Abstract
In this paper, we first present a novel secondary construction of bent functions (building new bent functions from two already defined ones). Furthermore, the algebraic degree and algebraic immunity of the constructed functions are analysed. Finally, we apply the construction using as initial functions some specific bent functions and then specify sufficient conditions for the resulting bent functions not to be contained in the completed Maiorana–McFarland class. In the second part of the paper, we present a corrigendum of “Constructions of bent–negabent functions and their relation to the completed Maiorana–McFarland Class” (IEEE Trans Inf Theory 61(3):1496–1506, 2015).
Similar content being viewed by others
Notes
A part of the results appeared in a pre-publication in arxiv arXiv:1211.4191.
References
Canteaut, A., Charpin, P.: Decomposing bent functions. IEEE Trans. Inf. Theory 49(8), 2004-2019 (2003)
Carlet, C.: Two new classes of bent functions. In: Helleseth, T. (ed.) Advances in EUROCRYPT’93, LNCS 765, 77-101 (1994)
Carlet, C.: Generalized partial spreads. IEEE Trans. Inf. Theory 41(5), 1482-1487 (1995)
Carlet, C.: A construction of bent functions. In: Cohen, S., Niederreiter, H. (eds) Proceedings of Third International Conference on Finite Fields and Applications, pp 47-58. Cambridge University Press, Cambridge (1996)
Carlet, C.: On the confusion and diffusion properties of Maiorana-McFarland’s and extended Maiorana-McFarland’s functions. J. Complex. 20(2-3), 182-204 (2004)
Carlet, C.: On the secondary constructions of resilient and bent functions. In: Feng, K., Niederreiter, H., Xing, C. (eds.) Proceedings of the Workshop on Coding, Cryptography and Combinatorics 2003, pp. 3-28. Birkhäuser Verlag (2004)
Carlet, C.: On bent and highly nonlinear balanced/resilient functions and their algebraic immunities. In: Fossorier, M. et al. (eds.) Proceedings of AAECC 2006, LNCS 3857, 1-28 (2006)
Carlet, C.: Boolean functions for cryptography and error correcting codes. In: Crama, Y., Hammer, P. (eds.) The Monography Boolean Models and Methods in Mathematics, Computer Science, and Engineering, pp. 257-397. Cambridge University Press, Cambridge (2010)
Carlet, C., Dobbertin, H., Leander, G.: Normal extensions of bent functions. IEEE Trans. Inf. Theory 50(11), 2880-2885 (2004)
Carlet, C., Yucas, J.L.: Piecewise constructions of bent and almost optimal Boolean functions. Des. Codes Cryptogr. 37(3), 449-464 (2005)
Carlet, C., Zhang, F., Hu, Y.: Secondary constructions of bent functions and their enforcement. Adv. Math. Commun. 6(3), 305-314 (2012)
Chepyzhov, V., Smeets, B.: On a fast correlation attack on certain stream ciphers. Proceedings of EUROCRYPT’91, Lecture Notes in Computer Science 547, pp. 176-185 (1992)
Dillon, J.: Elementary Hadamard difference sets. Ph.D. Dissertation, University of Maryland, College Park (1974)
Dobbertin, H.: Construction of bent functions and balanced Boolean functions with high nonlinearity. In: Gilbert, H., Handschuh, H. (eds.) Proceedings of FSE 1995, LNCS 1008, 61-74 (1995)
Dobbertin, H., Leander, G.: Bent functions embedded into the recursive framework of ${\bb Z\it }$-bent functions. Des. Codes Cryptogr. 49(1-3), 3-22 (2008)
Fu, S., Li, C., Matsuura, k, Qu, L.: Construction of odd-variable resilient Boolean functions with optimal degree. IEICE Trans. Fundam. E94-A, 265-267 (2011)
Guillot, P.: Completed GPS covers all bent functions. J. Comb. Theory Ser. A 93, 242-260 (2001)
Leander, G., McGuire, G.: Construction of bent functions from near-bent functions. J. Comb. Theory Ser. A 116, 960-970 (2009)
Lempel, A., Cohn, M.: Maximal families of bent sequences. IEEE Trans. Inf. Theory 28(6), 865-868 (1982)
MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland Publishing Company, Amsterdam (1977)
McFarland, R.I.: A family of difference sets in non-cyclic groups. J. Comb. Theory Ser. A. 15, 1-10 (1973)
No, J.S., Gil, G.M., Shin, D.J.: Generalized construction of binary bent sequences with optimal correlation property. IEEE Trans. Inf. Theory 49(7), 858-864 (1982)
Olsen, J.D., Scholtz, R.A., Welch, L.R.: Bent-function sequence. IEEE Trans. Inf. Theory 28(6), 1769-1780 (2003)
Rothaus, O.S.: On “bent” functions. J. Comb. Theory Ser. A 20, 300-305 (1976)
Wolfmann, J.: Bent functions and coding theory. In: Pott, A.P., Kumar, V., Helleseth, T., Jungnickel, D. (eds.) Difference Sets, Sequences and their Correlation Properties, pp. 395-417. Kluwer, Amsterdam (1999)
Zhang, F., Wei, Y., Pasalic, E.: Constructions of bent-negabent functions and their relation to the completed Maiorana-McFarland class. IEEE Trans. Inf. Theory 61(3), 1496-1506 (2015)
Zheng, D., Yu, L., Hu, L.: On a class of binomial bent functions over the finite fields of odd characteristic. Appl. Algebra Eng. Commun. Comput. 24(6), 461-475 (2013)
Zheng, Y., Zhang, X.-M.: Relationships between bent functions and complementary plateaued functions. In: Song, J. (ed.) Proceedings 2nd International Conference on Information Security and Cryptology (ICISC’99), LNCS 1787, 60-75 (1999)
Acknowledgments
The work of F. Zheng was supported in part by National Science Foundation of China (61303263), and in part by the Fundamental Research Funds for the Central Universities (2013QNA26), in part by the China Postdoctoral Science Foundation funded project (2014M562494, 2015T80600), and in part by the Jiangsu Planned Projects for Postdoctoral Research Funds (1401056B). The first author is grateful to Prof. Enes Pasalic and Yongzhuang Wei for valuable discussions and suggestions.
