# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021020
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## A new construction of weightwise perfectly balanced Boolean functions

 1 School of Mathematics and Statistics, Henan University, Kaifeng, 475004, China 2 Henan Engineering Research Center for Artificial Intelligence Theory and Algorithms, Henan University, Kaifeng, 475004, China

* Corresponding author: Sihong Su (E-mail: sush@henu.edu.cn)

Received  January 2021 Early access June 2021

Fund Project: The second author is supported by the Key Scientific Research Project of Colleges and Universities in Henan Province (Grant No. 21A413003) and the National Natural Science Foundation of China (Grant No. 61502147)

In this paper, we first introduce a class of quartic Boolean functions. And then, the construction of weightwise perfectly balanced Boolean functions on $2^m$ variables are given by modifying the support of the quartic functions, where $m$ is a positive integer. The algebraic degree, the weightwise nonlinearity, and the algebraic immunity of the newly constructed weightwise perfectly balanced functions are discussed at the end of this paper.

Citation: Rui Zhang, Sihong Su. A new construction of weightwise perfectly balanced Boolean functions. Advances in Mathematics of Communications, doi: 10.3934/amc.2021020
##### References:
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show all references

##### References:
 [1] C. Carlet, P. M$\mathrm{\acute{e}}$aux and Y. Rotella, Boolean functions with restricted input and their robustness: Application to the FLIP cipher, IACR Trans. Symm. Cryptol., 2017 (2017), 192-227.   Google Scholar [2] N. T. Coutois and W. Meier, Algebraic attacks on stream ciphers with linear feedback, Advances in Cryptology-EUROCRYPT 2003, 2656 (2003), 345-359.  doi: 10.1007/3-540-39200-9_21.  Google Scholar [3] S. Duval, V. Lallemand and Y. Rotella, Cryptanalysis of the FLIP family of stream ciphers, in Advances in Cryptology—CRYPTO 2016, Lecture Notes in Computer Science, 9814, Berlin, Springer-Verlag, 2016,457–475. doi: 10.1007/978-3-662-53018-4_17.  Google Scholar [4] Y. Filmus, Friedgut-Kalai-Naor theorem for slices of the Boolean cube, Chic. J. Theoret. Comput. Sci., 14 (2016), 1-17.  doi: 10.4086/cjtcs.2016.014.  Google Scholar [5] Y. Filmus, An orthogonal basis for functions over a slice of the Boolean hypercube, Electron. J. Combin., 23 (2016), 1-23.   Google Scholar [6] J. Li and S. Su, Construction of weightwise perfectly balanced Boolean functions with high weightwise nonlinearity, Discrete Appl. Math., 279 (2020), 218-227.  doi: 10.1016/j.dam.2020.01.020.  Google Scholar [7] P. M$\mathrm{\acute{e}}$aux, A. Journault, F.-X. Standaert and C. Carlet, Towards stream ciphers for efficient FHE with low-noise ciphertexts, in Advances in Cryptology-EUROCRYPT 2016, Lecture Notes in Computer Science, 9665, Berlin, Springer-Verlag, 2016,311–343. doi: 10.1007/978-3-662-49890-3_13.  Google Scholar [8] S. Mesnager and S. Su, On constructions of weightwise perfectly balanced Boolean functions, Cryptogr. Commun., (2021). doi: 10.1007/s12095-021-00481-32021.  Google Scholar [9] S. Mesnager, Z. Zhou and C. Ding, On the nonlinearity of Boolean functions with restricted input, Cryptogr. Commun., 11 (2019), 63-76.  doi: 10.1007/s12095-018-0293-6.  Google Scholar [10] A. Richard, Orthogonal polynomials and special functions, in Regional Conference Series in Applied Mathematics, 21, SIAM, Philadelphia, PA, 1975, 59–60. Google Scholar [11] S. Su, The lower bound of the weightwise nonlinearity profile of a class of weightwise perfectly balanced functions, Discrete Appl. Math., 297 (2021), 60-70.  doi: 10.1016/j.dam.2021.02.033.  Google Scholar [12] D. Tang and J. Liu, A family of weightwise perfectly balanced boolean functions with optimal algebraic immunity, Cryptogr. Commun., 11 (2019), 1185-1197.  doi: 10.1007/s12095-019-00374-6.  Google Scholar [13] E. W. Weisstein, Pascal's formula, From MathWorld-A Wolfram Web Resource. Available from: http://mathworld.wolfram.com/PascalsFormula.html. Google Scholar
The $k$-weight nonlinearity of $g_3$ in (11) for $k = 2,3,4,5,6$
 k $k$-weight nonlinearity of $g_3$ in (11) $\lfloor {8\choose k}/{2}-\sqrt{8\choose k}/{2}\rfloor$ 2 $\mathrm{NL}_{2}(g_{3})=2$ 11 3 $\mathrm{NL}_{3}(g_3)=12$ 24 4 $\mathrm{NL}_{4}(g_3)=19$ 30 5 $\mathrm{NL}_{5}(g_3)=12$ 24 6 $\mathrm{NL}_{6}(g_3)=6$ 11
 k $k$-weight nonlinearity of $g_3$ in (11) $\lfloor {8\choose k}/{2}-\sqrt{8\choose k}/{2}\rfloor$ 2 $\mathrm{NL}_{2}(g_{3})=2$ 11 3 $\mathrm{NL}_{3}(g_3)=12$ 24 4 $\mathrm{NL}_{4}(g_3)=19$ 30 5 $\mathrm{NL}_{5}(g_3)=12$ 24 6 $\mathrm{NL}_{6}(g_3)=6$ 11
The algebraic immunity of $g_{m}$ for $m = 2,3,4$
 $m$ algebraic immunity of $g_{m}$ in (11) optimal algebraic immunity 2 $AI(g_2)=2$ 2 3 $AI(g_3)=3$ 4 4 $AI(g_4)=3$ 8
 $m$ algebraic immunity of $g_{m}$ in (11) optimal algebraic immunity 2 $AI(g_2)=2$ 2 3 $AI(g_3)=3$ 4 4 $AI(g_4)=3$ 8
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