Two secondary constructions of bent functions without initial conditions

Abstract

In this article, we propose two secondary constructions of bent functions without any conditions on initial bent functions employed by these methods. It is shown that both methods generate bent functions that belong to the generalized Maiorana–McFarland (${\mathcal {GMM}}_{n/2+k}$) class of n-variable Boolean functions, with n even. The class ${\mathcal {GMM}}_{n/2+k}$ contains functions that can be viewed as a concatenation of $(n/2-k)$-variable (not necessarily distinct) affine functions, which was previously (mainly) used in the design of resilient Boolean functions. Most notably, we show that a subclass of bent functions generated by our first method is provably outside the completed Maiorana–McFarland class ${\mathcal {MM}}^\#$. This extremely large class of Boolean functions ${\mathcal {GMM}}_{n/2+k}$, which is shown to properly include the standard Maiorana–McFarland class ${\mathcal {MM}}$, may contain a significant subset of bent functions that are not the members of ${\mathcal {MM}}^\#$. In general, the inclusion of these bent functions, that are provably outside ${\mathcal {MM}}^\#$, into the completed partial spread class remains unknown.

Appendices

Appendix I: The derivation of $D_aD_bf$ in Lemma 3

$$\begin{aligned} D_a D_bf= & {} \bigoplus \limits _{ \varsigma =1}^{2}h_{\varsigma }(X_{({n}-2k+1)}^{(n)})\left( \bigoplus \limits _{\alpha ^{(\frac{n}{2}-k) }\in \Phi ^{-1}(H_{\varsigma }) }\prod \limits _{i=1}^{\frac{n}{2}-k}(x_i\oplus \alpha _i\oplus 1)\right) \\&\oplus \bigoplus \limits _{\begin{array}{c} \varsigma =1 \end{array}}^{2} h_{\varsigma }(X_{({n}-2k+1)}^{(n)}\oplus (a'_{[3]},a'_{[4]}))\left( \bigoplus \limits _{\alpha ^{(\frac{n}{2}-k) }\in \Phi ^{-1}(H_{\varsigma }) }\prod \limits _{i=1}^{\frac{n}{2}-k}(x_i\oplus \alpha _i\oplus a_i\oplus 1)\right) \\&\oplus \bigoplus \limits _{\begin{array}{c} \varsigma =1 \end{array}}^{2}h_{\varsigma }(X_{({n}-2k+1)}^{(n)}\oplus (b'_{[3]},b'_{[4]}))\left( \bigoplus \limits _{\alpha ^{(\frac{n}{2}-k) }\in \Phi ^{-1}(H_{\varsigma }) }\prod \limits _{i=1}^{\frac{n}{2}-k}(x_i\oplus \alpha _i\oplus b_i\oplus 1)\right) \\&\oplus \bigoplus \limits _{\begin{array}{c} \varsigma =1 \end{array}}^{2} h_{\varsigma }(X_{({n}-2k+1)}^{(n)}\oplus (a'_{[3]},a'_{[4]})\\&\oplus (b'_{[3]},b'_{[4]}))\left( \bigoplus \limits _{\alpha ^{(\frac{n}{2}-k) }\in \Phi ^{-1}(H_{\varsigma }) }\prod \limits _{i=1}^{\frac{n}{2}-k}(x_i\oplus \alpha _i\oplus a_i\oplus b_i\oplus 1)\right) \\&\oplus D_{(a_{[1]},a_{[2]})}D_{(b_{[1]},b_{[2]})}g(X^{(n-2k)})\\= & {} \bigoplus \limits _{ \varsigma =1}^{2}h_{\varsigma }(X_{({n}-2k+1)}^{(n)})\left( \bigoplus \limits _{\alpha ^{(\frac{n}{2}-k) }\in \Phi ^{-1}(H_{\varsigma }) }\prod \limits _{i=1}^{\frac{n}{2}-k}(x_i\oplus \alpha _i\oplus 1)\right) \\&\oplus \bigoplus \limits _{\begin{array}{c} \varsigma =1 \end{array}}^{2} h_{\varsigma }(X_{({n}-2k+1)}^{(n)}\oplus (a'_{[3]},a'_{[4]}))\left( \bigoplus \limits _{\alpha ^{(\frac{n}{2}-k) }\in \Phi ^{-1}(H_{\varsigma }) }\prod \limits _{i=1}^{\frac{n}{2}-k}(x_i\oplus \alpha _i \oplus 1)\right) \\&\oplus \bigoplus \limits _{\begin{array}{c} \varsigma =1 \end{array}}^{2} h_{\varsigma }(X_{({n}-2k+1)}^{(n)}\oplus (a'_{[3]},a'_{[4]}))\left( D_{a_{[1]}} \bigoplus \limits _{\alpha ^{(\frac{n}{2}-k) }\in \Phi ^{-1}(H_{\varsigma }) }\prod \limits _{i=1}^{\frac{n}{2}-k}(x_i\oplus \alpha _i\oplus 1)\right) \\&\oplus \bigoplus \limits _{\begin{array}{c} \varsigma =1 \end{array}}^{2} h_{\varsigma }(X_{({n}-2k+1)}^{(n)}\oplus (b'_{[3]},b'_{[4]}))\left( \bigoplus \limits _{\alpha ^{(\frac{n}{2}-k) }\in \Phi ^{-1}(H_{\varsigma }) }\prod \limits _{i=1}^{\frac{n}{2}-k}(x_i\oplus \alpha _i \oplus 1)\right) \\&\oplus \bigoplus \limits _{\begin{array}{c} \varsigma =1 \end{array}}^{2} h_{\varsigma }(X_{({n}-2k+1)}^{(n)}\oplus (b'_{[3]},b'_{[4]})) \left( D_{b_{[1]}}\bigoplus \limits _{\alpha ^{(\frac{n}{2}-k) }\in \Phi ^{-1}(H_{\varsigma }) }\prod \limits _{i=1}^{\frac{n}{2}-k}(x_i\oplus \alpha _i\oplus 1)\right) \\&\oplus \bigoplus \limits _{\begin{array}{c} \varsigma =1 \end{array}}^{2} h_{\varsigma }(X_{({n}-2k+1)}^{(n)}\oplus (a'_{[3]},a'_{[4]})\\&\oplus (b'_{[3]},b'_{[4]}))\left( \bigoplus \limits _{\alpha ^{(\frac{n}{2}-k) }\in \Phi ^{-1}(H_{\varsigma }) }\prod \limits _{i=1}^{\frac{n}{2}-k}(x_i\oplus \alpha _i \oplus 1)\right) \end{aligned}$$