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
Proof of Lemma 3
Set \(\textsc {f}_0(x)= {\mathop {\mathop {\bigoplus }\limits _{i=1}}\limits _{i\ne \mu }} ^{\frac{n}{2}}\phi _i(x^{(2)})\, x_{i}\) and \(\textsc {g}_0(y)= {\mathop {\mathop {\bigoplus }\limits _{j=1}}\limits _{j\ne \rho }} ^{\frac{m}{2}}\psi _j(y^{(2)})\, y_{j}\). We have then
-
1.
If there exist \((a_1, 0_{\frac{n}{2}}, 0_{\frac{m}{2}-1}, a_4),(b_1, 0_{\frac{n}{2}}, 0_{\frac{m}{2}-1}, b_4) \in \varDelta \) such that \(a_4\ne 0_{\frac{m}{2}}\), \(b_4\ne 0_{\frac{m}{2}}\) and \(a_4\ne b_4\). Then,
$$\begin{aligned} D_{(a_1, 0_{\frac{n}{2}})}D_{(b_1, 0_{\frac{n}{2}})}\left( \textsc {f}_0(x)\oplus \theta (x^{(2)})\right) =0. \end{aligned}$$We have
$$\begin{aligned}&D_{(a_1, 0_{\frac{n}{2}}, 0_{\frac{m}{2}-1}, a_4)}D_{(b_1, 0_{\frac{n}{2}}, 0_{\frac{m}{2}-1}, b_4)}h(x,y) \\&\quad = D_{( 0_{\frac{m}{2}-1},a_4)}D_{(0_{\frac{m}{2}-1},b_4)}\textsc {g}_0(y) \oplus D_{ a_4}D_{b_4}\varpi (y^{(2)}) \oplus \phi _{\mu }(x^{(2)}) D_{a_4}D_{b_4}\left( \psi _{\rho }(y^{(2)})\nonumber \right) . \end{aligned}$$(5)We know that \(D_{( 0_{\frac{m}{2}-1},a_4)}D_{(0_{\frac{m}{2}-1},b_4)}\textsc {g}_0(y)\) equals \({\mathop {\mathop {\bigoplus }\limits _{j=1}}\limits _{j\ne \rho }} ^{\frac{m}{2}}\left( D_{a_4}D_{b_4}\psi _j(y^{(2)})\right) \, y_{j}\), which equals 0 or must depend on \(y^{(1)}\). Clearly, \(\phi _{\mu }(x^{(2)}) D_{a_4}D_{b_4}\left( \psi _{\rho }(y^{(2)})\right) \) equals 0 or must depend on \(x^{(2)}\). However, \( D_{ a_4}D_{b_4}\varpi (y^{(2)})\) does not equal a constant and only depends on \(y^{(2)}\) since \(D_{u_4}D_{v_4}\varpi (y^{(2)}) \) is not a constant for any two different nonzero vectors \(u_4,v_4\in {\mathbb F}_2^{\frac{m}{2}}\setminus \{0_{\frac{m}{2}}\}\). Hence,
$$\begin{aligned} D_{(a_1, 0_{\frac{n}{2}}, 0_{\frac{m}{2}-1}, a_4)}D_{(b_1, 0_{\frac{n}{2}}, 0_{\frac{m}{2}-1}, b_4)}h(x,y)\ne 0. \end{aligned}$$ -
2.
