1 Introduction

In cryptography, Boolean functions are used extensively (block ciphers and stream ciphers). They are also elementary units of error-correcting codes. Let \(h:\mathbb {F}_{2^m} \rightarrow \mathbb {F}_2\) be a Boolean function in m-variable. An essential cryptographic technique that significantly contributes to the security of symmetric cryptosystems is the nonlinearity of Boolean functions [2, 5, 7]. They are also crucial in preventing Best Affine Approximation Attacks as well as Fast Correlation Attacks. In coding theory, nonlinearity and its property are vital as the covering radius of Reed-Muller code having length \(2^m\) and order r is equal to the maximum value of the \(r-th\) order nonlinearity of Boolean functions in m-variable [8]. Recently Wang et al. [25] showed that the covering radius of RM(2, 7) is at most 42. Also, RM(3, 7), i.e., the covering radius of third order Reed-Muller has been discussed by Gao et al. [13]. They corroborate that for a function \(f\in \mathcal {B}_{7,2}\), the third order nonlinearity \(nl_3(f)>20\) does not hold.

In 1976, Rothaus [23] was the first to established the concept of nonlinearity. A few known results exist about \(r-th\) order nonlinearity of a Boolean function h, i.e., \(nl_r(h)\), for \(r>1\). In 2006, Carlet et al. [6] provided the best asymptotic upper bound, which is given as:

$$\begin{aligned} nl_r(h)=2^{m-1}-\frac{\sqrt{15}}{2}(1+\sqrt{2})^{r-2}\cdot 2^{m/2}+O(m^{r-2}). \end{aligned}$$

Boolean functions with algebraic degrees strictly greater than r exhibit \(r-th\) order nonlinearity, which is difficult to compute. A lot of research has been done so far to compute the \(nl_r(h)\) for \(r>1\), as there is a relation between the nonlinearity and the Walsh Hadamard transform (WHT) of Boolean functions and the Walsh-Hadamard transform is easily computed by FFT (fast Fourier transform ). In [10, 21] Kabatiansky and Tavernier proposed an algorithm later enhanced and resolved by Fourquet et al. [11] for \(r=2\) and \(m \le 11\). It is also applicable for \(m \le 13\) (in some cases). A special algorithm is still missing in the literature for the computation of \(nl_r(h)\) when \(r\ge 3\). Iwata et al. [20] proposed a bent function with nonlinearity of \(r-th\) order having lower bound \(2^{m-r-3}(r+4)\), for \(r\le m-3\). In this area of research, Carlet had a great contribution. He developed the recursive approach to get a lower bound on the nonlinearity of the \(r-th\) order of a Boolean function [3]. To analyze the lower bounds of second as well as third order nonlinearity many authors contributed their work to distinct classes of Boolean functions [12, 14, 15, 18, 19, 24]. In this article, for \(m=2p\), we analyze the lower bounds on higher order nonlinearity of a Boolean function \(f_{\alpha }(u)=tr_1^m(\alpha u^{(2^p-1)3+1})\), for all \(u\in \mathbb {F}_{2^m},~\alpha \in \mathbb {F}_{2^m}^*\) and a Boolean function \(g_{\alpha }(u)=tr_1^m(\alpha u^{2^p-2})\), for all \(u\in \mathbb {F}_{2^m},~\alpha \in \mathbb {F}_{2^m}^*\).

2 Preliminaries

Assuming that \(\mathbb {F}_{2}\) is the finite field and \(\mathbb {F}_{2}^{m}\) is the vector space of all \(m-\)tuples over \(\mathbb {F}_{2}\). \(\mathcal {B}_{m,2}\) represents the set of all \(m-\)variable Boolean functions and is the function from \(\mathbb {F}_{2}^{m}\) to \(\mathbb {F}_{2}\). Since \(\mathbb {F}_{2}^{m}\) is isomorphic to the finite field \(\mathbb {F}_{2^m}\) therefore, a Boolean function can also be considered as a function from \(\mathbb {F}_{2^m}\) to \(\mathbb {F}_{2}\). The support of a Boolean function h is given by \(S_{h} =\{u\in \mathbb {F}_{2}^{m}:h(u)\ne 0 \}\), whose cardinality \( \mid S_{h}\mid \) is known as the Hamming weight of h.