$$\begin{aligned}&\oplus \bigoplus \limits _{\begin{array}{c} \varsigma =1 \end{array}}^{2}h_{\varsigma }(X_{({n}-2k+1)}^{(n)}\oplus (a'_{[3]},a'_{[4]})\\&\oplus (b'_{[3]},b'_{[4]})) \left( D_{a_{[1]}\oplus b_{[1]}}\bigoplus \limits _{\alpha ^{(\frac{n}{2}-k) }\in \Phi ^{-1}(H_{\varsigma }) }\prod \limits _{i=1}^{\frac{n}{2}-k}(x_i\oplus \alpha _i\oplus 1)\right) \\&\oplus D_{(a_{[1]},a_{[2]})}D_{(b_{[1]},b_{[2]})}g(X^{(n-2k)})\\= & {} \bigoplus \limits _{ \varsigma =1}^{2}D_{(a'_{[3]},a'_{[4]})}D_{(b'_{[3]},b'_{[4]})}h_{\varsigma }(X_{({n}-2k+1)}^{(n)})\left( \bigoplus \limits _{\alpha ^{(\frac{n}{2}-k) }\in \Phi ^{-1}(H_{\varsigma }) }\prod \limits _{i=1}^{\frac{n}{2}-k}(x_i\oplus \alpha _i\oplus 1)\right) \\&\oplus \bigoplus \limits _{\begin{array}{c} \varsigma =1 \end{array}}^{2} h_{\varsigma }(X_{({n}-2k+1)}^{(n)}\oplus (a'_{[3]},a'_{[4]}))\left( D_{a_{[1]}} \bigoplus \limits _{\alpha ^{(\frac{n}{2}-k) }\in \Phi ^{-1}(H_{\varsigma }) }\prod \limits _{i=1}^{\frac{n}{2}-k}(x_i\oplus \alpha _i\oplus 1)\right) \\&\oplus \bigoplus \limits _{\begin{array}{c} \varsigma =1 \end{array}}^{2} h_{\varsigma }(X_{({n}-2k+1)}^{(n)}\oplus (b'_{[3]},b'_{[4]})) \left( D_{b_{[1]}}\bigoplus \limits _{\alpha ^{(\frac{n}{2}-k) }\in \Phi ^{-1}(H_{\varsigma }) }\prod \limits _{i=1}^{\frac{n}{2}-k}(x_i\oplus \alpha _i\oplus 1)\right) \\&\oplus \bigoplus \limits _{\begin{array}{c} \varsigma =1 \end{array}}^{2}h_{\varsigma }(X_{({n}-2k+1)}^{(n)}\oplus (a'_{[3]},a'_{[4]})\\&\oplus (b'_{[3]},b'_{[4]})) \left( D_{a_{[1]}\oplus b_{[1]}}\bigoplus \limits _{\alpha ^{(\frac{n}{2}-k) }\in \Phi ^{-1}(H_{\varsigma }) }\prod \limits _{i=1}^{\frac{n}{2}-k}(x_i\oplus \alpha _i\oplus 1)\right) \\&\oplus D_{(a_{[1]},a_{[2]})}D_{(b_{[1]},b_{[2]})}g(X^{(n-2k)}). \end{aligned}$$

