({\upbeta }_0,{\upbeta }_1,\ldots ,{\upbeta }_{n-1})\) and \(c_{u}\in {\mathbb {F}}_2\). The number of variables in the highest order product term with nonzero coefficient is called its algebraic degree.

Definition 1

A Boolean function f over \({\mathbb {F}}_2^n\) or an algebraic normal form f in \({\mathbb {F}}_2[x_0,x_1,\ldots ,x_{n-1}]\) is called rotation symmetric if for each input \(x=(x_0,x_1,\ldots ,x_{n-1}) \in {\mathbb {F}}_2^n\), we have

$$\begin{aligned} f(x_1,x_2,\ldots ,x_{n-1},x_0) =f(x_0,x_1,\ldots ,x_{n-1}). \end{aligned}$$

The Walsh transform of a Boolean function calculates the correlations between the function and linear Boolean functions. The Walsh transform of f over \({\mathbb {F}}_{2}^n\) is

$$\begin{aligned} {\mathcal {W}}_f(b)= \sum _{x\in {\mathbb {F}}_{2}^n} (-1)^{f(x)+ b\cdot x}, \end{aligned}$$

where \(b=(b_0,b_1,\ldots ,b_{n-1}) \in {\mathbb {F}}_{2}^n, x= (x_0,x_1,\ldots ,x_{n-1})\), and \(b\cdot x= \sum _{i=0}^{n-1}x_ib_i\).

Definition 2

A Boolean function \(f:{\mathbb {F}}_{2}^n \longrightarrow {\mathbb {F}}_2\) is a bent function if \({\mathcal {W}}_f(b) =\pm 2^{n/2}\) for any \(b\in {\mathbb {F}}_2^n\).

A Boolean bent function only exists for even n. The algebraic degree of a bent function is no more than m for \(n=2m\ge 4\) and the algebraic degree of a bent function for \(n=2\) is 2.

Let \({\upsigma }\) be a permutation of \({\mathbb {F}}_2^n\) such that for any bent function \(f, f\circ {\upsigma }\) is also bent. Then \({\upsigma }(x)=xA+b\), where A is an \(n\times n\) nonsingular binary matrix over \({\mathbb {F}}_2, xA\) is the product of the row-vector x and A, and \(b\in {\mathbb {F}}_2^n\). All these permutations form an automorphism of the set of bent functions. Two functions f(x) and \(g(x)=f\circ {\upsigma }(x)\) are called linearly equivalent. If f(x) is bent and L(x) is an affine function, then \(f+L\) is also a bent function. Two functions f and \(f\circ {\upsigma } +L\) are called EA-equivalent. The completed version of a class is the set of all functions, which are EA-equivalent to the functions in the class.

Maiorana and McFarland [26] introduced independently a class of bent functions by concatenating affine functions. This class is called the Maiorana–McFarland class \({\mathcal {M}}\) of functions defined over \({\mathbb {F}}_2^m \times {\mathbb {F}}_2^m\) of the form

$$\begin{aligned} f(a,y)=y\cdot {\uppi }(a)+h(a), \end{aligned}$$
(3)

where \((a,y)\in {\mathbb {F}}_2^m \times {\mathbb {F}}_2^m, {\uppi }(a)\) is any mapping from \({\mathbb {F}}_2^m\) to \({\mathbb {F}}_2^m\), and h(a) is any Boolean function on \({\mathbb {F}}_2^m\). Then f is bent if and only if \({\uppi }\) is bijective.

3 Two infinite classes of rotation symmetric bent functions

In this section, we only present two infinite classes of rotation symmetric bent functions. The proofs of the main results will be given in the next section.

Theorem 1

Let \(n=2m\) and \({\upgamma }(X_0,X_1,\ldots , X_{m-1})\in {\mathbb {F}}_2[X_0,X_1,\ldots , X_{m-1}]\) be an algebraic normal form of algebraic degree d. Then the function

$$\begin{aligned} f(x)=\sum _{i=0}^{m-1}x_ix_{i+m} + {\upgamma }(x_0+x_m,\ldots , x_{m-1}+x_{2m-1}) \end{aligned}$$

is a bent function. Further, if \({\upgamma }(X_0,X_1,\ldots , X_{m-1})\) is rotation symmetric, then f is a rotation symmetric bent function. If \(d\ge 2\), then f has algebraic degree d.

