# American Institute of Mathematical Sciences

May  2017, 11(2): 339-345. doi: 10.3934/amc.2017026

## On constructions of bent, semi-bent and five valued spectrum functions from old bent functions

 1 Department of Mathematics, University of Paris Ⅷ and Paris ⅩⅢ and Télécom ParisTech, LAGA, UMR 7539, CNRS, Sorbonne Paris Cité 2 School of Computer Science and Technology, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China 3 Department of Mathematics, University of Paris Ⅷ and Paris ⅩⅢ, LAGA, UMR 7539, CNRS, Sorbonne Paris Cité

* Corresponding author

Received  February 2016 Revised  March 2016 Published  May 2017

Fund Project: This work was supported by National Science Foundation of China (Grant No. 61303263), and in part by the Fundamental Research Funds for the Central Universities (Grant No. 2015XKMS086), and in part by the China Postdoctoral Science Foundation funded project (Grant No. 2015T80600).

The paper presents methods for designing functions having many applications in particular to construct linear codes with few weights. The former codes have several applications in secret sharing, authentication codes, association schemes and strongly regular graphs. We firstly provide new secondary constructions of bent functions generalizing the well-known Rothaus' constructions as well as their dual functions. From our generalization, we show that we are able to compute the dual function of a bent function built from Rothaus' construction. Next we present a result leading to a new method for constructing semi-bent functions and few Walsh transform values functions built from bent functions.

Citation: Sihem Mesnager, Fengrong Zhang. On constructions of bent, semi-bent and five valued spectrum functions from old bent functions. Advances in Mathematics of Communications, 2017, 11 (2) : 339-345. doi: 10.3934/amc.2017026
##### References:
 [1] C. Carlet, A construction of bent functions, in Finite Fields and Applications, London Math. Soc. , 1996, 47-58. doi: 10.1017/CBO9780511525988.006. [2] C. Carlet, On the secondary constructions of resilient and bent functions, in Proc. Workshop Coding Crypt. Combin. 2003, Birkhäuser Verlag, 2004, 3-28. [3] C. Carlet, On bent and highly nonlinear balanced/resilient functions and their algebraic immunities, in Proc. AAECC 16, 2006, 1-28. doi: 10.1007/11617983_1. [4] C. Carlet, Boolean functions for cryptography and error correcting codes, in Boolean Models and Methods in Mathematics, Computer Science, and Engineering (eds. Y. Crama and P. Hammer), 2010,257-397. doi: 10.1017/CBO9780511780448. [5] C. Carlet and S. Mesnager, Four decades of research on bent functions, Des. Codes Crypt., 78 (2016), 5-50.  doi: 10.1007/s10623-015-0145-8. [6] C. Carlet, F. Zhang and Y. Hu, Secondary constructions of bent functions and their enforcement, Adv. Math. Commun., 6 (2012), 305-314.  doi: 10.3934/amc.2012.6.305. [7] J. Dillon, Elementary Hadamard Difference Sets, Ph. D thesis, Univ. Maryland, College Park, 1974. [8] C. Ding, Linear codes from some 2-designs, IEEE Trans. Inf. Theory, 61 (2015), 3265-3275.  doi: 10.1109/TIT.2015.2420118. [9] S. Mesnager, Several new infinite families of bent functions and their duals, IEEE Trans. Inf. Theory, 60 (2014), 4397-4407.  doi: 10.1109/TIT.2014.2320974. [10] S. Mesnager, Bent Functions: Fundamentals and Results, Springer-Verlag, 2016.  doi: 10.1007/978-3-319-32595-8. [11] O. S. Rothaus, On "bent" functions, J. Combin. Theory Ser. A, 20 (1976), 300-305. [12] F. Zhang, C. Carlet, Y. Hu and W. Zhang, New secondary constructions of bent functions, Appl. Algebra Eng. Commun. Comput., 27 (2016), 413-434.  doi: 10.1007/s00200-016-0287-6.

