# American Institute of Mathematical Sciences

August  2012, 6(3): 305-314. doi: 10.3934/amc.2012.6.305

## Secondary constructions of bent functions and their enforcement

 1 LAGA, Universities of Paris 8 and Paris 13; CNRS, UMR 7539, Department of Mathematics, University of Paris 8, 2 rue de la liberté, 93526 Saint-Denis cedex 02, France 2 School of Computer Science and Technology, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China, and ISN, Xidian University, Xi'an, Shannxi 710071, China 3 State Key Laboratory of Integrated Services Networks, Xidian university, P.O. Box 95, Taibai Road 2, Xi'an, Shannxi 710071, China

Received  July 2011 Revised  March 2012 Published  August 2012

Thirty years ago, Rothaus introduced the notion of bent function and presented a secondary construction (building new bent functions from already defined ones), which is now called the Rothaus construction. This construction has a strict requirement for its initial functions. In this paper, we first concentrate on the design of the initial functions in the Rothaus construction. We show how to construct Maiorana-McFarland's (M-M) bent functions, which can then be used as initial functions, from Boolean permutations and orthomorphic permutations. We deduce that at least $(2^n!\times 2^{2^n})(2^{2^n}\times2^{2^{n-1}})^2$ bent functions in $2n+2$ variables can be constructed by using Rothaus' construction. In the second part of the note, we present a new secondary construction of bent functions which generalizes the Rothaus construction. This construction requires initial functions with stronger conditions; we give examples of functions satisfying them. Further, we generalize the new secondary construction of bent functions and illustrate it with examples.
Citation: Claude Carlet, Fengrong Zhang, Yupu Hu. Secondary constructions of bent functions and their enforcement. Advances in Mathematics of Communications, 2012, 6 (3) : 305-314. doi: 10.3934/amc.2012.6.305
##### References:
 [1] A. Canteaut and M. Trabbia, Improved fast correlation attacks using parity-check equations of weight 4 and 5, in "EUROCRYPT 2000'' (ed. B. Preneel), Springer, (2000), 573-588. doi: 10.1007/3-540-45539-6_40. [2] C. Carlet, Two new classes of bent functions, in "EUROCRYPT'93'' (ed. T. Helleseth), Springer, (1994), 77-101. [3] C. Carlet, Generalized partial spreads, IEEE Trans. Inform. Theory, 41 (1995), 1482-1487. doi: 10.1109/18.412693. [4] C. Carlet, A construction of bent functions, in "Proceeding of the Third International Conference on Finite Fields and Applications'' (eds. S. Cohen and H. Niederreiter), Cambridge University Press, (1996), 47-58. doi: 10.1017/CBO9780511525988.006. [5] C. Carlet, On the confusion and diffusion properties of Maiorana-McFarland's and extended Maiorana-McFarland's functions, J. Complexity, 20 (2004), 182-204. doi: 10.1016/j.jco.2003.08.013. [6] C. Carlet, On the secondary constructions of resilient and bent functions, in "Proceedings of the Workshop on Coding, Cryptography and Combinatorics 2003'' (eds. K. Feng, H. Niederreiter and