Example 1

Let \(m=6\). Then the function

$$\begin{aligned} f(x)=\sum _{i=0}^{5}x_ix_{i+6} + \prod _{i=0}^5(x_i+x_{i+6}) \end{aligned}$$

is a rotation symmetric bent function of algebraic degree 6.

Theorem 2

Let \(n=2m, t\) be an integer such that \(1\le t\le m-1\) and m / gcd(m, t) is odd, and \({\upgamma }(X_0,X_1,\ldots , X_{m-1})\in {\mathbb {F}}_2[X_0,X_1,\ldots , X_{m-1}]\) be an algebraic normal form. Then the function

$$\begin{aligned} f_t(x)= \sum _{i=0}^{n-1}(x_ix_{i+t}x_{i+m} +x_{i}x_{i+t})+ \sum _{i=0}^{m-1}x_ix_{i+m}+ {\upgamma }(x_0+x_m,\ldots , x_{m-1}+x_{2m-1}) \end{aligned}$$

is a bent function. Further, if \({\upgamma }(X_0,X_1,\ldots , X_{m-1})\) is rotation symmetric of algebraic degree \(d\ge 3\), then f is a rotation symmetric bent function of algebraic degree d.

Example 2

Let \(m=6\) and \(t=2\). Then the function

$$\begin{aligned} f_2(x)= \sum _{i=0}^{11}(x_ix_{i+2}x_{i+6} +x_{i}x_{i+2})+ \sum _{i=0}^{5}x_ix_{i+6} +\prod _{i=0}^5(x_i+x_{i+6}) \end{aligned}$$

is a rotation symmetric bent function of algebraic degree 6.

Example 3

Let m be an odd positive integer with \(m\ge 3\) and \(t=1\). Then the function

$$\begin{aligned} f_1(x)= \sum _{i=0}^{2m-1}(x_ix_{i+1}x_{i+m} +x_{i}x_{i+1})+ \sum _{i=0}^{m-1}x_ix_{i+m}+\prod _{i=0}^{m-1}(x_i+x_{i+m}) \end{aligned}$$

is a rotation symmetric bent function with the greatest possible algebraic degree m.

The following lemma shows that the two classes of rotation symmetric bent functions constructed in Theorems 1 and 2 do not overlap.

Lemma 1

Let \(g(x_0,x_1,\ldots ,x_{n-1})\) be a Boolean function on \({\mathbb {F}}_2^n\) or an algebraic normal form in \({\mathbb {F}}_2 [x_0,x_1,\ldots ,x_{n-1}]\) such that

  1. (1)

    for any \(0\le i\le m-1, g(x_0,\ldots ,x_i,\ldots ,x_{i+m}, \ldots ,x_{n-1}) =g(x_0,\ldots ,x_{i+m}, \ldots , x_i,\ldots ,x_{n-1})\);

  2. (2)

    for any \(0\le i\le m-1, x_ix_{i+m}\) is not in the terms of g;

  3. (3)

    g is rotation symmetric.

Then there exists a rotation symmetric polynomial \({\upgamma }(X_0,X_1,\ldots ,X_{m-1}) \in {\mathbb {F}}_2[X_0,X_1,\ldots ,X_{m-1}]\) such that

$$\begin{aligned} g(x_0,x_1,\ldots ,x_{n-1}) ={\upgamma }(x_0+x_m,x_1+x_{m+1},\ldots , x_{m-1}+x_{2m-1}). \end{aligned}$$

Proof

If there exists \({\upgamma }(X_0,X_1,\ldots ,X_{m-1})\) such that

$$\begin{aligned} g(x_0,x_1,\ldots ,x_{n-1}) ={\upgamma }(x_0+x_m,x_1+x_{m+1},\ldots , x_{m-1}+x_{2m-1}). \end{aligned}$$

Since g is rotation symmetric, then \({\upgamma }(X_0,X_1,\ldots ,X_{m-1})\) is rotation symmetric.

Now we will give the proof by the induction on algebraic degree d of g, i.e, there exists such rotation symmetric polynomial \({\upgamma }\) from rotation symmetric g(x) of algebraic degree d satisfying conditions (1) and (2).

  1. (1)

    When \(g=0\) or \(g=1\), such \({\upgamma }\) obviously exists.