If there exist \( (a_1,a_2,a_3,a_4),(b_1,b_2,\) \(b_3,b_4)\in V\) such that \(D_{a_2}D_{b_2}\phi _{\mu }(x^{(2)})\ne 0\), \(a_3=b_3=0_{\frac{m}{2}-1}\). Then, according to \(a_4\) and \(b_4\), we have four cases: \(a_4=b_4\ne 0_{\frac{m}{2}}\), \(a_4=b_4= 0_{\frac{m}{2}}\), \(a_4\ne b_4\) and \(b_4= 0_{\frac{m}{2}}\) (or \(a_4= 0_{\frac{m}{2}}\)), \(a_4\ne b_4\) and \(b_4\ne 0_{\frac{m}{2}}\) and \(a_4\ne 0_{\frac{m}{2}}\).
-
(a)
\(a_4=b_4\ne 0_{\frac{m}{2}}\): In this case \((c_1,c_2,0_{\frac{m}{2}-1},0_{\frac{m}{2}})\in V\), where \( (c_1,c_2,0_{\frac{m}{2}-1},0_{\frac{m}{2}}) =(a_1,a_2,a_3,a_4)\oplus (b_1,b_2,b_3,b_4)\). We have
$$\begin{aligned}&D_{(a_1, a_2, a_3, a_4)}D_{(c_1,c_2,0_{\frac{m}{2}-1},0_{\frac{m}{2}})}h(x,y) \nonumber \\&\quad = D_{(a_1,a_2)}D_{(c_1,c_2)}\textsc {f}_0(x)\oplus D_{a_2}D_{c_2}\theta (x^{(2)})\\&\qquad \oplus D_{c_2}\phi _{\mu }(x^{(2)})\psi _{\rho }(y^{(2)})\oplus D_{c_2}\phi _{\mu }(x^{(2)}\oplus a_2)\psi _{\rho }(y^{(2)}\oplus a_4) \nonumber . \end{aligned}$$(6)Note that, denoting \(\psi _{\rho }(y^{(2)})=\bigoplus \limits _{I\subset \{\frac{m}{2}+1,\ldots ,m\}}d_{I}\prod \limits _{i\in I}y_i,\) and \( \psi _{\rho }(y^{(2)}\oplus a_4)= \) \(\bigoplus \limits _{I\subset \{\frac{m}{2}+1,\ldots ,m\}}d_{I}^{(a_4)}\prod \limits _{i\in I}y_i \), we have \(D_{c_2}\phi _{\mu }(x^{(2)})\psi _{\rho }(y^{(2)})\oplus D_{c_2}\phi _{\mu }(x^{(2)}\oplus a_2)\psi _{\rho }(y^{(2)}\oplus a_4) =\bigoplus \limits _{I\subset \{\frac{m}{2}+1,\ldots ,m\}}\left( d_{I}D_{c_2}\phi _{\mu }(x^{(2)})\oplus d_{I}^{(a_4)} D_{c_2}\phi _{\mu }(x^{(2)}\oplus a_2)\right) \prod \limits _{i\in I}y_i \). Since \(\psi (y^{(2)})\) and \(\phi (x^{(2)})\) have no nonzero linear structure, we have \(\psi _{\rho }(y^{(2)})\ne \psi _{\rho }(y^{(2)}\oplus a_4)\) and \(D_{a_2}D_{b_2}\phi _{\mu }(x^{(2)})= D_{c_2}\phi _{\mu }(x^{(2)})\oplus D_{c_2}\phi _{\mu }(x^{(2)}\oplus a_2) \ne 0\), then \(D_{c_2}\phi _{\mu }(x^{(2)})\psi _{\rho }(y^{(2)})\oplus D_{c_2}\phi _{\mu }(x^{(2)}\oplus a_2)\psi _{\rho }(y^{(2)}\oplus a_4)\) must depend on \(y^{(2)}\). However, \(D_{(a_1,a_2)}D_{(c_1,c_2)}\textsc {f}_0(x)\oplus D_{a_2}D_{c_2}\theta (x^{(2)})\) can not depend on \(y^{(2)}\), that is, it cannot contribute in the cancellation of the terms in \(y^{(2)}\). Hence, we have
$$\begin{aligned} D_{(a_1, a_2, a_3, a_4)}D_{\left( c_1,c_2,0_{\frac{m}{2}-1},0_{\frac{m}{2}}\right) }h(x,y)\ne 0. \end{aligned}$$ -
(b)
\(a_4=b_4= 0_{\frac{m}{2}}\): In this case we have
$$\begin{aligned}&D_{\left( a_1, a_2,0_{\frac{m}{2}-1},0_{\frac{m}{2}}\right) } D_{\left( b_1,b_2,0_{\frac{m}{2}-1},0_{\frac{m}{2}}\right) }h(x,y) \nonumber \\&\quad = D_{(a_1,a_2)}D_{(b_1,b_2)}\textsc {f}_0(x)\oplus D_{a_2}D_{b_2}\theta (x^{(2)})\\&\qquad \oplus D_{a_2} D_{b_2}\phi _{\mu }(x^{(2)})\psi _{\rho }(y^{(2)})\nonumber . \end{aligned}$$(7)Since \(D_{a_2} D_{b_2}\phi _{\mu }(x^{(2)})\ne 0\), we have
$$\begin{aligned} D_{\left( a_1, a_2, ,0_{\frac{m}{2}-1},0_{\frac{m}{2}}\right) } D_{\left( b_1,b_2,0_{\frac{m}{2}-1},0_{\frac{m}{2}}\right) }h(x,y)\ne 0. \end{aligned}$$ -