Boolean functions are represented by their truth table representation in which all the \(2^m\) elements of \(\mathbb {F}_2^m\) are in lexicographically increasing order that is

$$\begin{aligned}{}[h(0,0,...,0),h(0,0,...,1),...,h(1,1,...,1)]. \end{aligned}$$

If the truth table of a Boolean function has the same number of \(0's\) and \(1's\), then the Boolean function is known as a balanced Boolean function. The balancedness of a Boolean function can also be defined with the help of its Hamming weight. That is, for a Boolean function \(h \in \mathcal {B}_{m,2}\), if \(wt(h)=2^{m-1}\), then it is considered to be balanced . To study the cryptographic properties of Boolean functions it is not always favorable to represent Boolean functions by their truth table, so we study an another representation of Boolean functions which is known as Algebraic Normal Form (ANF).

The Algebraic Normal Form of a Boolean function h can be defined as

$$\begin{aligned} h(u_1,u_2,...,u_m)=\sum \limits _{a=(a_1,a_2,...,a_m)\in \mathbb {F}_2^m} \mu _a \Big ( \prod _{i=1}^m u_i^{a_i} \Big ), \end{aligned}$$

where \(\mu _a \in \mathbb {F}_2\). The number of variables in the highest order term of the Algebraic Normal Form of a Boolean function h for which \(\mu _a \ne 0\) is known as the algebraic degree of h and is denoted by deg(h).The trace function \(tr_s^m: \mathbb {F}_{2^m}\rightarrow \mathbb {F}_{2^s}\), where \(m=qs\) for some integer q can be defined as

$$\begin{aligned} tr_s^m(u)=u+u^{2^s}+u^{2^{2s}}+...+u^{2^{m-s}}, \end{aligned}$$

for all \(u \in \mathbb {F}_{2^m}\). When \(s=1\), the trace function \(tr_1^m:\mathbb {F}_{2^m}\rightarrow \mathbb {F}_2\) is known as the absolute trace function. The absolute trace function \(tr_1^m:\mathbb {F}_{2^m}\rightarrow \mathbb {F}_2\) has some properties which are as follows:

  1. 1.

    \(tr_1^m\) is a linear transformation from \(\mathbb {F}_{2^m}\) into \(\mathbb {F}_2\).

  2. 2.

    \(tr_1^m(u^{2^i})=tr_1^m(u)\), for all \(u \in \mathbb {F}_{2^m}\) and i is any positive integer.

Suppose \(u,v \in \mathbb {F}_2^m\). Let us define \(\langle u,v \rangle = u_1v_1+u_2v_2+...+u_mv_m\) which is known as the inner product of u and v. Also, \(\langle u,v \rangle = tr_1^m(uv)\), for all \(u,v \in \mathbb {F}_{2^m}\) is known as the trace of the product of u and v.

Let \(h_1,h_2 \in \mathcal {B}_{m,2}\), then the Hamming distance \(d(h_1,h_2)\) is defined by

$$\begin{aligned} d(h_1,h_2)=\mid \{u \in \mathbb {F}_{2^m}:h_1(u)\ne h_2(u)\} \mid . \end{aligned}$$

The \(r-th\) order nonlinearity of \(h_1\) is the least hamming distance between \(h_1\) and all the Boolean functions having algebraic degree at most r i.e.,

$$\begin{aligned} nl_r(h_1)= min_{h_2\in \mathcal {B}_{m,2},~ deg(h_2)\le r}d(h_1,h_2). \end{aligned}$$

When \(r=1\), it is simply denoted by \(nl(h_1)\).

Let \(h \in \mathcal {B}_{m,2}\) and \(\alpha \in \mathbb {F}_2^m\) then the Walsh-Hadamard transform of h at \(\alpha \) is defined as

$$\begin{aligned} W_h(\alpha )=\sum \limits _{u\in \mathbb {F}_2^{m}}(-1)^{h(u)+\langle u, v\rangle }. \end{aligned}$$

In terms of \(tr_1^m(\alpha u)\), for all \(\alpha ,u \in \mathbb {F}_{2^m}\), the Walsh-Hadamard transform of a Boolean function \(h \in \mathcal {B}_{m,2}\) can also be defined as

$$\begin{aligned} W_h(\alpha )=\sum \limits _{u\in \mathbb {F}_{2^m}}(-1)^{h(u)+tr_1^m(\alpha u)}. \end{aligned}$$