  2. (2)

    When \(d=1\), such \({\upgamma }\) obviously exists.

  3. (3)

    Suppose \(d\ge 2\). From the conditions (1) and (2), there exists i such that

    $$\begin{aligned}&g(x_0,x_1,\ldots ,x_{n-1})\\&\quad = x_ig'(x_0,\ldots ,x_{i-1},x_{i+1}, \ldots ,x_{i+m-1},x_{i+m+1}, \ldots ,x_{n-1})\\&\qquad +x_{i+m}g''(x_0,\ldots ,x_{i-1},x_{i+1}, \ldots ,x_{i+m-1},x_{i+m+1}, \ldots ,x_{n-1}), \end{aligned}$$

    where \(g',g''\in {\mathbb {F}}_2(x_0,\ldots ,x_{i-1},x_{i+1}, \ldots ,x_{i+m-1},x_{i+m+1}, \ldots ,x_{n-1})\). From the condition (1), we have \(g'=g''\). From the induction of algebraic degree d, for \(g'\) and \(g''\), there exists \({\upgamma }' (X_0,\ldots ,X_{i-1},X_{i+1},\ldots , X_{m-1})\) such that

    $$\begin{aligned} g'=g''={\upgamma }'(x_0+x_m, \ldots ,x_{i-1}+x_{i+m-1}, x_{i+1}+x_{i+m+1},\ldots , x_{m-1}+x_{2m-1}). \end{aligned}$$

    Take \({\upgamma }(X_0,X_1,\ldots ,X_{m}) =X_i{\upgamma }' (X_0,\ldots ,X_{i-1},X_{i+1},\ldots , X_{m-1})\). Then

    $$\begin{aligned} g(x_0,x_1,\ldots ,x_{n-1}) ={\upgamma }(x_0+x_m,x_1+x_{m+1},\ldots , x_{m-1}+x_{2m-1}). \end{aligned}$$

Hence, this lemma follows. \(\square \)

Remark 1

Let \(f(x)=\sum _{i=0}^{m-1} x_ix_{i+m}+g(x) \) defined in Eq. (1), where

$$\begin{aligned}g(x)=\sum _{{\updelta }\in A} \sum _{{\upbeta }'\boxplus {\upbeta }''\in {\mathcal {O}}_m({\updelta }) }\prod _{i=0}^{m-1}x_i^{{\upbeta }'} x_{i+m}^{{\upbeta }''}. \end{aligned}$$

We can verify that g(x) satisfies all the three conditions in Lemma 1. There exists \({\upgamma }\in {\mathbb {F}}_2[X_0,X_1,\ldots , X_{m-1}]\) such that

$$\begin{aligned} f(x)=\sum _{i=0}^{m-1}x_ix_{i+m} + {\upgamma }(x_0+x_m,\ldots , x_{m-1}+x_{2m-1}) \end{aligned}$$

is bent. This shows that rotation symmetric bent functions constructed by Su and Tang [40] are contained in functions in Theorem 1. Let \(h(x)=\sum _{i=0}^{n-1}(x_ix_{i+t}x_{i+m} +x_{i}x_{i+t})\). From Lemma 1, h can not be expressed as \(h(x)={\upgamma }(x_0+x_m,\ldots , x_{m-1}+x_{2m-1})\) with \({\upgamma }\in {\mathbb {F}}_2[X_0,X_1,\ldots , X_{m-1}]\) being a rotation symmetric polynomial. Thus, any function f(x) constructed in Theorem 2 can not be written in \(f(x)=\sum _{i=0}^{m-1}x_ix_{i+m} + {\upgamma }(x_0+x_m,\ldots , x_{m-1}+x_{2m-1})\), where \({\upgamma }\) is a rotation symmetric polynomial in \({\mathbb {F}}_2[X_0,X_1,\ldots , X_{m-1}]\). Hence, the class of rotation symmetric bent functions constructed in Theorem 2 is completely disjoint from the one constructed in Theorem 1. In particular, any rotation symmetric bent functions in Theorem 2 are different from rotation symmetric bent functions constructed by Su and Tang [40].

