# American Institute of Mathematical Sciences

December  2020, 10(4): 571-578. doi: 10.3934/naco.2020052

## The research on the properties of Fourier matrix and bent function

 1 School of Mathematics and Statistics, Changshu Institute of Technology, Suzhou, 215500, China 2 Shanghai Seed Power Enterprise Management Cosulting Co., LTD, Minhang District, Shanghai, 201200, China 3 Asset Management Department, Jiangsu Zijin Rural Commercial Bank Co., LTD, Nanjing, 210023, China

* Corresponding author: Li Zhang

Received  April 2020 Revised  September 2020 Published  September 2020

Fund Project: The first author is supported by NSF grant 10231060

This paper first gives out basic background and some definitions and propositions for Fourier matrix and bent function. Secondly we construct an standard orthogonal basis by the eigenvectors of the corresponding Fourier matrix. At last the diagonalization work of Fourier matrix is completed and some theorems about them are proved.

Citation: Li Zhang, Xiaofeng Zhou, Min Chen. The research on the properties of Fourier matrix and bent function. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 571-578. doi: 10.3934/naco.2020052
##### References:
 [1] C. Carlet, Boolen Function For Cryptography And Error Correcting Codes, Cambridge University, 2007. [2] C. Hughes, Partially-bent functions, Proceedings of The 12th Annual International Cryptography Conference on Advances Incryptology, 74 (1992), 280-291. [3] J. Ding, H. Zhen, Q. Wen and Y. Yang, Construction and enumeration of multiple-output orthogonal Boolean functions, Journal of Beijing University of Posts and Telecommunications, 4 (2005), 9-11. [4] V. Kumear, A. Scholtz and R. Welch, Generalized bent functions and their properties, Journal of Combinatorial Theory, 4 (1985), 90-107.  doi: 10.1016/0097-3165(85)90049-4. [5] R.Lial and H. Niederreiter, Finite Fields, Addison-Wesley Publishing Company, New York, 1984. [6] O. Rothaus, On bent fuctions, Journal of Combination Theory, 20 (1976), 300-305.  doi: 10.1016/0097-3165(76)90024-8. [7] X. Hou and P. Langevin, Results on bent fuctions, Journal of Combination Theory, 80 (1997), 232-246.  doi: 10.1006/jcta.1997.2804. [8] P. Langevin, On generalized bent fuctions,, CISM Course and Lectures, 339 (1992), 147-159. [9] P. Langevin and X. Hou, Computing partial spread functions in eight variables, IEEE Trans. Inf. Theory, 57 (2011), 2263-2269.  doi: 10.1109/TIT.2011.2112230. [10] P. Langevin, P. Rabizzoni, P. Veron and J. Zanotti, On the number of bent functions with 8 variables, Proceedings of the Conference BFCA 2006, Publications des universites de Rousen et du Havre(2006), 125–136. [11] Y. Yuan, D. Jin, Y. Zhao and S. Zhang, Relationship between generalized partially bent functions and generalized bent functions over finite fields, Journal of Information Engineering University, 9 (2009), 313-317. [12] Y. Zhao and S. Li, Generalized partially-bent functions and characteristics of the auto-correlation functions and spectrum, Journal of Engineering Mathematics, 4 (1999), 91-96. [13] W. Sun and Y. Yuan, Optimization Theory and Methods: Nonlinear Programming, Springer, New York, 2006. [14] Y. Yuan and W. Sun, Optimization Theory and Methods, Science Press, Beijing, 1997.

show all references

##### References:
 [1] C. Carlet, Boolen Function For Cryptography And Error Correcting Codes, Cambridge University, 2007. [2] C. Hughes, Partially-bent functions, Proceedings of The 12th Annual International Cryptography Conference on Advances Incryptology, 74 (1992), 280-291. [3] J. Ding, H. Zhen, Q. Wen and Y. Yang, Construction and enumeration of multiple-output orthogonal Boolean functions, Journal of Beijing University of Posts and Telecommunications, 4 (2005), 9-11. [4] V. Kumear, A. Scholtz and R. Welch, Generalized bent functions and their properties, Journal of Combinatorial Theory, 4 (1985), 90-107.  doi: 10.1016/0097-3165(85)90049-4. [5] R.Lial and H. Niederreiter, Finite Fields, Addison-Wesley Publishing Company, New York, 1984. [6] O. Rothaus, On bent fuctions, Journal of Combination Theory, 20 (1976), 300-305.  doi: 10.1016/0097-3165(76)90024-8. [7] X. Hou and P. Langevin, Results on bent fuctions, Journal of Combination Theory, 80 (1997), 232-246.  doi: 10.1006/jcta.1997.2804. [8] P. Langevin, On generalized bent fuctions,, CISM Course and Lectures, 339 (1992), 147-159. [9] P. Langevin and X. Hou, Computing partial spread functions in eight variables, IEEE Trans. Inf. Theory, 57 (2011), 2263-2269.  doi: 10.1109/TIT.2011.2112230. [10] P. Langevin, P. Rabizzoni, P. Veron and J. Zanotti, On the number of bent functions with 8 variables, Proceedings of the Conference BFCA 2006, Publications des universites de Rousen et du Havre(2006), 125–136. [11] Y. Yuan, D. Jin, Y. Zhao and S. Zhang, Relationship between generalized partially bent functions and generalized bent functions over finite fields, Journal of Information Engineering University, 9 (2009), 313-317. [12] Y. Zhao and S. Li, Generalized partially-bent functions and characteristics of the auto-correlation functions and spectrum, Journal of Engineering Mathematics, 4 (1999), 91-96. [13] W. Sun and Y. Yuan, Optimization Theory and Methods: Nonlinear Programming, Springer, New York, 2006. [14] Y. Yuan and W. Sun, Optimization Theory and Methods, Science Press, Beijing, 1997.
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