The multiset \([W_h(\alpha ):\alpha \in \mathbb {F}_2^m]\) is known as the Walsh spectrum of h. For a Boolean function \(h \in \mathcal {B}_{m,2}\), the nonlinearity and the Walsh-Hadamard transform are correlated as follows:

$$\begin{aligned} nl(h)=2^{m-1}-\frac{1}{2}\max _{\alpha \in \mathbb {F}_2^m} \mid W_h(\alpha ) \mid . \end{aligned}$$

A Boolean function \(h \in \mathcal {B}_{m,2}\) is known as bent function if h has maximum nonlinearity with \(r=1\) i.e., \(nl(h)=2^{m-1}-2^{\frac{m}{2}-1}\).

The derivative of a Boolean function \(h \in \mathcal {B}_{m,2}\) with respect to \(x\in \mathbb {F}_2^m\) is defined as

$$\begin{aligned} D_xh(u)=h(u)+h(u+x), \end{aligned}$$

for all \(u \in \mathbb {F}_2^m\). The higher-order derivative can be obtained by extending the definition of the derivative of a Boolean function. Let \(W_j\) be a \(j-\)dimensional subspace of \(\mathbb {F}_2^m\) having \(x_1,x_2,...,x_j\) as a basis. Then the \(j-\)th order derivative of h with respect to \(W_j\) is defined as

$$\begin{aligned} D_{W_j}h(u)=D_{x_1}D_{x_2}...D_{x_j}h(u)= \sum \limits _{w\in \mathbb {F}_2^j}h\left( u+\sum \limits _{i=1}^jw_ix_i\right) , \end{aligned}$$

for all \(u \in \mathbb {F}_2^m\).

Let the set of all invertible matrices of order \(m \times m\) is denoted by \(GL(m,\mathbb {F}_2)\). The entries of matrices of \(GL(m,\mathbb {F}_2)\) are either 0 or 1. Let \(g,h \in \mathcal {B}_{m,2}\) are two Boolean functions. For a matrix \(M \in GL(m,\mathbb {F}_2),~y,~\alpha \in \mathbb {F}_2^m\) and \(\epsilon \in \mathbb {F}_2\) if

$$\begin{aligned} h(u)=g(Mu+y)+\langle \alpha , u \rangle + \epsilon , \end{aligned}$$

for all \(u \in \mathbb {F}_2^m\), then g and h are known as affine equivalent.

Let \(d\in \{1,2,...,2^m-2\}\) be any integer. Then d is known as Niho exponent and if \(u^d\) is restricted to \(\mathbb {F}_{2^p}\) linearly, it is known as Niho power function or we can say that if \(d\equiv 2^i(mod~ 2^p-1)\).

For any prime power k, a polynomial

$$\begin{aligned} P(u)=\sum \limits _{i=0}^m\zeta _i u^{k^i}, \end{aligned}$$

where each \(\zeta _i\in \mathbb {F}_{k^s}\) (an extension field of \(\mathbb {F}_k\)) is known as a linearized polynomial over \(\mathbb {F}_{k^s}\).

Proposition 2.1

[3] Let \(h\in \mathcal {B}_{m,2}\) and \(r<m\) be any positive integer. Then for all \(0<j<r\),

$$\begin{aligned} nl_r(h)\ge \frac{1}{2^j} \max _{x_1,x_2,...,x_j \in \mathbb {F}_{2^m}}nl_{r-j}(D_{x_1}D_{x_2}...D_{x_j}h). \end{aligned}$$

We shall use the notation \([l,x_1,x_2,...,x_m]\) for any collection of integers with the aforementioned characteristics:

  1. 1.

    \(\sum _{j=1}^m x_j=l\).

  2. 2.

    \(x_j>0\), for all \(j=1,2,...,m\).

  3. 3.

    \(x_j \wedge x_k=0\), for all \(1\le j < k \le m\), where \(\wedge \) is the bitwise AND operation.

It indicates that m non-empty disjoint groups are formed from one bit of the binary representation of l.