II. Theorem 4: Proof for the remaining cases when $|V\cap \Delta |>1$.

(B2) Case $||V\cap \Delta ||= 2$: In this case there exists one vector $(0_{\frac{n}{2}-k},a_{[2]}, a_{[3]}, 0_{k})\in V$ such that $(a_{[2]}, a_{[3]})\ne 0_{ \frac{n}{2}}$. We have $||\{(v_1^{(1)},v_4^{(1)}),\ldots , (v_1^{(2^{\frac{n}{2}})}, v_4^{(2^{\frac{n}{2}})})\}||\ge 2^{n/2-1}$ (counting distinct vectors). Assuming, on contrary, that $||\{(v_1^{(1)},v_4^{(1)}),\dots , (v_1^{(2^{\frac{n}{2}})}, v_4^{(2^{\frac{n}{2}})})\}||< 2^{n/2-1}$ would imply the existence of (at least) 4 vectors

$$\begin{aligned} (v_1^{(i_1)},v_4^{(i_1)}), (v_1^{(i_2)},v_4^{(i_2)}), (v_1^{(i_3)},v_4^{(i_3)}),(v_1^{(i_4)},v_4^{(i_4)}) \in \{(v_1^{(1)},v_4^{(1)}),\dots , (v_1^{(2^{\frac{n}{2}})}, v_4^{(2^{\frac{n}{2}})})\} \end{aligned}$$

such that $(v_1^{(i_1)},v_4^{(i_1)})= (v_1^{(i_2)},v_4^{(i_2)})=(v_1^{(i_3)},v_4^{(i_3)})=(v_1^{(i_4)},v_4^{(i_4)})$ (since V is a subspace). This contradicts the fact that $||V\cap \Delta ||= 2$, since the addition of above vectors gives at least 3 vectors of the form $(0_{\frac{n}{2}-k},*,*,0_{k}) \in V \cap \Delta$. Note that

$$\begin{aligned}&\dim (\{v_1^{(1)},\ldots , v_1^{(2^{\frac{n}{2}})} \}) + \dim (\{v_4^{(1)},\ldots , v_4^{(2^{\frac{n}{2}})} \}) \\&\quad \ge \dim (\{(v_1^{(1)},v_4^{(1)}),\ldots , (v_1^{(2^{\frac{n}{2}})}, v_4^{(2^{\frac{n}{2}})}) \}) \ge \frac{n}{2}-1, \end{aligned}$$

so we either have $\{v_1^{(1)},\dots ,$ $v_1^{(2^{\frac{n}{2}})}\}= {\mathbb {F}}_2^{n/2-k}$ or alternatively $\{v_4^{(1)},\dots , v_4^{(2^{\frac{n}{2}})}\}= {\mathbb {F}}_2^{k}$ (up to repetition of some $v_1^{(i)}$ and $v_4^{(j)}$). We consider these two cases separately.

(B2) i) Case $\{v_1^{(1)},\dots ,$ $v_1^{(2^{\frac{n}{2}})}\}= {\mathbb {F}}_2^{n/2-k}$: Since $||\{(v_1^{(1)},v_4^{(1)}),\ldots , (v_1^{(2^{\frac{n}{2}})}, v_4^{(2^{\frac{n}{2}})})\}||\ge 2^{n/2-1}$, we have $||\{v_4^{(1)},\ldots , v_4^{(2^{\frac{n}{2}})}\}||\ge 2^{k-1}$. There are two subcases to be considered now.

• (1) When $\dim (\{(v_3^{(1)},v_4^{(1)}),\ldots , (v_3^{(2^{\frac{n}{2}})}, v_4^{(2^{\frac{n}{2}})}) \})=k$, from item (B1), we have $D_aD_bf\ne 0$.