Remark 2

From the proof of Lemma 1, it is observed that a Boolean function \(g(x_0,\ldots , x_{2m-1})\) on \({\mathbb {F}}_2^{2m}\) can be written as \(g={\upgamma }(x_0+x_m,\ldots ,x_{m-1}+x_{2m-1})\) with \({\upgamma }\in {\mathbb {F}}_2[X_0,X_1,\ldots , X_{m-1}]\) if and only if the following two conditions hold

  1. (1)

    for any permutation \({\upsigma }\) over \(\{0,1, \ldots , 2m-1\}\) with \(\{{\upsigma }(i), {\upsigma }(i+m)\}=\{i, i+m\}\), where \(0\le i\le m-1, g(x_{0},\ldots ,x_{2m-1})=g(x_{{\upsigma } (0)},\ldots ,x_{{\upsigma } (2m-1)})\);

  2. (2)

    for any \(0\le i\le m-1, x_ix_{i+m}\) is not in the terms of g.

For any boolean function \(g(x_{0},\ldots ,x_{2m-1})\) on \({\mathbb {F}}_2^{2m}\) and \(i\in \{0,1, \ldots , {2m-1}\}\), let \(D_ig\) be the functions defined as

$$\begin{aligned} D_ig(x_{0},\ldots ,x_{2m-1})=g(x_{0},\ldots ,x_i+1,\ldots , x_{2m-1})+g(x_{0},\ldots ,x_i,\ldots , x_{2m-1}). \end{aligned}$$

Then, one has the following proposition.

Proposition 1

Let \(g(x_0,\ldots , x_{2m-1})\) be a Boolean function on \({\mathbb {F}}_2^{2m}\). Then, \(g={\upgamma }(x_0+x_m,\ldots ,x_{m-1}+x_{2m-1})\) with \({\upgamma }\in {\mathbb {F}}_2[X_0,X_1,\ldots , X_{m-1}]\) if and only if \(D_ig=D_{i+m}g\) for any \(i\in \{0,1, \ldots , m-1\}\).

Proof

First, assume \(g={\upgamma }(x_0+x_m,\ldots ,x_{m-1}+x_{2m-1})\). Then,

$$\begin{aligned} D_{i+m}g(x_0,\ldots , x_{2m-1})= & {} {\upgamma }(x_0+x_m,\ldots ,x_{i}+x_{i+m} +1,\ldots ,x_{m-1} +x_{2m-1})\\&+{\upgamma }(x_0+x_m,\ldots ,x_{i}+x_{i+m},\ldots ,x_{m-1}+x_{2m-1})\\= & {} D_{i} g(x_0,\ldots , x_{2m-1}). \end{aligned}$$

Conversely, assume \(D_ig=D_{i+m}g\). Then, for any \(i\in \{0,1, \ldots , m-1\}\), there exist Boolean functions \(A_i, B_i, C_i\) and \(D_i\), whose values are completely independent with \(x_i\) and \(x_{i+m}\) such that

$$\begin{aligned} g(x_0,\ldots ,x_{2m-1})= A_ix_ix_{i+m}+B_ix_i +C_ix_{i+m}+D_i. \end{aligned}$$
(4)

Thus, \(D_{i+m}g(x_0,\ldots ,x_{2m-1})=A_ix_i+C_i\) and \(D_{i}g(x_0,\ldots ,x_{2m-1})=A_ix_{i+m}+B_i\). One gets \(A_i=0\) and \(B_i=C_i\) from \(D_ig=D_{i+m}g\). By Eq. (4), one has

$$\begin{aligned} g(x_0,\ldots ,x_{2m-1})= B_i(x_i +x_{i+m})+D_i. \end{aligned}$$

Hence, the function g satisfies the conditions (1) and (2) in Remark 2, which completes the proof.

4 Proofs

In this section, we give the proofs of our main results on rotation symmetric bent functions. We first give the following lemma on rotation symmetric functions.

Lemma 2

Let \({\upgamma } \in {\mathbb {F}}_2[X_0, \ldots , X_{m-1}]\), then \(g(x_0, \ldots , x_{2m-1})={\upgamma }(x_0+x_{m}, \ldots , x_{m-1}+x_{2m-1})\) over \({\mathbb {F}}_2^{2m}\) is rotation symmetric if and only if \({\upgamma }(X_0, \ldots , X_{m-1})\) over \({\mathbb {F}}_2^{m}\) is rotation symmetric.