C. Xing), Birkhäuser Verlag, (2004), 3-28. [7] C. Carlet, On bent and highly nonlinear balanced/resilient functions and their algebaric immunities, in "AAECC 2006'' (eds. M. Fossorier et al.), Springer, (2006), 1-28. [8] C. Carlet, Boolean functions for cryptography and error correcting codes, in "Boolean Models and Methods in Mathematics, Computer Science, and Engineering'' (eds. Y. Crama and P. Hammer), Cambridge University Press, (2010), 257-397. [9] C. Carlet, H. Dobbertin and G. Leander, Normal extensions of bent functions, IEEE Trans. Inform. Theory, 50 (2004), 2880-2885. doi: 10.1109/TIT.2004.836681. [10] J. Dillon, "Elementary Hadamard Difference Sets,'' Ph.D thesis, Univ. Maryland, College Park, 1974. [11] H. Dobbertin, Construction of bent functions and balanced Boolean functions with high nonlinearity, in "Fast Software Encryption,'' Springer, (1995), 61-74. doi: 10.1007/3-540-60590-8_5. [12] H. Dobbertin and G. Leander, Bent functions embedded into the recursive framework of $\mathbb Z$-bent functions, Des. Codes Cryptogr., 49 (2008), 3-22. doi: 10.1007/s10623-008-9189-3. [13] P. Guillo, Completed GPS covers all bent functions, J. Combin. Theory Ser. A, 93 (2001), 242-260. doi: 10.1006/jcta.2000.3076. [14] X.-D. Hou, New constructions of bent functions, J. Combin. Inform. System Sci., 25 (2000), 173-189. [15] P. Langevin, G. Leander, P. Rabizzoni, P. Veron and J.-P. Zanotti, Classification of Boolean quartics forms in eight variables, availabel at http://langevin.univ-tln.fr/project/quartics/quartics.html [16] G. Leander, Monomial bent functions, IEEE Trans. Inform. Theory, 52 (2006), 738-743. doi: 10.1109/TIT.2005.862121. [17] G. Leander and G. McGuire, Construction of bent functions from near-bent functions, J. Combin. Theory Ser. A, 116 (2009), 960-970. doi: 10.1016/j.jcta.2008.12.004. [18] Q. Liu, Y. Zhang, C. Cheng and W. Lü, Construction and counting orthomorphism based on transversal, in "2008 International Conference on Computational Intelligence and Security,'' IEEE Computer Society, (2008), 369-373. [19] F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland, Amsterdam, 1977. [20] R. I. McFarland, A family of difference sets in non-cyclic groups, J. Comb. Theory Ser. A, 15 (1973), 1-10. doi: 10.1016/0097-3165(73)90031-9. [21] Q. Meng, L. Chen and F. Fu, On homogeneous rotation symmetric bent functions, Discrete Appl. Math., 158 (2010), 1111-1117. doi: 10.1016/j.dam.2010.02.009. [22] J. D. Olsen, R. A. Scholtz and L. R. Welch, Bent-function sequence, IEEE Trans. Inform. Theory, 28 (1982), 858-864. doi: 10.1109/TIT.1982.1056589. [23] O. S. Rothaus, On "bent'' functions, J. Combin. Theory Ser. A, 20 (1976), 300-305. doi: 10.1016/0097-3165(76)90024-8. [24] J. Wolfmann, Bent functions and coding theory, in "Difference Sets, Sequences and their Correlation Properties'' (eds. A. Pott, P.V. Kumar, T. Helleseth and D. Jungnickel), Amsterdam, Kluwer, (1999), 393-417. [25] H. Zhen, H. Zhang, T. Cui and X. Du, A new method for construction of orthomorphic permutations (in Chinese), J. Electr. Inform. Tech., 31 (2009), 1438-1441.

