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. Google Scholar

[2]

C. Carlet, On the secondary constructions of resilient and bent functions, in Proc. Workshop Coding Crypt. Combin. 2003, Birkhäuser Verlag, 2004, 3-28. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

C. CarletF. 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. Google Scholar

[7]

J. Dillon, Elementary Hadamard Difference Sets, Ph. D thesis, Univ. Maryland, College Park, 1974. Google Scholar

[8]

C. Ding, Linear codes from some 2-designs, IEEE Trans. Inf. Theory, 61 (2015), 3265-3275. doi: 10.1109/TIT.2015.2420118. Google Scholar

[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. Google Scholar

[10] S. Mesnager, Bent Functions: Fundamentals and Results, Springer-Verlag, 2016. doi: 10.1007/978-3-319-32595-8. Google Scholar
[11]

O. S. Rothaus, On "bent" functions, J. Combin. Theory Ser. A, 20 (1976), 300-305. Google Scholar

[12]

F. ZhangC. CarletY. 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. Google Scholar

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. Google Scholar

[2]

C. Carlet, On the secondary constructions of resilient and bent functions, in Proc. Workshop Coding Crypt. Combin. 2003, Birkhäuser Verlag, 2004, 3-28. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

C. CarletF. 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. Google Scholar

[7]

J. Dillon, Elementary Hadamard Difference Sets, Ph. D thesis, Univ. Maryland, College Park, 1974. Google Scholar

[8]

C. Ding, Linear codes from some 2-designs, IEEE Trans. Inf. Theory, 61 (2015), 3265-3275. doi: 10.1109/TIT.2015.2420118. Google Scholar

[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. Google Scholar

[10] S. Mesnager, Bent Functions: Fundamentals and Results, Springer-Verlag, 2016. doi: 10.1007/978-3-319-32595-8. Google Scholar
[11]

O. S. Rothaus, On "bent" functions, J. Combin. Theory Ser. A, 20 (1976), 300-305. Google Scholar

[12]

F. ZhangC. CarletY. 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. Google Scholar

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