(c)
\(a_4\ne b_4\) and \(b_4= 0_{\frac{m}{2}}\) (or \(a_4= 0_{\frac{m}{2}}\)), without loss of generality, set \(b_4= 0_{\frac{m}{2}}\): In this case we have
$$\begin{aligned}&D_{(a_1, a_2, a_3, a_4)}D_{\left( b_1,b_2,0_{\frac{m}{2}-1},0_{\frac{m}{2}}\right) }h(x,y) \nonumber \\&\quad = D_{(a_1,a_2)}D_{(b_1,b_2)}\textsc {f}_0(x)\oplus D_{a_2}D_{b_2}\theta (x^{(2)})\\&\qquad \oplus \left( \phi _{\mu }(x^{(2)})\oplus \phi _{\mu }(x^{(2)}\oplus b_2) \right) \psi _{\rho }(y^{(2)})\nonumber \\&\qquad \oplus \left( \phi _{\mu }(x^{(2)}\oplus a_2)\oplus \phi _{\mu }(x^{(2)}\oplus b_2\oplus a_2) \right) \psi _{\rho }(y^{(2)}\oplus a_4) \nonumber . \end{aligned}$$(8)Thus, utilizing the method of item 2, we have \(D_{(a_1, a_2, a_3, a_4)}D_{(b_1,b_2,0_{\frac{m}{2}-1},0_{\frac{m}{2}})}h(x,y)\ne 0\).
-
(d)
\(a_4\ne b_4\), \(b_4\ne 0_{\frac{m}{2}}\) and \(a_4\ne 0_{\frac{m}{2}}\): In this case we have
(9)Note that \(\phi _{\mu }(x^{(2)})\psi _{\rho }(y^{(2)}) \oplus \phi _{\mu }(x^{(2)}\oplus a_2)\psi _{\rho }(y^{(2)}\oplus a_4)\) \( \oplus \phi _{\mu }(x^{(2)}\oplus b_2)\psi _{\rho }(y^{(2)}\oplus b_4)\oplus \) \(\phi _{\mu }(x^{(2)}\oplus b_2\oplus a_2)\psi _{\rho }(y^{(2)}\oplus b_4\oplus a_4)\) \(=\bigoplus \limits _{I\subset \{\frac{m}{2}+1,\ldots ,m\}}\left( d_{I}\phi _{\mu }(x^{(2)})\oplus d_{I}^{(a_4)}\phi _{\mu }(x^{(2)}\oplus a_2)\oplus d_{I}^{(b_4)}\phi _{\mu }(x^{(2)}\oplus b_2) \oplus d_{I}^{(b_4\oplus a_4)}\phi _{\mu }(x^{(2)}\oplus \right. \left. b_2\oplus a_2)\right) \prod \limits _{i\in I}y_i \), where \(\psi _{\rho }(y^{(2)})=\bigoplus \limits _{I\subset \{\frac{m}{2}+1,\ldots ,m\}}d_{I}\prod \limits _{i\in I}y_i,\) \( \psi _{\rho }(y^{(2)}\oplus a_4)\) \(= \bigoplus \limits _{I\subset \{\frac{m}{2}+1,\ldots ,m\}}d_{I}^{(a_4)}\prod \limits _{i\in I}y_i \), \(\psi _{\rho }(y^{(2)}\oplus b_4)=\) \( \bigoplus \limits _{I\subset \{\frac{m}{2}+1,\ldots ,m\}}d_{I}^{(b_4)}\prod \limits _{i\in I}y_i \), \(\psi _{\rho }(y^{(2)}\oplus a_4\oplus b_4)=\) \( \bigoplus \limits _{I\subset \{\frac{m}{2}+1,\ldots ,m\}}d_{I}^{(a_4\oplus b_4)}\prod \limits _{i\in I}y_i\). Since \(D_{a_2}D_{b_2}\phi _{\mu }(x^{(2)})\ne 0\) and \(\phi _{\mu }(x^{(2)})\) has no nonzero linear structure, there must exist a set J such that \(d_{J} \ne d_{J}^{(a_4)}\), that is, \(d_{J}\phi _{\mu }(x^{(2)})\oplus d_{J}^{(a_4)}\phi _{\mu }(x^{(2)}\oplus a_2)\oplus d_{J}^{(b_4)}\phi _{\mu }(x^{(2)}\oplus b_2) \oplus d_{J}^{(b_4\oplus a_4)}\phi _{\mu }(x^{(2)}\oplus b_2\oplus a_2)\) depends on \(x^{(2)}\). Hence, \( \phi _{\mu }(x^{(2)})\psi _{\rho }(y^{(2)})\oplus \phi _{\mu }(x^{(2)}\oplus a_2)\psi _{\rho }(y^{(2)}\oplus a_4) \) \( \oplus \phi _{\mu }(x^{(2)}\oplus b_2)\psi _{\rho }(y^{(2)}\oplus b_4)\oplus \) \( \phi _{\mu }(x^{(2)}\oplus b_2\oplus a_2)\psi _{\rho }(y^{(2)}\oplus b_4\oplus a_4) \) depends on \(x^{(2)}\) and \(y^{(2)}\). Further, from Relation (9), we have \(D_{(a_1, a_2, a_3, a_4)}D_{(b_1,b_2,b_3,b_4)}h(x,y)\ne 0\).