Lemma 2.2

[16] Let \(u,v \in \mathbb {F}_{2^m}\) and \(l>0\) then

$$\begin{aligned} (u+v)^l=\sum \limits _{[l,j,k]}u^jv^k+u^l+v^l. \end{aligned}$$

Lemma 2.3

[16] Let t,  l and \(x_j>0\) are positive integers then for all \(j=1,2,...,t\)

$$\begin{aligned} D_{x_1}D_{x_2}...D_{x_t}u^l=\sum \limits _{[l,\beta _0,\beta _1,...,\beta _t]}u^{\beta _0}x_1^{\beta _1}...x_t^{\beta _t}+ constant. \end{aligned}$$

Let us assume a quadratic Boolean function \(h\in \mathcal {B}_{m,2}\). It is said that the bilinear form B(a, b) associated with h is \(B(a,b)=h(0)+h(a)+h(b)+h(a+b)\). The kernel of B(a, b) [1] is the subspace of \(\mathbb {F}_{2^m}\) and is defined by

$$\begin{aligned} \varepsilon _h=\{a \in \mathbb {F}_{2^m}:B(a,b)=0,~~for~all~b\in \mathbb {F}_{2^m}\}. \end{aligned}$$

Lemma 2.4

[1] Let \(\mathbb {F}_q\) be the field of characteristic 2 and W be the vector space of \(\mathbb {F}_q\). If \(Q:W\rightarrow \mathbb {F}_q\) is a quadratic form on W then the parity of the dimensions of W and the kernel of B(u, v) is the same.

Lemma 2.5

[1, 22] Let \(h:\mathbb {F}_{2^m}\rightarrow \mathbb {F}_2\) be the Boolean quadratic form and the bilinear form associated with h is B(u, v). Let b be the dimension of the kernel of B(u, v). The Walsh spectrum of h depends only on the dimension b. Also the Walsh spectrum of h is given by:

\(W_h(\alpha )\)

Number of \(\alpha \)

0

\(2^m-2^{m-b}\)

\(2^{(m+b)/2}\)

\(2^{(m-b-1)/2}+(-1)^{h(0)}2^{(m-b-2)/2}\)

\(-2^{(m+b)/2}\)

\(2^{(m-b-1)/2}-(-1)^{h(0)}2^{(m-b-2)/2}\)

3 Main result

In the below given results, we contributed our work to find the general lower bounds on the higher order nonlinearity of some classes of Boolean functions. In Theorem 3.1, we study the nonlinearity of Boolean function with higher order by using the Niho power function, which was given by Dobbertin et al. [9]. In Theorem 3.2, we compute the lower bound on higher order nonlinearity of the inverse Boolean function.

Theorem 3.1

Let \(m=2p\), where \(p>0\) and a Boolean function \(f_\alpha \in \mathcal {B}_{m,2}\) of the form

$$\begin{aligned} f_{\alpha }(u)=tr_1^m(\alpha u^d), \end{aligned}$$

where \(d=(2^p-1)3+1\), for all \(u\in \mathbb {F}_{2^m},~\alpha \in \mathbb {F}_{2^m}^*\). Then

$$\begin{aligned} nl_{(r=p-1)}f_{\alpha }(u) \ge 2^{p+1}-2^{\frac{p+1}{2}}. \end{aligned}$$

Proof

The Boolean function \(f_{\alpha }(u)\) is of algebraic degree p [9]. By applying Proposition 2.1, we obtain

$$\begin{aligned} nl_{(r=p-1)}f_{\alpha }(u)\ge \frac{1}{2^{p-2}}\max \limits _{x_1,x_2,...,x_{p-2}}nl(D_{x_1}D_{x_2}...D_{x_{p-2}}(f_{\alpha }(u))). \end{aligned}$$
(3.1)

Let \(x_1,x_2,...,x_{p-2}\in \mathbb {F}_{2^m}\), then observe that

$$\begin{aligned} D_{x_1}D_{x_2}...D_{x_{p-2}}(f_{\alpha }(u))=D_{x_1}D_{x_2}...D_{x_{p-2}}(tr_1^m(\alpha u^{d})). \end{aligned}$$

Hence,

$$\begin{aligned} D_{x_1}D_{x_2}...D_{x_{p-2}}(f_{\alpha }(u))=tr_1^m\left( \alpha \left( D_{x_1}D_{x_2}...D_{x_{p-2}}(u^d)\right) \right) . \end{aligned}$$
(3.2)