• (2) When $\dim (\{(v_3^{(1)},v_4^{(1)}),\ldots , (v_3^{(2^{\frac{n}{2}})}, v_4^{(2^{\frac{n}{2}})}) \})=k-1$, we also know $\dim (\{v_4^{(1)},\ldots ,$ $v_4^{(2^{\frac{n}{2}})} \})=k-1$ since $||\{(v_1^{(1)},v_4^{(1)}), \ldots , (v_1^{(2^{\frac{n}{2}})}, v_4^{(2^{\frac{n}{2}})})\}||\ge 2^{n/2-1}$. This can be confirmed using the fact that

  $$\begin{aligned} \{(v_1^{(1)},v_3^{(1)},v_4^{(1)}),\ldots , (v_1^{(2^{\frac{n}{2}})},v_3^{(2^{\frac{n}{2}})}, v_4^{(2^{\frac{n}{2}})})\}=2^{n/2-k}\times 2^{k-1} \end{aligned}$$

  and because V is a subspace then

  $$\begin{aligned} 2^{n/2-k}\times 2^{k-1}\ge & {} \Vert \{(v_1^{(1)},v_3^{(1)},v_4^{(1)}),\ldots , (v_1^{(2^{\frac{n}{2}})},v_3^{(2^{\frac{n}{2}})}, v_4^{(2^{\frac{n}{2}})})\}\Vert \\\ge & {} \Vert \{(v_1^{(1)},v_4^{(1)}),\ldots , (v_1^{(2^{\frac{n}{2}})}, v_4^{(2^{\frac{n}{2}})})\}\Vert \\\ge & {} 2^{n/2-1}. \end{aligned}$$

  Further, ${\mathbb {F}}_2^{n/2-k} \times 0_{2k}\subseteq \{(v_1^{(1)},v_3^{(1)},v_4^{(1)}),\ldots ,$ $(v_1^{(2^{\frac{n}{2}})}, v_3^{(2^{\frac{n}{2}})},v_4^{(2^{\frac{n}{2}})})\}$, since $\{v_1^{(1)},\dots ,$ $v_1^{(2^{\frac{n}{2}})}\}= {\mathbb {F}}_2^{n/2-k}$ and $\{(v_1^{(1)},v_3^{(1)},v_4^{(1)}),\ldots ,$ $(v_1^{(2^{\frac{n}{2}})}, v_3^{(2^{\frac{n}{2}})},v_4^{(2^{\frac{n}{2}})})\}$ is a subspace of ${{\mathbb {F}}}_2^{n/2+k}$. Now, since $\Vert \Phi ^{-1}(H_{1})\Vert$ is odd, there must exist the term $x_1x_2\cdots x_{\frac{n}{2}-k}h_{1}(X_{({n}-2k+1)}^{(n)})$ in the ANF of f. We also know that there must exist two vectors $a=(a_{[1]}, a_{[2]}, 0_{2k}),b=(b_{[1]}, b_{[2]}, 0_{2k})\in V$ such that $a_{[1]}\ne b_{[1]}$, $a_{[1]}\ne 0_{n/2-k}$ and $b_{[1]}\ne 0_{n/2-k}$ since $\{v_1^{(1)},\dots ,$ $v_1^{(2^{\frac{n}{2}})}\}= {\mathbb {F}}_2^{n/2-k}$. Hence, the term $(h_{1}\oplus h_2)(X_{({n}-2k+1)}^{(n)})$ must be included in the ANF of

  $$\begin{aligned} D_{a_{[1]}}D_{b_{[1]}} \bigoplus \limits _{\varsigma =1}^{2}\bigoplus \limits _{\alpha ^{(\frac{n}{2}-k) }\in \Phi ^{-1}(H_{2}) }\prod \limits _{i=1}^{\frac{n}{2}-k}(x_i\oplus \alpha _i\oplus 1)h_{\varsigma }(X_{({n}-2k+1)}^{(n)}). \end{aligned}$$

  Thus,

  $$\begin{aligned} \begin{aligned} D_aD_bf=&D_{a_{[1]}}D_{b_{[1]}}\bigoplus \limits _{\varsigma =1}^{2}\bigoplus \limits _{\alpha ^{(\frac{n}{2}-k) }\in \Phi ^{-1}(H_{\varsigma }) }\prod \limits _{i=1}^{\frac{n}{2}-k}(x_i\oplus \alpha _i\oplus 1)h_{\varsigma }(X_{({n}-2k+1)}^{(n)})\\&\oplus D_{(a_{[1]},a_{[2]})}D_{(b_{[1]},b_{[2]})}g(X^{({n}-2k)}) \ne 0\end{aligned}. \end{aligned}$$

(B2) ii) Case $\{v_4^{(1)},\ldots , v_4^{(2^{\frac{n}{2}})}\}= {\mathbb {F}}_2^{k}$: From item (B1), we directly get $D_aD_bf\ne 0$.