-
(a)
-
3.
If there exist \((a_1, a_2, a_3, a_4)=(a_1,0_{\frac{n}{2}},0_{\frac{m}{2}-1},0_{\frac{m}{2}})\ne 0_{n+m-2}\), \((b_1,b_2,b_3,b_4)\) such that \(b_2 \ne 0_{\frac{n}{2}}\). We have
$$\begin{aligned}&D_{(a_1, a_2, a_3, a_4)}D_{(b_1,b_2,b_3,b_4)}h(x,y) = D_{\left( a_1,0_{\frac{n}{2}}\right) }D_{(b_1,b_2)}\textsc {f}_0(x)\ne 0 \end{aligned}$$(10)since \(\phi (x^{(2)})\) has no nonzero linear structure and \(b_2\ne 0_{\frac{n}{2}}\).
-
4.
If there exist \((a_1, 0_{\frac{n}{2}}, 0_{\frac{m}{2}-1},a_4)\in V\cap \varDelta \) such that \(a_4\ne 0_{\frac{m}{2}}\), \((b_1,b_2,b_3,b_4)\in V\) and \(b_2\ne 0_{\frac{n}{2}}\). We have
$$\begin{aligned}&D_{(a_1, a_2, a_3, a_4)}D_{(b_1,b_2,b_3,b_4)}h(x,y) \nonumber \\&\quad = D_{\left( a_1,0_{\frac{n}{2}}\right) }D_{(b_1,b_2)}\textsc {f}_0(x) \oplus D_{a_4}D_{b_4}\varpi (y^{(2)}) \oplus D_{\left( 0_{\frac{m}{2}-1},a_4\right) }D_{(b_3,b_4)}\textsc {g}_0(y) \nonumber \\&\qquad \oplus \phi _{\mu }(x^{(2)})D_{a_4}\psi _{\rho }(y^{(2)}) \oplus \phi _{\mu }(x^{(2)}\oplus b_2) D_{a_4}\psi _{\rho }(y^{(2)}\oplus b_4) . \end{aligned}$$(11)There are also two cases to be considered.
-
(a)
\(b_4= 0_{\frac{m}{2}}\): In this case \(\phi _{\mu }(x^{(2)})D_{a_4}\psi _{\rho }(y^{(2)}) \oplus \phi _{\mu }(x^{(2)}\oplus b_2) D_{a_4}\psi _{\rho }(y^{(2)}\oplus b_4)=D_{b_2}\phi _{\mu }(x^{(2)})D_{a_4}\psi _{\rho }(y^{(2)})\) depends on \(x^{(2)}, y^{(2)}\) since both \(\phi (x^{(2)})\) and \(\psi (y^{(2)} )\) have no nonzero linear structure and \(a_4\ne 0_{\frac{m}{2}}, b_2\ne 0_{\frac{n}{2}}\). Further, from Relation (11), we have
$$\begin{aligned} D_{(a_1, a_2, a_3, a_4)}D_{(b_1,b_2,b_3,b_4)}h(x,y) \ne 0. \end{aligned}$$ -
(b)
\(b_4\ne 0_{\frac{m}{2}}\): In this case we have \(\phi _{\mu }(x^{(2)})D_{a_4}\psi _{\rho }(y^{(2)}) \oplus \phi _{\mu }(x^{(2)}\oplus b_2) D_{a_4}\psi _{\rho }(y^{(2)}\oplus b_4)\) \(=\bigoplus \limits _{I\subset \{\frac{m}{2}+1,\ldots ,m\}}\left( (d_{I}\oplus d_{I}^{(a_4)})\phi _{\mu }(x^{(2)})\oplus \ (d_{I}^{(b_4)}\oplus d_{I}^{(b_4\oplus a_4)}) \phi _{\mu }(x^{(2)}\oplus b_2) \right) \prod \limits _{i\in I}y_i \), where \(\psi _{\rho }(y^{(2)})=\bigoplus \limits _{I\subset \{\frac{m}{2}+1,\ldots ,m\}}d_{I}\prod \limits _{i\in I}y_i,\) \( \psi _{\rho }(y^{(2)}\oplus a_4)=\) \( \bigoplus \limits _{I\subset \{\frac{m}{2}+1,\ldots ,m\}}d_{I}^{(a_4)}\prod \limits _{i\in I}y_i \), \(\psi _{\rho }(y^{(2)}\oplus b_4)=\) \( \bigoplus \limits _{I\subset \{\frac{m}{2}+1,\ldots ,m\}}d_{I}^{(b_4)}\prod \limits _{i\in I}y_i \), \(\psi _{\rho }(y^{(2)}\oplus a_4\oplus b_4)=\) \( \bigoplus \limits _{I\subset \{\frac{m}{2}+1,\ldots ,m\}}d_{I}^{(a_4\oplus b_4)}\prod \limits _{i\in I}y_i\). Since \(a_4\ne 0_{\frac{m}{2}}, b_4\ne 0_{\frac{m}{2}}, b_2\ne 0_{\frac{n}{2}}\) and both \(\phi (x^{(2)})\) and \(\psi (y^{(2)} )\) have no nonzero linear structure, we know \( \phi _{\mu }(x^{(2)})D_{a_4}\psi _{\rho }(y^{(2)}) \oplus \phi _{\mu }(x^{(2)}\oplus b_2) D_{a_4}\psi _{\rho }(y^{(2)}\oplus b_4)\) depends on \(x^{(2)}, y^{(2)}\). Further, from Relation (11), we have
$$\begin{aligned} D_{(a_1, a_2, a_3, a_4)}D_{(b_1,b_2,b_3,b_4)}h(x,y) \ne 0. \end{aligned}$$
-
(a)
-
5.
