# American Institute of Mathematical Sciences

May  2018, 12(2): 303-315. doi: 10.3934/amc.2018019

## Several infinite families of p-ary weakly regular bent functions

 1 School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, China 2 School of Mathematics and Information, China West Normal University, Sichuan Nanchong, 637002, China 3 School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China

* Corresponding author: Chunming Tang

Received  November 2016 Published  March 2018

Fund Project: This work is supported by the National Natural Science Foundation of China (Grant No. 11401480, 11701129, 11571285, 11531002). C. Tang also acknowledges support from 14E013, CXTD2014-4 and the Meritocracy Research Funds of China West Normal University. Z. Zhou and C. Fan are in part supported by the National Cryptography Development Fund under Grant MMJJ20170119. Y. Qi also acknowledges support from Zhejiang provincial Natural Science Foundation of China (LQ17A010008, LQ16A010005).

As an optimal combinatorial object, bent functions have been an interesting research object due to their important applications in cryptography, coding theory, and sequence design. The characterization and construction of bent functions are challenging problems in general. The objective of this paper is to present a construction of p-ary weakly regular bent functions from known weakly regular bent functions. This generalizes some earlier constructions of Boolean bent functions and p-ary bent functions, and produces several infinite families of p-ary weakly regular bent functions from known ones. Some infinite families of p-ary rotation symmetric bent functions are obtained as well.

Citation: Yanfeng Qi, Chunming Tang, Zhengchun Zhou, Cuiling Fan. Several infinite families of p-ary weakly regular bent functions. Advances in Mathematics of Communications, 2018, 12 (2) : 303-315. doi: 10.3934/amc.2018019
##### References:
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Fontaine, Highly nonlinear balanced Boolean functions with a good correlationimmunity, Proceedings of EUROCRYPT '98. Lecture Notes in Computer Science, 1403 (1998), 475-488. Google Scholar [12] C. Fontaine, On some cosets of the first-order Reed-Muller code with high minimum weight, IEEE Trans. Inform. Theory, 45 (1999), 1237-1243. doi: 10.1109/18.761276. Google Scholar [13] R. Gold, Maximal recursive sequences with 3-valued recursive crosscorrelation functions (Corresp.), IEEE Trans. Inf. Theory, 14 (1968), 154-156. Google Scholar [14] T. Helleseth and A. Kholosha, Monomial and quadratic bent functions over the finite fields of odd characteristic, IEEE Trans. Inf. Theory, 52 (2006), 2018-2032. doi: 10.1109/TIT.2006.872854. Google Scholar [15] T. Helleseth and A. Kholosha, New binomial bent functions over the finite fields of odd characteristic, IEEE Trans. Inf. Theory, 56 (2010), 4646-4652. doi: 10.1109/TIT.2010.2055130. Google Scholar [16] P. V. Kumar, R. A. Scholtz and L. R. Welch, Generalized bent functions and their properties, J. Combin. Theory Ser. A, 40 (1985), 90-107. doi: 10.1016/0097-3165(85)90049-4. Google Scholar [17] N. Li, T. Helleseth, X. Tang and A. Kholosha, Several new classes of bent functions from Dillon exponents, IEEE Trans. Inf. Theory, 59 (2013), 1818-1831. doi: 10.1109/TIT.2012.2229782. Google Scholar [18] S. C. Liu and J. J. Komo, Nonbinary Kasami sequences over GF(p), IEEE Trans. Inf. Theory, 38 (1992), 1409-1412. doi: 10.1109/18.144728. Google Scholar [19] 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 [20] S. Mesnager, Bent Functions: Fundamentals and Results, Springer Verlag, New York, 2016. Google Scholar [21] J. D. Olsen, R. A. Scholtz and L. R. Welch, Bent-function sequences, IEEE Trans. Inf. Theory, 28 (1982), 858-864. doi: 10.1109/TIT.1982.1056589. Google Scholar [22] J. Pieprzyk and C. Qu, Fast Hashing and rotation symmetric functions, J. Univ. Comput. Sci., 5 (1999), 20-31. Google Scholar [23] A. Pott, Y. Tan and T. Feng, Strongly regular graphs associated with ternary bent functions, J. Combinat. Theory, Ser. A, 117 (2010), 668-682. doi: 10.1016/j.jcta.2009.05.003. Google Scholar [24] A. Pott, Y. Tan, T. Feng and S. Ling, Association schemes arising from bent functions, Des., Codes Cryptography, 59 (2011), 319-331. doi: 10.1007/s10623-010-9463-z. Google Scholar [25] O. Rothaus, On bent functions, J. Combinat. Theory, Ser. A, 20 (1976), 300-305. doi: 10.1016/0097-3165(76)90024-8. Google Scholar [26] S. Su and X. Tang, Construction of rotation symmetric boolean functions with optimal algebraic immunity and high nonlinearity, Des., Codes Cryptography, 71 (2014), 183-199. doi: 10.1007/s10623-012-9727-x. Google Scholar [27] C. Tang, N. Li, Y. Qi, Z. Zhou and T. Helleseth, Linear codes with two or three weights from weakly regular bent functions, IEEE Trans. Inf. Theory., 62 (2016), 1166-1176. doi: 10.1109/TIT.2016.2518678. Google Scholar [28] G. Xu, X. Cao and S. Xu., Several new classes of Boolean functions with few Walsh transform values, Applicable Algebra in Engineering, Communication and Computing, 28 (2017), 155-176. doi: 10.1007/s00200-016-0298-3. Google Scholar [29] G. Xu and X. Cao, Several classes of bent, near-bent and 2-plateaued functions over finite fields of odd characteristic, arXiv: 1508.03415.Google Scholar [30] N. Y. Yu and G. Gong, Construction of quadratic bent functions in polynomial forms, EEE Trans. Inf. Theory, 52 (2006), 3291-3299. doi: 10.1109/TIT.2006.876251. Google Scholar [31] Z. Zhou, N. Li, C. Fan and T. Helleseth, Linear codes with two or three weights from quadratic Bent functions, Des. Codes Cryptography, 81 (2016), 283-295. doi: 10.1007/s10623-015-0144-9. Google Scholar