Now by applying Lemma 2.3, we get

$$\begin{aligned} D_{x_1}D_{x_2}...D_{x_{p-2}}(u^d)=\sum \limits _{[d,\gamma _0,\gamma _1,...,\gamma _{p-2}]} u^{\gamma _{0}}x_1^{\gamma _{1}}...x_{p-2}^{\gamma _{p-2}}+~constant. \end{aligned}$$
(3.3)

From Eqs. (3.2) and (3.3), we have

$$\begin{aligned} D_{x_1}D_{x_2}...D_{x_{p-2}}(f_{\alpha }(u))= tr_1^m(\alpha \sum \limits _{[d,\gamma _0,\gamma _1,...,\gamma _{p-2}]} u^{\gamma _{0}}x_1^{\gamma _{1}}...x_{p-2}^{\gamma _{p-2}}+~constant). \end{aligned}$$

Also, it is very well known that in the binary form of \(d=(2^{p}-1)3+1\) there exist p ones and each \(\gamma _i>0\), for all i, must possess at least one of them. Again, in the binary form of \(\gamma _0\) there are \(p-(p-2)=2\) ones. Therefore, the above Boolean function is a quadratic Boolean function. Let us assume that \(g_{\alpha }(u)\) be an affine equivalent Boolean function to \(D_{x_1}D_{x_2}...D_{x_{p-2}}(f_{\alpha }(u))\) which can be obtained from the above expression by eliminating all the terms with \(wt(\gamma _0)=1\) and the constant terms in the sum. Since \(D_{x_1}D_{x_2}...D_{x_{p-2}}(f_{\alpha }(u))\) and \(g_{\alpha }(u)\) are affine equivalent, therefore they both have the same nonlinearity. The bilinear form B(u, v) associated with \(g_\alpha (u)\) is

$$\begin{aligned} B(u,v)=g_{\alpha }(0)+g_{\alpha }(u)+g_{\alpha }(v)+g_{\alpha }(u+v). \end{aligned}$$

Then by applying Lemma 2.2, we have

$$\begin{aligned} B(u,v)=tr_1^m\left( \sum \limits _{i=1}^p v^{\beta _i}\left( \sum \limits _{[d,\lambda ,\beta _i,\gamma _1,...,\gamma _{p-2}]} \alpha u^{\lambda }x_1^{\gamma _1}...x_{p-2}^{\gamma _ {p-2}}\right) \right) . \end{aligned}$$

Then,

$$\begin{aligned} B(u,v)=tr_1^m(\sum \limits _{i=1}^p v^{\beta _i}Q_i(u)), \end{aligned}$$

where,

$$\begin{aligned} Q_i(u)=\sum \limits _{[d,\lambda ,\beta _i,\gamma _1,...,\gamma _{p-2}]} \alpha u^{\lambda }x_1^{\gamma _1}...x_{p-2}^{\gamma _ {p-2}}. \end{aligned}$$

Since in the binary form of d there are p ones and \(\lambda , \beta _i, \gamma _i,..., \gamma _{p-2}\) are p in number for \(i=1,2,...,p\). Consequently, they are all powers of 2. By applying the properties of linearity \(v^{2^m}=v\) and the property of trace function, \(tr_1^m(u^{2^i})=tr_1^m(u)\), for all \(u\in \mathbb {F}_{2^m}\), B(u, v) becomes to

$$\begin{aligned} B(u,v)=\sum \limits _{i=1}^{p}tr_1^m(vQ_i(u)^ {\frac{2^m}{\beta _i}}), \end{aligned}$$

or

$$\begin{aligned} B(u,v)=tr_1^m(v\sum \limits _{i=1}^{p}Q_i(u)^{\frac{2^m}{\beta _i}}). \end{aligned}$$

Therefore,

$$\begin{aligned} B(u,v)=tr_1^m(v Q(u)), \end{aligned}$$

where, \(Q(u)=\sum _{i=1}^p Q_i(u)^{\frac{2^m}{\beta _i}}\).

