# American Institute of Mathematical Sciences

November  2016, 10(4): 725-741. doi: 10.3934/amc.2016037

## Further results on semi-bent functions in polynomial form

 1 School of Mathematical Sciences, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China, China, China

Received  August 2014 Revised  June 2015 Published  November 2016

Plateaued functions have been introduced by Zheng and Zhang in 1999 as good candidates for designing cryptographic functions since they possess many desirable cryptographic characteristics. Plateaued functions bring together various nonlinear characteristics and include two important classes of Boolean functions defined in even dimension: the well-known bent functions ($0$-plateaued functions) and the semi-bent functions ($2$-plateaued functions). Bent functions have been extensively investigated since 1976. Very recently, the study of semi-bent functions has attracted a lot of attention in symmetric cryptography. Many intensive progresses in the design of such functions have been made especially in recent years. The paper is devoted to the construction of semi-bent functions on the finite field $\mathbb{F}_{2^n}$ ($n=2m$) in the line of a recent work of S. Mesnager [IEEE Transactions on Information Theory, Vol 57, No 11, 2011]. We extend Mesnager's results and present a new construction of infinite classes of binary semi-bent functions in polynomial trace. The extension is achieved by inserting mappings $h$ on $\mathbb{F}_{2^n}$ which can be expressed as $h(0) = 0$ and $h(uy) = h_1(u)h_2(y)$ with $u$ ranging over the circle $U$ of unity of $\mathbb{F}_{2^n}$, $y \in \mathbb{F}_{2^m}^{*}$ and $uy \in \mathbb{F}_{2^n}^{*}$, where $h_1$ is a isomorphism on $U$ and $h_2$ is an arbitrary mapping on $\mathbb{F}_{2^m}^{*}$. We then characterize the semi-bentness property of the extended family in terms of classical binary exponential sums and binary polynomials.
Citation: 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
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##### References:
 [1] L. Carlitz, Explicit evaluation of certain exponential sums, Math. Scand, 44 (1979), 5-16.  Google Scholar [2] C. Carlet, Boolean Functions for Cryptography and Error Correcting Codes, in Boolean Models and Methods in Mathematics, Computer Science, and Engineering, Cambridge Univ. Press, 2010, 257-397. Google Scholar [3] P. Charpin, T. Helleseth and V. Zinoviev, The divisibility modulo $24$ of Kloosterman sums of $GF(2^m)$, $m$ odd, J. Comb. Theory A, 114 (2007), 322-338. doi: 10.1016/j.jcta.2006.06.002.  Google Scholar [4] S. Chee, S. Lee and K. Kim, Semi-bent Functions, in Int. Conf. Theory Appl. Crypt., Springer, Berlin, 1994, 105-118.  Google Scholar [5] J. F. Dillon and H. Dobbertin, New cyclic difference sets with Singer parameters, Finite Fields Appl., 10 (2004), 342-389. doi: 10.1016/j.ffa.2003.09.003.  Google Scholar [6] G. Lachaud and J. Wolfmann, The weights of the orthogonals of the extended quadratic binary Goppa codes, IEEE Trans. IT, 36 (1990), 686-692. doi: 10.1109/18.54892.  Google Scholar [7] R. Lidl, G. L. Mullen and G. Turnwald, Dickson Polynomials, Addison-Wesley, Reading, MA, 1993, 186-199.  Google Scholar [8] G. Leander, Monomial bent functions, IEEE Trans. IT, 52 (2006), 738-743. doi: 10.1109/TIT.2005.862121.  Google Scholar [9] S. Mesnager, Bent and hyper-bent functions in polynomial form and their link with some exponential sums and Dickson polynomials, IEEE Trans. IT, 57 (2011), 5996-6009. doi: 10.1109/TIT.2011.2124439.  Google Scholar [10] S. Mesnager, Semi-bent functions from Dillon and Niho exponents, Kloosterman sums and Dickson polynomials, IEEE Trans. IT, 57 (2011), 7443-7458. doi: 10.1109/TIT.2011.2160039.  Google Scholar [11] S. Mesnager, Contributions on Boolean functions for symmetric cryptography and error correcting codes, Habilitation to Direct Research in Mathematics (HDR thesis), 2012. Google Scholar [12] S. Mesnager, Several new infinite families of bent functions and their duals, IEEE Trans. IT, 60 (2014), 4397-4407. doi: 10.1109/TIT.2014.2320974.  Google Scholar [13] S. Mesnager, On semi-bent functions and related plateaued functions over the Galois field $\mathbb F_{2^n}$, in Open Problems in Mathematics and Computational Science, Springer, 2014, 243-273.  Google Scholar [14] S. Mesnager and J. P. Flori, Hyper-bent functions via Dillon-like exponents, IEEE Trans. IT, 59 (2013), 3215-3232. doi: 10.1109/TIT.2013.2238580.  Google Scholar [15] O. S. Rothaus, On bent functions, J. Combin.Theory, Ser A, 20 (1976), 300-305.  Google Scholar [16] Y. Zheng and X. M. Zhang, Relationships between bent functions and complementary plateaued functions, in Int. Conf. Inform. Secur. Crypt., Springer, Berlin, 1999, 60-75. Google Scholar [17] Y. Zheng and X. M. Zhang, Plateaued functions, in Int. Conf. Inform. Commun. Secur., Springer, Berlin, 1999, 284-300. doi: 10.1007/3-540-48892-8_22.  Google Scholar
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