Proof

If \({\upgamma }(X_0, \ldots , X_{m-1})\) over \({\mathbb {F}}_2^{m}\) is rotation symmetric, then

$$\begin{aligned} g(x_1,\ldots ,x_{m}, x_{m+1} \ldots , x_{0})&={\upgamma }(x_1+x_{m+1}, \ldots , x_{0}+x_{m})\\&={\upgamma }(x_0+x_{m}, \ldots , x_{m-1}+x_{2m-1})\\&= g(x_0,\ldots ,x_{m-1}, x_{m}, \ldots , x_{2m-1}). \end{aligned}$$

Thus, \({\upgamma }(x_0+x_{m}, \ldots , x_{m-1}+x_{2m-1})\) is rotation symmetric.

Conversely, let \(g(x_0, \ldots , x_{2m-1})\) be rotation symmetric. Set \(x_i=X_i\) for \(0\le i\le m-1\) and \(x_i=0\) for \(m\le i \le 2m-1\).Then,

$$\begin{aligned} {\upgamma }(X_1, \ldots , X_{0})&={\upgamma }(x_1+x_{m+1}, \ldots , x_{0}+x_{m})\\&=g(x_1,\ldots ,x_{m}, x_{m+1}, \ldots , x_{0})\\&=g(x_0,\ldots ,x_{m-1}, x_{m}, \ldots , x_{2m-1})\\&={\upgamma }(x_0+x_{m}, \ldots , x_{m-1}+x_{2m-1})\\&={\upgamma }(X_0, \ldots , X_{m-1}). \end{aligned}$$

Thus, \({\upgamma }(X_0, \ldots , X_{m-1})\) is rotation symmetric. \(\square \)

4.1 The proof of Theorem 1

For any function \({\upgamma }\) on \({\mathbb {F}}_2^m\), the function

$$\begin{aligned} f_0(a,y) =\sum _{i=0}^{m-1}y_ia_i+ {\upgamma }(a_0,a_1,\ldots ,a_{m-1}) \end{aligned}$$

is a bent function on \({\mathbb {F}}_2^m\times {\mathbb {F}}_2^m\) in the Maiorana–McFarland class \({\mathcal {M}}\) of functions defined in Eq. (3). Take the nondegenerate linear transform on \(f_0(a,y)\) as

$$\begin{aligned} y_i&=x_i,\\ a_i&=x_i+x_{i+m}, \end{aligned}$$

where \(0\le i\le m-1\). We have a bent function

$$\begin{aligned} f_1(x_0,x_1,\ldots ,x_{n-1})&= \sum _{i=0}^{m-1}x_i(x_i+x_{i+m}) +{\upgamma }(x_0+x_m,\ldots ,x_{m-1}+x_{2m-1})\\&= \sum _{i=0}^{m-1}x_ix_{i+m} +{\upgamma }(x_0+x_m,\ldots ,x_{m-1}+x_{2m-1}) + \sum _{i=0}^{m-1}x_i. \end{aligned}$$

Since \( \sum _{i=0}^{m-1}x_i\) is a linear function, then \(f(x)= f_1+ \sum _{i=0}^{m-1}x_i\) is a bent function. Since Lemma 2, if \({\upgamma }\) is a rotation symmetric polynomial in \({\mathbb {F}}_2[X_0,X_1, \ldots ,X_{m-1}]\), then f(x) is also rotation symmetric.

If \({\upgamma }(X_0,X_1,\ldots ,X_{m-1}) \) has algebraic degree d, then \({\upgamma }(x_0+x_m,\ldots ,x_{m-1}+x_{2m-1})\) has algebraic degree d. If \(d\ge 3\), then the algebraic degree of f is d. Otherwise, f has algebraic degree less than 2. Thus, f has algebraic degree 2 since f is bent. Hence, Theorem 1 follows.

