Further results on semi-bent functions in polynomial form

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

Further results on semi-bent functions in polynomial form

Xiwang Cao ^1, , Hao Chen ^1, and Sihem Mesnager ^1,

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

Abstract
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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.

Keywords: Kloosterman sums, Dickson polynomials., exponential sums, Walsh-Hadamard transformation, Boolean functions, semi-bent functions, cubic sums.

Mathematics Subject Classification: Primary: 11T71; Secondary: 11T2.

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

References:

[1]	L. Carlitz, Explicit evaluation of certain exponential sums, Math. Scand, 44 (1979), 5-16.
[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.
[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.
[4]	S. Chee, S. Lee and K. Kim, Semi-bent Functions, in Int. Conf. Theory Appl. Crypt., Springer, Berlin, 1994, 105-118.
[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.
[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.
[7]	R. Lidl, G. L. Mullen and G. Turnwald, Dickson Polynomials, Addison-Wesley, Reading, MA, 1993, 186-199.
[8]	G. Leander, Monomial bent functions, IEEE Trans. IT, 52 (2006), 738-743. doi: 10.1109/TIT.2005.862121.
[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.
[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.
[11]	S. Mesnager, Contributions on Boolean functions for symmetric cryptography and error correcting codes, Habilitation to Direct Research in Mathematics (HDR thesis), 2012.
[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.
[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.
[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.
[15]	O. S. Rothaus, On bent functions, J. Combin.Theory, Ser A, 20 (1976), 300-305.
[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.
[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.

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References:

[1]	L. Carlitz, Explicit evaluation of certain exponential sums, Math. Scand, 44 (1979), 5-16.
[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.
[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.
[4]	S. Chee, S. Lee and K. Kim, Semi-bent Functions, in Int. Conf. Theory Appl. Crypt., Springer, Berlin, 1994, 105-118.
[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.
[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.
[7]	R. Lidl, G. L. Mullen and G. Turnwald, Dickson Polynomials, Addison-Wesley, Reading, MA, 1993, 186-199.
[8]	G. Leander, Monomial bent functions, IEEE Trans. IT, 52 (2006), 738-743. doi: 10.1109/TIT.2005.862121.
[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.
[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.
[11]	S. Mesnager, Contributions on Boolean functions for symmetric cryptography and error correcting codes, Habilitation to Direct Research in Mathematics (HDR thesis), 2012.
[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.
[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.
[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.
[15]	O. S. Rothaus, On bent functions, J. Combin.Theory, Ser A, 20 (1976), 300-305.
[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.
[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.

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