Abstract
We give a necessary and sufficient condition such that the class of \(p\)-ary binomial functions proposed by Jia et al. (IEEE Trans Inf Theory 58(9):6054–6063, 2012) are regular bent functions, and thus settle the open problem raised at the end of that paper. Moreover, we investigate the bentness of the proposed binomials under the case \(\gcd (\frac{t}{2}, p^{\frac{n}{2}}+1)=1\) for some even integers \(t\) and \(n\). Computer experiments show that the new class contains bent functions that are affinely inequivalent to known monomial and binomial ones.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Canteaut, A., Charpin, P., Kyureghyan, G.: A new class of monomial bent functions. Finite Fields Appl. 14(1), 221–241 (2008)
Carlet, C.: Boolean functions for cryptography and error correcting codes. In: Crama, Y., Hammer, P. (eds.) The Monography Boolean Models and Methods in Mathematics, Computer Science, and Engineering, pp. 257–397. Cambridge University Press, Cambridge (2010)
Carlet, C., Ding, C.: Highly nonlinear mappings. J. Complex. 20(2–3), 205–244 (2004)
Charpin, P., Kyureghyan, G.: Cubic monomial bent functions: a subclass of \({\cal M}\). SIAM. J. Discret. Math. 22(2), 650–665 (2008)
Dillon, J.F.: Elementary Hadamard difference sets. Ph. D. these, University Maryland, Collage Park (1974)
Dobbertin, H., Leander, G., Canteaut, A., Gabort, P.: Construction of bent functions via Niho power functions. J. Comb. Theory Ser. 113, 779–798 (2006)
Ding, C., Yuan, J.: A family of skew Hadamard difference sets. J. Comb. Theory Ser. A 113, 1526–1535 (2006)
Golomb, S.W., Gong, G.: Signal Designs With Good Correlation: For Wireless Communications. Cryptography and Radar Applications. Cambridge University Press, Cambridge (2005)
Helleseth, T., Kholosha, A.: Monomial and quadratic bent functions over the finite fields of odd characteristic. IEEE Trans. Inf. Theory 52(5), 2018–2032 (2006)
Helleseth, T., Kholosha, A.: New binomial bent functions over finite fields of odd characteristic. IEEE Trans. Inf. Theory 56(9), 4646–4652 (2010)
Hou, X.D.: \(p\)-ary and \(q\)-ary versions of certain results about bent functions and resilient functions. Finite Fields Appl. 10(4), 566–582 (2004)
Jia, W., Zeng, X., Helleseth, T., Li, C.: A class of binomial bent functions over the finite fields of odd characteristic. IEEE Trans. Inf. Theory 58(9), 6054–6063 (2012)
Kumar, P.V., Scholtz, R.A., Welch, L.R.: Generalized bent functions and their properties. J. Comb. Theory Ser. A 40, 90–107 (1985)
Leander, G.: Monomial bent functions. IEEE Trans. Inf. Theory 52(2), 738–743 (2006)
Lidl, R., Niederreiter, H.: Finite fields ser. In: Encyclopedia of Mathematics and its Applications. Addison-Wesley, Amsterdam (1983)
Liu, S.C., Komo, J.J.: Nonbinary Kasami sequence over GF(p). IEEE Trans. Inf. Theory 38(4), 1409–1412 (1983)
MacWilliams, F.J., Sloane, N.J.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)
Mesnager, S.: Bent and hyper-bent functions in polynomial form their link with some exponential sums and Dickson polynomials. IEEE Trans. Inf. Theory 57(9), 5996–6009 (2011)
Rothaus, O.S.: On bent functions. J. Comb. Theory Ser. A 20, 300–305 (1976)
Xiang, Q.: Maximally nonlinear functions and bent functions. Des. Codes Cryptogr. 17, 211–218 (1999)
Acknowledgments
The authors wish to thank Xiangyong Zeng, Xiwang Cao and two anonymous referees for their helpful comments. The work of D. Zheng was supported by National Natural Science Foundation of China (NSFC) under Grant 11101131. The work of L. Hu was supported by the NSFC (61070172 and 10990011), and the National Basic Research Program of China (2013CB834203).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zheng, D., Yu, L. & Hu, L. On a class of binomial bent functions over the finite fields of odd characteristic. AAECC 24, 461–475 (2013). https://doi.org/10.1007/s00200-013-0202-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00200-013-0202-3