(B3) Case $\Vert V\cap \Delta \Vert > 2$: This implies that $\Vert V\cap \Delta \Vert \ge 4$ and we consider two cases:

$$\begin{aligned} i) \; \dim (\{(v_3^{(1)},v_4^{(1)}), \ldots , (v_3^{(2^{\frac{n}{2}})}, v_4^{(2^{\frac{n}{2}})}) \}) \ge k, \end{aligned}$$

and

$$\begin{aligned} ii) \;\dim (\{(v_3^{(1)},v_4^{(1)}),\ldots , (v_3^{(2^{\frac{n}{2}})}, v_4^{(2^{\frac{n}{2}})}) \})<k. \end{aligned}$$

The case (B3) i) is easily dealt with, since again using (B1) we get $D_aD_bf\ne 0$.

For the case (B3) ii) we have that $\dim (\{(v_1^{(1)},v_2^{(1)}),\ldots , (v_1^{(2^{\frac{n}{2}})}, v_2^{(2^{\frac{n}{2}})}) \})> n/2-k$ since

$$\begin{aligned} \dim \{(v_1^{(1)},v_2^{(1)}),\ldots , (v_1^{(2^{\frac{n}{2}})}, v_2^{(2^{\frac{n}{2}})}) \}+\dim \{(v_3^{(1)},v_4^{(1)}),\ldots , (v_3^{(2^{\frac{n}{2}})}, v_4^{(2^{\frac{n}{2}})})\}\ge \dim (V)=\frac{n}{2}. \end{aligned}$$

Further, since V is a subspace and $\Vert V\Vert =2^{n/2}$, we have

$$\begin{aligned} \Vert S\Vert =\Vert \{(v_1,v_2, 0_{2k}): (v_1,v_2, 0_{2k})\in V \} \Vert >2^{n/2-k}, \end{aligned}$$

for a linear subspace $S \subset {{\mathbb {F}}}_2^n$ because $\dim (\{(v_1^{(1)},v_2^{(1)}),$ $\ldots , (v_1^{(2^{\frac{n}{2}})}, v_2^{(2^{\frac{n}{2}})}) \})> n/2-k$.

Assume now that for any $v_1,v_2$ such that $(v_1,v_2, 0_{2k})\in V$, we have that $v_2\ne 0_{n/2-k}$ implies that $v_1\ne 0_{n/2-k}$. This would contradict the fact that $\Vert S\Vert >2^{n/2-k}$. To see this, without loss of generality, we set $\{ v_1^{i_1},\ldots , v_1^{i_\mu }\}=\{ v_1: (v_1,v_2, 0_{2k})\in S\}$. Since S is a subspace, we must have $0_{n/2-k}\in \{ v_1^{i_1},\ldots , v_1^{i_\mu }\}$. We also know $i_\mu \le 2^{n/2-k}<\Vert S\Vert$. Hence, there must exist $(0_{n/2-k}, v_2^{i_\rho },0_{2k})\in V$. Therefore, there must exist one non-zero vector $a \in V$ of the form $a=(0_{n/2-k},a_{[2]}, 0_{2k})\in V$. Then, taking some $b=(b_{[1]},b_{[2]}, 0_{2k})\in V$ we obtain

$$\begin{aligned} \begin{aligned} D_{a}D_bf =&D_{(0_{\frac{n}{2}-k},a_{ [2]})} D_{(b_{[1]},b_{[2]})}g(X^{(n-2k)})\\ =&D_{(b_{[1]},b_{[2]})}(\Phi (X^{(n/2-k)} )\cdot a_{ [2]} )\\ =&D_{b_{[1]}}(\Phi (X^{(n/2-k)} )\cdot a_{ [2]} )\ne 0, \end{aligned} \end{aligned}$$

since $\Phi \cdot \nu$ has no nonzero linear structure for any $\nu \in {{\mathbb {F}}}_2^n\setminus \{\mathbf{0 }_{n/2-k}\}$. Combining the cases (A) and (B), we deduce that f does not belong to ${\mathcal {MM}}^{\#}$. $\square$