show all references

##### References:
 [1] A. Canteaut and M. Trabbia, Improved fast correlation attacks using parity-check equations of weight 4 and 5, in "EUROCRYPT 2000'' (ed. B. Preneel), Springer, (2000), 573-588. doi: 10.1007/3-540-45539-6_40. [2] C. Carlet, Two new classes of bent functions, in "EUROCRYPT'93'' (ed. T. Helleseth), Springer, (1994), 77-101. [3] C. Carlet, Generalized partial spreads, IEEE Trans. Inform. Theory, 41 (1995), 1482-1487. doi: 10.1109/18.412693. [4] C. Carlet, A construction of bent functions, in "Proceeding of the Third International Conference on Finite Fields and Applications'' (eds. S. Cohen and H. Niederreiter), Cambridge University Press, (1996), 47-58. doi: 10.1017/CBO9780511525988.006. [5] C. Carlet, On the confusion and diffusion properties of Maiorana-McFarland's and extended Maiorana-McFarland's functions, J. Complexity, 20 (2004), 182-204. doi: 10.1016/j.jco.2003.08.013. [6] C. Carlet, On the secondary constructions of resilient and bent functions, in "Proceedings of the Workshop on Coding, Cryptography and Combinatorics 2003'' (eds. K. Feng, H. Niederreiter and C. Xing), Birkhäuser Verlag, (2004), 3-28. [7] C. Carlet, On bent and highly nonlinear balanced/resilient functions and their algebaric immunities, in "AAECC 2006'' (eds. M. Fossorier et al.), Springer, (2006), 1-28. [8] C. Carlet, Boolean functions for cryptography and error correcting codes, in "Boolean Models and Methods in Mathematics, Computer Science, and Engineering'' (eds. Y. Crama and P. Hammer), Cambridge University Press, (2010), 257-397. [9] C. Carlet, H. Dobbertin and G. Leander, Normal extensions of bent functions, IEEE Trans. Inform. Theory, 50 (2004), 2880-2885. doi: 10.1109/TIT.2004.836681. [10] J. Dillon, "Elementary Hadamard Difference Sets,'' Ph.D thesis, Univ. Maryland, College Park, 1974. [11] H. Dobbertin, Construction of bent functions and balanced Boolean functions with high nonlinearity, in "Fast Software Encryption,'' Springer, (1995), 61-74. doi: 10.1007/3-540-60590-8_5. [12] H. Dobbertin and G. Leander, Bent functions embedded into the recursive framework of $\mathbb Z$-bent functions, Des. Codes Cryptogr., 49 (2008), 3-22. doi: 10.1007/s10623-008-9189-3. [13] P. Guillo, Completed GPS covers all bent functions, J. Combin. Theory Ser. A, 93 (2001), 242-260. doi: 10.1006/jcta.2000.3076. [14] X.-D. Hou, New constructions of bent functions, J. Combin. Inform. System Sci., 25 (2000), 173-189. [15] P. Langevin, G. Leander, P. Rabizzoni, P. Veron and J.-P. Zanotti, Classification of Boolean quartics forms in eight variables, availabel at http://langevin.univ-tln.fr/project/quartics/quartics.html [16] G. Leander, Monomial bent functions, IEEE Trans. Inform. Theory, 52 (2006), 738-743. doi: 10.1109/TIT.2005.862121. [17] G. Leander and G. McGuire, Construction of bent functions from near-bent functions, J. Combin. Theory Ser. A, 116 (2009), 960-970. doi: 10.1016/j.jcta.2008.12.004. [18] Q. Liu, Y. Zhang, C. Cheng and W. Lü, Construction and counting orthomorphism based on transversal, in "2008 International Conference on Computational Intelligence and Security,'' IEEE Computer Society, (2008), 369-373. [19] F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland, Amsterdam, 1977. [20] R. I. McFarland, A family of difference sets in non-cyclic groups, J. Comb. Theory Ser. A, 15 (1973), 1-10. doi: 10.1016/0097-3165(73)90031-9. [21] Q. Meng, L. Chen and F. Fu, On homogeneous rotation symmetric bent functions, Discrete Appl. Math., 158 (2010), 1111-1117. doi: 10.1016/j.dam.2010.02.009. [22] J. D. Olsen, R. A. Scholtz and L. R. Welch, Bent-function sequence, IEEE Trans. Inform. Theory, 28 (1982), 858-864. doi: 10.1109/TIT.1982.1056589. [23] O. S. Rothaus, On "bent'' functions, J. Combin. Theory Ser. A, 20 (1976), 300-305. doi: 10.1016/0097-3165(76)90024-8. [24] J. Wolfmann, Bent functions and coding theory, in "Difference Sets, Sequences and their Correlation Properties'' (eds. A. Pott, P.V. Kumar, T. Helleseth and D. Jungnickel), Amsterdam, Kluwer, (1999), 393-417. [25] H. Zhen, H. Zhang, T. Cui and X. Du, A new method for construction of orthomorphic permutations (in Chinese), J. Electr. Inform. Tech., 31 (2009), 1438-1441.
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