If there exist \((a_1, a_2, a_3, a_4)=(a_1,0_{\frac{n}{2}},0_{\frac{m}{2}-1},a_4), (b_1,b_2,\) \(b_3,b_4)=(b_1,0_{\frac{n}{2}},b_3,b_4)\) such that \(a_4=b_4\ne 0_{\frac{m}{2}}\) and \(b_3\ne 0_{\frac{m}{2}-1}\). We have
$$\begin{aligned}&D_{(a_1, a_2, a_3, a_4)}D_{(b_1,b_2,b_3,b_4)}h(x,y) \nonumber \\&\quad = D_{\left( 0_{\frac{m}{2}-1},a_4\right) }D_{(b_3,a_4)}\textsc {g}_0(y)\nonumber \\&\quad = b_3\cdot \left( D_{a_4}\psi _1(y^{(2)}), \ldots , D_{a_4}\psi _{\rho -1}(y^{(2)}),\right. \left. D_{a_4}\psi _{\rho +1}(y^{(2)}),\ldots ,D_{a_4}\psi _{\frac{m}{2}}(y^{(2)})\right) \ne 0\nonumber \\ \end{aligned}$$(12)since \(\psi (y^{(2)})\) has no nonzero linear structure and \(b_3 \ne 0_{\frac{m}{2}-1}\).
Proof of Theorem 3
Functions f and g being bent, h is bent, according to Theorem 2. We prove that h does not belong to \({M}^{\#}\), by using Lemma 2. We need to show that there does not exist an \((\frac{n+m-2}{2})\)-dimensional subspace V of \( {\mathbb F}_2^{\frac{n}{2}-1} \times {\mathbb F}_2^{\frac{n}{2}}\times {\mathbb F}_2^{\frac{m}{2}-1}\times {\mathbb F}_2^{\frac{m}{2}}\) such that, for any \( (a_1, a_2, a_3, a_4),\) \((b_1, b_2, b_3, b_4)\in V\):
We denote the elements of V by \((v_1^{(1)},v_2^{(1)},v_3^{(1)},v_4^{(1)}),(v_1^{(2)},v_2^{(2)},v_3^{(2)},v_4^{(2)}),\ldots ,\)
\( (v_1^{(2^{\frac{n+m-2}{2}})},v_2^{(2^{\frac{n+m-2}{2}})},v_3^{(2^{\frac{n+m-2}{2}})},v_4^{(2^{\frac{n+m-2}{2}})})\). Let the subspace of \({\mathbb F}_2^{n+m-2}\) given by \(\{(x^{(1)},0_{\frac{n}{2}},0_{\frac{m}{2}-1}, y^{(2)})\mid x^{(1)}\in {\mathbb F}_2^{\frac{n}{2}-1}, y^{(2)}\in {\mathbb F}_2^{\frac{m}{2}}\}\) be denoted by \(\varDelta \). We consider two cases: \(V= \varDelta \) and \(V \ne \varDelta \).
-
1.
For \(V=\varDelta \), we can find two vectors \((a_1, 0_{\frac{n}{2}}, 0_{\frac{m}{2}-1}, a_4),(b_1, 0_{\frac{n}{2}}, 0_{\frac{m}{2}-1}, b_4) \in \varDelta \) such that \(a_4\ne 0_{\frac{m}{2}}\), \(b_4\ne 0_{\frac{m}{2}}\) and \(a_4\ne b_4\). Thus, a, b satisfy the item 1 of Lemma 3.
-
2.
For \(V\ne \varDelta \), we split the proof into three cases depending on the cardinality of \(V\cap \varDelta \).