By the definition, the kernel of B(u, v) is \(\varepsilon _f=\{u \in \mathbb {F}_{2^m}:B(u,v)=0,~for ~all~v\in \mathbb {F}_{2^m}\}\). So, \(\varepsilon _f\) has equal number of zeros of the polynomial Q(u), where

$$\begin{aligned} Q(u)=\sum \limits _{i=1}^p\sum \limits _{[d,\lambda ,\beta _i,\gamma _1,...,\gamma _{p-2}]}\alpha ^{\frac{2 ^m}{\beta _i}}u^{\frac{2 ^m \lambda }{\beta _i}}x_1^{\frac{2 ^m \gamma _1}{\beta _i}}...x_{p-2}^{\frac{2 ^m \gamma _{p-2}}{\beta _i}}, \end{aligned}$$

or

$$\begin{aligned} Q(u)=\sum \limits _{i=1}^p\sum \limits _{[d,\lambda ,\beta _i,\gamma _1,...,\gamma _{p-2}]}\alpha ^{\frac{1}{\beta _i}}u^{\frac{\lambda }{\beta _i}}x_1^{\frac{\gamma _1}{\beta _i}}... x_{p-2}^{\frac{\gamma _{p-2}}{\beta _i}}. \end{aligned}$$

Since, there are same number of zeroes in Q(u) and \(Q(u)^{2^{p-1}}\). Therefore, the number of elements in \(\varepsilon _f\) and the number of zeroes of the polynomial \(Q(u)^{2^{p-1}}\) are also same. Hence, we have,

$$\begin{aligned} Q(u)^{2^{p-1}}= \sum \limits _{i=1}^p\sum \limits _{[d,\lambda ,\beta _i,\gamma _1,...,\gamma _{p-2}]}\alpha ^{\frac{2 ^{p-1}}{\beta _i}}u^{\frac{2 ^{p-1} \lambda }{\beta _i}}x_1^{\frac{2 ^{p-1} \gamma _1}{\beta _i}}...x_{p-2}^{\frac{2 ^{p-1} \gamma _{p-2}}{\beta _i}}, \end{aligned}$$

a linearized polynomial in u whose degree is at most \(2^{p-1}\). Also, the kernel \(\varepsilon _f\) has at most \(2^{p-1}\) number of elements. Let the kernel of B(u, v) has dimension k, by applying Lemma 2.4, \(k\le p-1\), since m is even. Therefore, by using Lemma 2.5, for all \(u\in \mathbb {F}_{2^m}\), we obtain,

$$\begin{aligned} W_{D_{x_1}D_{x_2}...D_{x_{p-2}}f_{\alpha }}(u) \le 2^{\frac{2^m+k}{2}} \le 2^{\frac{3p-1}{2}}. \end{aligned}$$

So,

$$\begin{aligned} nl_{D_{x_1}D_{x_2}...D_{x_{p-2}}f_{\alpha }}(u) \ge 2^{2p-1}- 2^{\frac{3p-3}{2}}. \end{aligned}$$
(3.4)

From Eqs. (3.1) and (3.4), we have,

$$\begin{aligned} nl_{(r=p-1)}f_{\alpha }(u) \ge 2^{p+1}-2^{\frac{p+1}{2}}. \end{aligned}$$

\(\square \)

Theorem 3.2

Let \(m=2p\), where \(p>0\) and let \(g_\alpha \) be a Boolean function of the form

$$\begin{aligned} g_{\alpha }(u)=tr_1^m(\alpha u^d), \end{aligned}$$

where \(d=(2^p-2)\), for all \(u\in \mathbb {F}_{2^m},~\alpha \in \mathbb {F}_{2^m}^*\). Then

$$\begin{aligned} nl_{(r=p-2)}g_{\alpha }(u) \ge 2^{p+2}-2^{\frac{p+3}{2}}. \end{aligned}$$
Table 1 Comparison of the lower bounds of the higher order nonlinearity
Full size table

Proof

For any positive integer p, we know that \(2^p-2=2^{p-1}+2^{p-2}+...+2^2+2\). Therefore, the binary expansion of \(2^p-2\) has \(p-1\) ones. Hence the algebraic degree of \(g_{\alpha }(u)\) is \(p-1\). By Proposition 2.1, we obtain

$$\begin{aligned} nl_{(r=p-2)}g_{\alpha }(u)\ge \frac{1}{2^{p-3}}\max \limits _{x_1,x_2,...,x_{p-3}}nl(D_{x_1}D_{x_2}...D_{x_{p-3}}(g_{\alpha }(u))). \end{aligned}$$
(3.5)

