Abstract
In this paper, we first discuss the bentness of a large class of quadratic Boolean functions in polynomial form \(f(x)=\sum _{i=1}^{{n}/{2}-1}\mathrm {Tr}^n_1(c_ix^{1+2^i})+ \mathrm {Tr}_1^{n/2}(c_{n/2}x^{1+2^{n/2}})\), where n is even, \(c_i\in \mathrm {GF}(2^n)\) for \(1\le i \le {n}/{2}-1\) and \(c_{n/2}\in \mathrm {GF}(2^{n/2})\). The bentness of these functions can be connected with linearized permutation polynomials. Hence, methods for constructing quadratic bent functions are given. Further, we consider a subclass of quadratic Boolean functions of the form \(f(x)=\sum _{i=1}^{{m}/{2}-1}\mathrm {Tr}^n_1(c_ix^{1+2^{ei}})+ \mathrm {Tr}_1^{n/2}(c_{m/2}x^{1+2^{n/2}})\), where \(n=em\), m is even, and \(c_i\in \mathrm {GF}(2^e)\). The bentness of these functions is characterized and some methods for deriving new quadratic bent functions are given. Finally, when m and e satisfy some conditions, we determine the number of these quadratic bent functions.
Similar content being viewed by others
References
Berlekamp, E.R.: Algebraic Coding Theory, revised edn. Aegean Park, Laguna Hills (1984)
Boztas, S., Kumar, P.V.: Binary sequences with Gold-like correlation but larger linear span. IEEE Trans. Inf. Theory 40, 532–537 (1994)
Canteaut, A., Charpin, P.: Decomposing bent functions. IEEE Trans. Inf. Theory 49(8), 2004–2019 (2003)
Carlet, C.: A larger class of cryptographic Boolean functions via a study of the Maiorana–McFarland construction. In: Yung, M. (ed.) Advances in Cryptology-CRYPTO 2002. Lecture Notes in Computer Science, vol. 2442, pp. 549–564. Springer, Berlin (2002)
Carlet, C., Charpin, P., Zinoviev, V.A.: Codes, bent functions and permutations suitable for DES-like cryptosystem. Des. Codes. Cryptogr. 15, 125–156 (1998)
Charpin, P., Pasalic, E., Tavernier, C.: On bent and semi-bent quadratic Boolean functions. IEEE Trans. Inf. Theory 51(12), 4286–4298 (2005)
Dobbertin, H., Leander, G., Canteaut, A., Carlet, C., Felke, P., Gaborit, P.: Construction of bent functions via Niho power functions. J. Comb. Theory, Ser. A 113, 779–798 (2006)
Gold, R.: Maximal recursive sequences with 3-valued recursive crosscorrelation functions. IEEE Trans. Inf. Theory 14(1), 154–156 (1968)
Golomb, S.W., Gong, G.: Signal Design for Good Correlation: for Wireless Communication, Cryptography and Radar. Cambridge University Press, Cambridge (2005)
Hu, H., Feng, D.: On quadratic bent functions in polynomial forms. IEEE Trans. Inf. Theory 53(7), 2610–2615 (2007)
Helleseth, T., Kumar, P.V.: Sequences with low correlation. In: Pless, V.S., Huffman, W.C. (eds.) Handbook of Coding Theory, vol. 2, pp. 1765–1853. North-Holland, Amsterdam (1998)
Leander, G.: Monomial bent functions. IEEE Trans. Inf. Theory 52(2), 738–743 (2006)
Lidl, R., Niederreiter, H.: Finite fields. In: Lidl, R., Niederreiter, H., Cohn, P.M. (eds.).). Encyclopedia of Mathematics and Its Applications, 20th edn. Addison-Wesley, Reading (1983)
Khoo, K., Gong, G., Stinson, D.R.: A new family of Gold-like sequences. In: Proceedings of IEEE International Symposium Information Theory, Lausanne, Switzerland, p. 181 (2002)
Khoo, K., Gong, G., Stinson, D.R.: A new characterization of semi-bent and bent functions on finite fields. Des. Codes. Cryptogr. 38(2), 279–295 (2006)
Kim, S.H., No, J.S.: New families of binary sequences with low correlation. IEEE Trans. Inf. Theory 49(11), 3059–3065 (2003)
Lempel, A., Cohn, M.: Maximal families of bent sequences. IEEE Trans. Inf. Theory 28, 865–868 (1982)
Ma, W., Lee, M., Zhang, F.: A new class of bent functions. IEICE Trans. Fundam. E88–A(7), 2039–2040 (2005)
MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)
McEliece, R.J.: Finite Fields for Computer Scientists and Engineers. Kluwer, Dordrecht (1987)
Ore, O.: Theory of non-commutative polynomials. Ann. Math. 34, 480–508 (1933)
Ore, O.: On a special class of polynomials. Trans. Am. Math. Soc. 35, 559–584 (1933)
Olsen, J.D., Scholtz, R.A., Welch, L.R.: Bent-function sequences. IEEE Trans. Inf. Theory 28(6), 858–864 (1982)
Rothaus, O.S.: On bent functions. J. Comb. Theory A 20, 300–305 (1976)
Udaya, P.: Polyphase and frequency hopping sequences obtained from finite rings, Ph.D. Dissertation, Department of Electrical Engineering, Indian Institute of Technology, Kanpur, India (1992)
Wu, B., Liu, Z.: Linearized polynomials over finite fields revisited. Finite Fields Appl. 22, 79–100 (2013)
Yu, N.Y., Gong, G.: Constructions of quadratic bent functions in polynomial forms. IEEE Trans. Inf. Theory 52(7), 3291–3299 (2006)
Acknowledgements
The authors are very grateful to the anonymous reviewers and Prof. Teo Mora for their valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11871058, 11531002, 11701129, 61672059). C. Tang also acknowledges support from 14E013, CXTD2014-4 and the Meritocracy Research Funds of China West Normal University. Y. Qi also acknowledges support from Zhejiang provincial Natural Science Foundation of China (LQ17A010008, LQ16A010005).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Huang, D., Tang, C., Qi, Y. et al. New quadratic bent functions in polynomial forms with coefficients in extension fields. AAECC 30, 333–347 (2019). https://doi.org/10.1007/s00200-018-0376-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00200-018-0376-9
Keywords
- Bent function
- Boolean function
- Linearized permutation polynomial
- Cyclotomic polynomial
- Semi-bent function