-
(a)
For \(|V\cap \varDelta |= 1\), that is, \(V\cap \varDelta =\{0_{n+m-2}\}\), we have \((v_2^{(i)},v_3^{(i)})\ne (v_2^{(j)},v_3^{(j)})\) for every \(i\ne j\). Indeed, if there exist indices \(i_1,j_1\) such that \((v_2^{(i_1)},v_3^{(i_1)})= (v_2^{(j_1)},v_3^{(j_1)})\), then we have \((v_1^{(i_1)},v_2^{(i_1)},v_3^{(i_1)},v_4^{(i_1)})\oplus (v_1^{(j_1)},v_2^{(j_1)},v_3^{(j_1)},v_4^{(j_1)})\in V\cap \varDelta \), that is, \((v_1^{(i_1)},v_2^{(i_1)},v_3^{(i_1)},v_4^{(i_1)})= (v_1^{(j_1)},v_2^{(j_1)},v_3^{(j_1)},v_4^{(j_1)})\). This implies then that \(|\{(v_2^{(1)},v_3^{(1)}),(v_2^{(2)},v_3^{(2)}),\ldots , (v_2^{(2^{\frac{n+m-2}{2}})}, v_3^{(2^{\frac{n+m-2}{2}})})\}|=|V|= 2^{\frac{n+m-2}{2}}\), that is, \(\{(v_2^{(1)},v_3^{(1)}),(v_2^{(2)},v_3^{(2)}),\ldots , (v_2^{(2^{\frac{n+m-2}{2}})}, v_3^{(2^{\frac{n+m-2}{2}})})\}= {\mathbb F}_2^{\frac{n}{2}}\times {\mathbb F}_2^{\frac{m}{2}-1} \). Hence, according to the hypothesis that \(\phi (x^{(2)})\) has no nonzero linear structure, we have \(\deg (\phi _{\mu }(x^{(2)}))\ge 2\), and there must exist two vectors \( (a_1,a_2,a_3,a_4),(b_1,b_2,\) \(b_3,b_4)\in V\) such that \(D_{a_2}D_{b_2}\phi _{\mu }(x^{(2)})\ne 0\), and \(a_3=b_3=0_{\frac{m}{2}-1}\). Thus a, b satisfy the item 2 of Lemma 3.
-
(b)
For \(|V\cap \varDelta |= 2\), there exists one vector \((a_1, 0_{\frac{n}{2}}, 0_{\frac{m}{2}-1},a_4)\in V\) such that \((a_1,a_4)\ne 0_{ \frac{n}{2}-1+ \frac{m}{2}}\). Additionally, there must exist one vector \((v_1^{(l)},v_2^{(l)},v_3^{(l)},v_4^{(l)})\) such that \(v_2^{(l)}\ne 0_{\frac{n}{2}}\). Indeed, since \(|V|= 2^{\frac{n+m-2}{2}}\) and \(n>2\), then there are at least four vectors \(v^{(i_1)},v^{(i_2)},v^{(i_3)},v^{(i_4)}\) such that \(v_3^{(i_1)}=v_3^{(i_2)}=v_3^{(i_3)}=v_3^{(i_4)}\). Suppose \(v_2^{(i)}= 0_{\frac{n}{2}}\) for \(i=1,2,\ldots ,2^{\frac{n+m-2}{2}}\), then there are at least three vectors \(v^{(i_2)}\oplus v^{(i_1)},v^{(i_3)}\oplus v^{(i_1)}, v^{(i_4)}\oplus v^{(i_1)}\in V\cap \varDelta \), which is in contradiction with \(|V\cap \varDelta |= 2\). Hence, there must exist one vector \((v_1^{(l)},v_2^{(l)},v_3^{(l)},v_4^{(l)})\) such that \(v_2^{(l)}\ne 0_{\frac{n}{2}}\). We set \((b_1,b_2,b_3,b_4)=(v_1^{(l)},v_2^{(l)},v_3^{(l)},v_4^{(l)})\). There are two cases to be considered.
-
(c)
For \(|V\cap \varDelta |=t> 2\) (i.e., \(|V\cap \varDelta |=t\ge 4\) ), without loss of generality, set \(V\cap \varDelta =\{v^{(1)},v^{(2)},\ldots , v^{(t)}\}\), we consider three cases: \(|\{v^{(1)}_4,\ldots , v^{(t)}_4\}|>2\), \(|\{v^{(1)}_4,\ldots , \) \( v^{(t)}_4\}|=2\) and \(|\{v^{(1)}_4,\ldots , v^{(t)}_4\}|=0\).
-
i.
If there exist at least two vectors \((a_1,0_{\frac{n}{2}},0_{\frac{m}{2}-1},a_4), (b_1,0_{\frac{n}{2}},0_{\frac{m}{2}-1},b_4)\in V\cap \varDelta \) such that \(a_4\ne 0_{\frac{m}{2}},b_4\ne 0_{\frac{m}{2}}\) and \(a_4\ne b_4\), then, a, b satisfy the item 1 of Lemma 3.