4.2 The proof of Theorem 2

In order to prove Theorem 2, we first recall some notations and results in [21]. Define the sets

$$\begin{aligned} E&=\{x\in {\mathbb {F}}_2^n: x_i+x_{i+m}=1 \text { for } 0\le i\le m-1\}\\&=\{(y_0,\ldots , y_{m-1}, 1+y_0,\ldots , 1+y_{m-1})\in {\mathbb {F}}_2^n: (y_0,\ldots , y_{m-1})\in {\mathbb {F}}_2^m\} \end{aligned}$$

and

$$\begin{aligned} W&=\{x\in {\mathbb {F}}_2^n: x_i=0 \text { for } m\le i\le n-1\}\\&=\{(a_0,\ldots , a_{m-1},0,\ldots , 0)\in {\mathbb {F}}_2^n: (a_0,\ldots , a_{m-1})\in {\mathbb {F}}_2^m\}. \end{aligned}$$

Then

$$\begin{aligned} {\mathbb {F}}_2^n= \bigcup _{a\in W} (a+E). \end{aligned}$$

Thus, for any \(x\in {\mathbb {F}}_2^n\), there exists a unique pair (a, y) (\(a\in W\) and \(y\in E\)), such that \(x=a+y\). Furthermore, if \(x=(x_0, x_1, \ldots , x_{n-1})\), then

$$\begin{aligned} y&=(x_m+1, x_{m+1}+1, \ldots , x_{n-1}+1, x_m, x_{m+1}, \ldots , x_{n-1})\nonumber \\ a&=(x_0+x_m+1, x_1+x_{m+1}+1, \ldots , x_{m-1}+x_{n-1}+1, 0, 0, \ldots , 0). \end{aligned}$$
(5)

Gao et al. [21] proved that \(F_t(x) =\sum _{i=0}^{n-1}(x_ix_{i+t}x_{i+m} +x_ix_{i+t})+\sum _{i=0}^{m-1} x_ix_{i+m} \) can be expressed in Maiorana–McFarland’s form

$$\begin{aligned} F_t(x)=F_t(a+y)={\uppi } (a) \cdot y +h_0(a), \end{aligned}$$
(6)

where \(a\in W, y\in E\) , \(h_0(a) =\sum _{i=m-t}^{m-1}a_ia_{i+t}\), and \({\uppi }(a)=({\uppi }_0(a),{\uppi }_1(a),\ldots , {\uppi }_{m-1}(a),0,0, \ldots , 0)\) with \({\uppi }_i(a)=a_ia_{(i+t) \mod m}+a_{(i+t) \mod m} +a_{(i+m-t) \mod m}\). Since m / gcd(m, t) is odd, then from Gao et al. [21][Proof in Theorem 1], \(a \mapsto ({\uppi }_0(a),{\uppi }_1(a),\ldots , {\uppi }_{m-1}(a), 0, 0, \ldots , 0) \) is a permutation of W. Then \(F_t(x)\) is a bent function.

Let \(f_t(x)=F_t(x)+{\upgamma }(x_0+x_m,\ldots , x_{m-1}+x_{2m-1})=\sum _{i=0}^{n-1}(x_ix_{i+t}x_{i+m}+x_{i}x_{i+t})+ \sum _{i=0}^{m-1}x_ix_{i+m}+{\upgamma }(x_0+x_m,\ldots , x_{m-1}+x_{2m-1})\), where \({\upgamma } \in {\mathbb {F}}_2[X_0, X_1, \ldots , X_{m-1}]\). From Eq. (5), for any \({\upgamma } \in {\mathbb {F}}_2[X_0, X_1, \ldots , X_{m-1}]\), there exists a function \(h_1\) on W such that \({\upgamma }(x_0+x_m,\ldots , x_{m-1}+x_{2m-1})=h_1(a)\). Then, from Eq. (6), we can express \(f_t\) in the form

$$\begin{aligned} f_t(x)=f_t(a+y)={\uppi } (a) \cdot y +(h_0(a)+h_1(a)). \end{aligned}$$

Hence, \(f_t(x)\) is a bent function. From Lemma 2, if \({\upgamma }\) is a rotation symmetric polynomial in \({\mathbb {F}}_2[X_0,X_1, \ldots ,X_{m-1}]\), then \(f_t(x)\) is also rotation symmetric. Obviously, if \({\upgamma }\) has algebraic degree \(d\ge 3\), then f is also a function of algebraic degree d. Hence, Theorem 2 follows.

Remark 3

From the proofs of Theorems 1 and 2, bent functions in both theorems are in the completed Maiorana–McFarland class of bent functions.

5 Conclusion

In this paper, we propose a systematic method for constructing n-variable rotation symmetric bent functions from some functions in the Maiorana–McFarland class. One class of rotation symmetric bent functions has algebraic degree ranging from 2 to m and the other class has algebraic degree ranging from 3 to m.