Let \(x_1,x_2,...,x_{p-3}\in \mathbb {F}_{2^m}\), then observe that

$$\begin{aligned} D_{x_1}D_{x_2}...D_{x_{p-3}}(g_{\alpha }(u))=D_{x_1}D_{x_2}...D_{x_{p-3}}(tr_1^m(\alpha u^{d})). \end{aligned}$$

Hence,

$$\begin{aligned} D_{x_1}D_{x_2}...D_{x_{p-3}}(g_{\alpha }(u))=tr_1^m\left( \alpha \left( D_{x_1}D_{x_2}...D_{x_{p-3}}(u^d)\right) \right) . \end{aligned}$$
(3.6)

Now by applying Lemma 2.3, we get

$$\begin{aligned} D_{x_1}D_{x_2}...D_{x_{p-3}}(u^d)=\sum \limits _{[d,\gamma _0,\gamma _1,...,\gamma _{p-3}]} u^{\gamma _{0}}x_1^{\gamma _{1}}...x_{p-3}^{\gamma _{p-3}}+~constant. \end{aligned}$$
(3.7)

From Eqs. (3.6) and (3.7), we have

$$\begin{aligned} D_{x_1}D_{x_2}...D_{x_{p-3}}(g_{\alpha }(u))= tr_1^m(\alpha \sum \limits _{[d,\gamma _0,\gamma _1,...,\gamma _{p-3}]} u^{\gamma _{0}}x_1^{\gamma _{1}}...x_{p-3}^{\gamma _{p-3}}+~constant). \end{aligned}$$

Since in the binary form of \(d=(2^{p}-2)\) there exist \(p-1\) ones and each \(\gamma _i>0\), for all i, must have at least one of them. Also, in the binary form of \(\gamma _0\) there are 2 ones. Therefore, the Boolean function \(g_{\alpha }(u)\)is quadratic. Let us assume that \(h_{\alpha }(u)\) be an affine equivalent Boolean function to \(D_{x_1}D_{x_2}...D_{x_{p-3}}(g_{\alpha }(u))\) which can be obtained from the above expression by eliminating all the terms with \(wt(\gamma _0)=1\) and the constant terms in the sum. Since \(D_{x_1}D_{x_2}...D_{x_{p-3}}(g_{\alpha }(u))\) and \(h_{\alpha }(u)\) are affine equivalent, therefore they both have the same nonlinearity. The bilinear form B(u, v) associated with \(h_\alpha (u)\) is

$$\begin{aligned} B(u,v)=h_{\alpha }(0)+h_{\alpha }(u)+h_{\alpha }(v)+h_{\alpha }(u+v). \end{aligned}$$

Also, we have

$$\begin{aligned} B(u,v)=tr_1^m(v Q'(u)), \end{aligned}$$

where, \(Q'(u)=\sum _{i=1}^{p-1} Q'_i(u)^{\frac{2^m}{\beta _i}}\).

Now on proceeding as Theorem 3.1, we can get

$$\begin{aligned} nl_{(r=p-2)}g_{\alpha }(u) \ge 2^{p+2}-2^{\frac{p+3}{2}}. \end{aligned}$$

\(\square \)

4 Comparison

Carlet [4] has given the lower bounds of the higher order nonlinearity of the Dillon bent function. Also Garg [17] has computed lower bounds of the higher order nonlinearity of the monomial partial spread Boolean function. In the below Table 1, a comparison of the lower bounds acquired in Theorem 3.1 with those gained by Carlet and Garg is given.

5 Conclusion

The lower bounds on the higher order nonlinearity of Boolean function with Niho exponent and inverse Boolean function have been discussed in the article. Also, after the comparison, it has been discovered that the lower bounds obtained in this paper are better than the lower bounds obtained by Carlet and Garg. Hence, after analyzing the other properties of cryptography of these Boolean functions, they can be used in ciphers (block cipher and stream cipher) as combiners or filters. The future aspect of this paper is to study and compute the lower bounds of higher order nonlinearity of some other classes of Boolean functions.