-
ii.
If \(\{v^{(1)}_4,\ldots ,v^{(t)}_4 \}=\{0_{\frac{m}{2}}, v^{(\ell )}_4\}\), where \((v^{(\ell )}_1,0_{\frac{n}{2}},0_{\frac{m}{2}-1},v^{(\ell )}_4)\in V\cap \varDelta \) such that \(v^{(\ell )}_4\ne 0_{\frac{m}{2}}\), \(\ell \le t\), then \(|V\cap \varDelta |=t\le 2^{\frac{n}{2}}\) since \(x^{(1)}\in {\mathbb F}_2^{\frac{n}{2}-1}\). Further, there are two cases to be considered.
-
A.
If there exists one vector \(v^{(p)} \) such that \(v_2^{(p)}\ne 0_{\frac{n}{2}}\), we set \((a_1, a_2, a_3, a_4)=(v^{(\ell )}_1,0_{\frac{n}{2}},\) \(0_{\frac{m}{2}-1},v^{(\ell )}_4),\) \( (b_1,b_2,b_3,b_4)=v^{(p)}\). then, a, b satisfy the item 4 of Lemma 3.
-
B.
If \( v_2^{(i)}= 0_{\frac{n}{2}}\) for \(i=1,2,\ldots , 2^{\frac{n+m}{2}-1}\), then there must exist a vector \(v^{(i_1)} \) such that \( v_3^{(i_1)}\ne 0_{\frac{m}{2}-1}\) and \( v_4^{(i_1)}= v^{(\ell )}_4(\ne 0_{\frac{m}{2}})\). Indeed, since \(n>2,m>4\), we have \(2^{\frac{n+m-2}{2}}-2^{\frac{n}{2}}=2^{\frac{n}{2}-1}\cdot (2^{\frac{m}{2}}-2)>2^{\frac{m}{2}}\), there exist two vectors \(v^{(j_1)},v^{(j_2)}\) such that \(v_4^{(j_1)}= v_4^{(j_2)} \) (i.e., \(v_3^{(j_1)}\ne v_3^{(j_2)} \) since \( v_2^{(j_1)}=v_2^{(j_2)}= 0_{\frac{n}{2}}\)), where \(j_1,j_2>t\). Thus, we can set \(v^{(i_1)}=v^{(j_1)}\oplus v^{(j_2)}\oplus (v^{(\ell )}_1,0_{\frac{n}{2}},0_{\frac{m}{2}-1},v^{(\ell )}_4)\). Hence, we set \((a_1, a_2, a_3, a_4)=(v^{(\ell )}_1,0_{\frac{n}{2}},0_{\frac{m}{2}-1},v^{(\ell )}_4),\) \( (b_1,b_2,b_3,b_4)=(v^{(i_1)}_1,0_{\frac{n}{2}},v_3^{(i_1)},v^{(\ell )}_4)\). Thus, a, b satisfy the item 5 of Lemma 3.
-
A.
-
iii.
If \(v^{(i)}_4= 0_{\frac{m}{2}}\) for \(i=1,2,\ldots ,t\), then there must exist one vector \(v^{(l)}\in V\backslash \varDelta \) such that \(v^{(l)}_2\ne 0_{\frac{n}{2}}\), where \(l> t\). To show this, if \(v^{(j)}_2= 0_{\frac{m}{2}}\) for \(j=1,2,\ldots , 2^{\frac{n+m-2}{2}}\), and if \(v^{(j_1)}_3=v^{(j_2)}_3\), we have \(v^{(j_1)}_4=v^{(j_2)}_4\) since \(v^{(i)}_4= 0_{\frac{m}{2}}\) for \(i=1,2,\ldots ,t\), where \(j_1,j_2 >t\). Moreover,
$$\begin{aligned} 2^{\frac{n+m-2}{2}}>2^{\frac{n-2}{2}}\cdot 2^{\frac{m-2}{2}}=|{\mathbb F}_2^{\frac{n}{2}-1}|\cdot |{\mathbb F}_2^{\frac{m}{2}-1}|, \end{aligned}$$so there is at least one vector \(v^{(l)}\in V\backslash \varDelta \) such that \(v^{(l)}_2\ne 0_{\frac{n}{2}}\). We set \((a_1, a_2, a_3, a_4)=(a_1,0_{\frac{n}{2}},0_{\frac{m}{2}-1},0_{\frac{m}{2}})\in (V\cap \varDelta )\backslash \{0_{n+m-2}\}, (b_1,b_2,b_3,b_4)=v^{(l)}\). Thus, a, b satisfy the item 2 of Lemma 3.
-
i.
-
(a)
Combining items 1 and 2, we deduce that h does not belong to \({M}^{\#}\).
Rights and permissions
About this article
Cite this article
Zhang, F., Carlet, C., Hu, Y. et al. New secondary constructions of Bent functions. AAECC 27, 413–434 (2016). https://doi.org/10.1007/s00200-016-0287-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00200-016-0287-6