show all references

##### References:
 [1] C. Carlet, A construction of bent functions, in Finite Fields and Applications, London Math. Soc. , 1996, 47-58. doi: 10.1017/CBO9780511525988.006. [2] C. Carlet, On the secondary constructions of resilient and bent functions, in Proc. Workshop Coding Crypt. Combin. 2003, Birkhäuser Verlag, 2004, 3-28. [3] C. Carlet, On bent and highly nonlinear balanced/resilient functions and their algebraic immunities, in Proc. AAECC 16, 2006, 1-28. doi: 10.1007/11617983_1. [4] C. Carlet, Boolean functions for cryptography and error correcting codes, in Boolean Models and Methods in Mathematics, Computer Science, and Engineering (eds. Y. Crama and P. Hammer), 2010,257-397. doi: 10.1017/CBO9780511780448. [5] C. Carlet and S. Mesnager, Four decades of research on bent functions, Des. Codes Crypt., 78 (2016), 5-50.  doi: 10.1007/s10623-015-0145-8. [6] C. Carlet, F. Zhang and Y. Hu, Secondary constructions of bent functions and their enforcement, Adv. Math. Commun., 6 (2012), 305-314.  doi: 10.3934/amc.2012.6.305. [7] J. Dillon, Elementary Hadamard Difference Sets, Ph. D thesis, Univ. Maryland, College Park, 1974. [8] C. Ding, Linear codes from some 2-designs, IEEE Trans. Inf. Theory, 61 (2015), 3265-3275.  doi: 10.1109/TIT.2015.2420118. [9] S. Mesnager, Several new infinite families of bent functions and their duals, IEEE Trans. Inf. Theory, 60 (2014), 4397-4407.  doi: 10.1109/TIT.2014.2320974. [10] S. Mesnager, Bent Functions: Fundamentals and Results, Springer-Verlag, 2016.  doi: 10.1007/978-3-319-32595-8. [11] O. S. Rothaus, On "bent" functions, J. Combin. Theory Ser. A, 20 (1976), 300-305. [12] F. Zhang, C. Carlet, Y. Hu and W. Zhang, New secondary constructions of bent functions, Appl. Algebra Eng. Commun. Comput., 27 (2016), 413-434.  doi: 10.1007/s00200-016-0287-6.
 [1] Xiwang Cao, Hao Chen, Sihem Mesnager. Further results on semi-bent functions in polynomial form. Advances in Mathematics of Communications, 2016, 10 (4) : 725-741. doi: 10.3934/amc.2016037 [2] Bimal Mandal, Aditi Kar Gangopadhyay. A note on generalization of bent boolean functions. Advances in Mathematics of Communications, 2021, 15 (2) : 329-346. doi: 10.3934/amc.2020069 [3] Tingting Pang, Nian Li, Li Zhang, Xiangyong Zeng. Several new classes of (balanced) Boolean functions with few Walsh transform values. Advances in Mathematics of Communications, 2021, 15 (4) : 757-775. doi: 10.3934/amc.2020095 [4] Ayça Çeşmelioǧlu, Wilfried Meidl, Alexander Pott. On the dual of (non)-weakly regular bent functions and self-dual bent functions. Advances in Mathematics of Communications, 2013, 7 (4) : 425-440. doi: 10.3934/amc.2013.7.425 [5] Jacques Wolfmann. Special bent and near-bent functions. Advances in Mathematics of Communications, 2014, 8 (1) : 21-33. doi: 10.3934/amc.2014.8.21 [6] Claude Carlet, Fengrong Zhang, Yupu Hu. Secondary constructions of bent functions and their enforcement. Advances in Mathematics of Communications, 2012, 6 (3) : 305-314. doi: 10.3934/amc.2012.6.305 [7] Sihem Mesnager, Fengrong Zhang, Yong Zhou. On construction of bent functions involving symmetric functions and their duals. Advances in Mathematics of Communications, 2017, 11 (2) : 347-352. doi: 10.3934/amc.2017027 [8] Ayça Çeşmelioğlu, Wilfried Meidl. Bent and vectorial bent functions, partial difference sets, and strongly regular graphs. Advances in Mathematics of Communications, 2018, 12 (4) : 691-705. doi: 10.3934/amc.2018041 [9] Junchao Zhou, Yunge Xu, Lisha Wang, Nian Li. Nearly optimal codebooks from generalized Boolean bent functions over $\mathbb{Z}_{4}$. Advances in Mathematics of Communications, 2022, 16 (3) : 485-501. doi: 10.3934/amc.2020121 [10] Samir Hodžić, Enes Pasalic. Generalized bent functions -sufficient conditions and related constructions. Advances in Mathematics of Communications, 2017, 11 (3) : 549-566. doi: 10.3934/amc.2017043 [11] Claude Carlet, Juan Carlos Ku-Cauich, Horacio Tapia-Recillas. Bent functions on a Galois ring and systematic authentication codes. Advances in Mathematics of Communications, 2012, 6 (2) : 249-258. doi: 10.3934/amc.2012.6.249 [12] Junchao Zhou, Nian Li, Xiangyong Zeng, Yunge Xu. A generic construction of rotation symmetric bent functions. Advances in Mathematics of Communications, 2021, 15 (4) : 721-736. doi: 10.3934/amc.2020092 [13] Kanat Abdukhalikov, Duy Ho. Vandermonde sets, hyperovals and Niho bent functions. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021048 [14] Jyrki Lahtonen, Gary McGuire, Harold N. Ward. Gold and Kasami-Welch functions, quadratic forms, and bent functions. Advances in Mathematics of Communications, 2007, 1 (2) : 243-250. doi: 10.3934/amc.2007.1.243 [15] Kanat Abdukhalikov, Sihem Mesnager. Explicit constructions of bent functions from pseudo-planar functions. Advances in Mathematics of Communications, 2017, 11 (2) : 293-299. doi: 10.3934/amc.2017021 [16] Joan-Josep Climent, Francisco J. García, Verónica Requena. On the construction of bent functions of $n+2$ variables from bent functions of $n$ variables. Advances in Mathematics of Communications, 2008, 2 (4) : 421-431. doi: 10.3934/amc.2008.2.421 [17] Sihong Su. A new construction of rotation symmetric bent functions with maximal algebraic degree. Advances in Mathematics of Communications, 2019, 13 (2) : 253-265. doi: 10.3934/amc.2019017 [18] Wenying Zhang, Zhaohui Xing, Keqin Feng. A construction of bent functions with optimal algebraic degree and large symmetric group. Advances in Mathematics of Communications, 2020, 14 (1) : 23-33. doi: 10.3934/amc.2020003 [19] Natalia Tokareva. On the number of bent functions from iterative constructions: lower bounds and hypotheses. Advances in Mathematics of Communications, 2011, 5 (4) : 609-621. doi: 10.3934/amc.2011.5.609 [20] Yanfeng Qi, Chunming Tang, Zhengchun Zhou, Cuiling Fan. Several infinite families of p-ary weakly regular bent functions. Advances in Mathematics of Communications, 2018, 12 (2) : 303-315. doi: 10.3934/amc.2018019

2021 Impact Factor: 1.015