show all references

##### References:
 [1] R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., 18 (1986), 97-122. doi: 10.1112/blms/18.2.97. Google Scholar [2] C. Carlet, Boolean functions for cryptography and error correcting codes, in Boolean Models and Methods in Mathematics, Computer Science, and Engineering, Y. Crama and P. L. Hammer, Eds. Cambridge, U.K.: Cambridge Univ. Press, 2010,257–397.Google Scholar [3] C. Carlet, Open problems on binary bent functions, Open Problems in Mathematics and Computational Science 2014, 2014 (2014), 203-241. Google Scholar [4] G. Cohen, I. Honkala, S. Litsyn and A. Lobstein, Covering Codes, Amsterdam, The Netherlands: North Holland, 1997. Google Scholar [5] J. Dillon, Elementary Hadamard Difference Sets, Ph. D. dissertation, Netw. Commun. Lab., Univ. Maryland, College Park, MD, USA, 1974 Google Scholar [6] J. F. Dillon and H. Dobbertin, New cyclic difference sets with Singer parameters, Finite Fields Their Appl., 10 (2004), 342-389. doi: 10.1016/j.ffa.2003.09.003. Google Scholar [7] C. Ding, Linear codes from some 2-designs, IEEE Trans. Inform. Theory, 61 (2015), 3265-3275. doi: 10.1109/TIT.2015.2420118. Google Scholar [8] C. Ding, C. Fan and Z. Zhou, The dimension and minimum distance of two classes of primitive BCH codes, Finite Fields and Their Applications, 45 (2017), 237-263. doi: 10.1016/j.ffa.2016.12.009. Google Scholar [9] H. Dobbertin, G. Leander, A. Canteaut, C. Carlet, P. Felke and P. Gaborit, Construction of bent functions via Niho power functions, J. Combinat. Theory, Ser. A, 113 (2006), 779-798. doi: 10.1016/j.jcta.2005.07.009. Google Scholar [10] K. Feng and J. Luo, Value distributions of exponential sums from perfect nonlinear functions and their applications, IEEE Trans. Inform. Theory, 53 (2007), 3035-3041. doi: 10.1109/TIT.2007.903153. Google Scholar [11] E. Filiol and C. Fontaine, Highly nonlinear balanced Boolean functions with a good correlationimmunity, Proceedings of EUROCRYPT '98. Lecture Notes in Computer Science, 1403 (1998), 475-488. Google Scholar [12] C. Fontaine, On some cosets of the first-order Reed-Muller code with high minimum weight, IEEE Trans. Inform. Theory, 45 (1999), 1237-1243. doi: 10.1109/18.761276. Google Scholar [13] R. Gold, Maximal recursive sequences with 3-valued recursive crosscorrelation functions (Corresp.), IEEE Trans. Inf. Theory, 14 (1968), 154-156. Google Scholar [14] T. Helleseth and A. Kholosha, Monomial and quadratic bent functions over the finite fields of odd characteristic, IEEE Trans. Inf. Theory, 52 (2006), 2018-2032. doi: 10.1109/TIT.2006.872854. Google Scholar [15] T. Helleseth and A. Kholosha, New binomial bent functions over the finite fields of odd characteristic, IEEE Trans. Inf. Theory, 56 (2010), 4646-4652. doi: 10.1109/TIT.2010.2055130. Google Scholar [16] P. V. Kumar, R. A. Scholtz and L. R. Welch, Generalized bent functions and their properties, J. Combin. Theory Ser. A, 40 (1985), 90-107. doi: 10.1016/0097-3165(85)90049-4. Google Scholar [17] N. Li, T. Helleseth, X. Tang and A. Kholosha, Several new classes of bent functions from Dillon exponents, IEEE Trans. Inf. Theory, 59 (2013), 1818-1831. doi: 10.1109/TIT.2012.2229782. Google Scholar [18] S. C. Liu and J. J. Komo, Nonbinary Kasami sequences over GF(p), IEEE Trans. Inf. Theory, 38 (1992), 1409-1412. doi: 10.1109/18.144728. Google Scholar [19] 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 [20] S. Mesnager, Bent Functions: Fundamentals and Results, Springer Verlag, New York, 2016. Google Scholar [21] J. D. Olsen, R. A. Scholtz and L. R. Welch, Bent-function sequences, IEEE Trans. Inf. Theory, 28 (1982), 858-864. doi: 10.1109/TIT.1982.1056589. Google Scholar [22] J. Pieprzyk and C. Qu, Fast Hashing and rotation symmetric functions, J. Univ. Comput. Sci., 5 (1999), 20-31. Google Scholar [23] A. Pott, Y. Tan and T. Feng, Strongly regular graphs associated with ternary bent functions, J. Combinat. Theory, Ser. A, 117 (2010), 668-682. doi: 10.1016/j.jcta.2009.05.003. Google Scholar [24] A. Pott, Y. Tan, T. Feng and S. Ling, Association schemes arising from bent functions, Des., Codes Cryptography, 59 (2011), 319-331. doi: 10.1007/s10623-010-9463-z. Google Scholar [25] O. Rothaus, On bent functions, J. Combinat. Theory, Ser. A, 20 (1976), 300-305. doi: 10.1016/0097-3165(76)90024-8. Google Scholar [26] S. Su and X. Tang, Construction of rotation symmetric boolean functions with optimal algebraic immunity and high nonlinearity, Des., Codes Cryptography, 71 (2014), 183-199. doi: 10.1007/s10623-012-9727-x. Google Scholar [27] C. Tang, N. Li, Y. Qi, Z. Zhou and T. Helleseth, Linear codes with two or three weights from weakly regular bent functions, IEEE Trans. Inf. Theory., 62 (2016), 1166-1176. doi: 10.1109/TIT.2016.2518678. Google Scholar [28] G. Xu, X. Cao and S. Xu., Several new classes of Boolean functions with few Walsh transform values, Applicable Algebra in Engineering, Communication and Computing, 28 (2017), 155-176. doi: 10.1007/s00200-016-0298-3. Google Scholar [29] G. Xu and X. Cao, Several classes of bent, near-bent and 2-plateaued functions over finite fields of odd characteristic, arXiv: 1508.03415.Google Scholar [30] N. Y. Yu and G. Gong, Construction of quadratic bent functions in polynomial forms, EEE Trans. Inf. Theory, 52 (2006), 3291-3299. doi: 10.1109/TIT.2006.876251. Google Scholar [31] Z. Zhou, N. Li, C. Fan and T. Helleseth, Linear codes with two or three weights from quadratic Bent functions, Des. Codes Cryptography, 81 (2016), 283-295. doi: 10.1007/s10623-015-0144-9. Google Scholar
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