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May  2021, 15(2): 329-346. doi: 10.3934/amc.2020069

A note on generalization of bent boolean functions

Bimal Mandal 1, and  Aditi Kar Gangopadhyay 2,,

1. 

CARAMBA, INRIA Nancy-Grand Est., 54600, France
2. 

Department of Mathematics, Indian Institute of Technology Roorkee, 247667, India

* Corresponding author: Aditi Kar Gangopadhyay

Received  April 2019 Revised  February 2020 Published  April 2020

Suppose that $ \mu_p $ is a probability measure defined on the input space of Boolean functions. We consider a generalization of Walsh–Hadamard transform on Boolean functions to $ \mu_p $-Walsh–Hadamard transforms. In this paper, first, we derive the properties of $ \mu_p $-Walsh–Hadamard transformation for some classes of Boolean functions and specify a class of nonsingular affine transformations that preserve the $ \mu_p $-bent property. We further derive the results on $ \mu_p $-Walsh–Hadamard transform of concatenation of Boolean functions and provide some secondary constructions of $ \mu_p $-bent functions. Finally, we discuss the $ \mu_p $-bentness for Maiorana–McFarland class of bent functions.

Keywords: Boolean function, $ \mu_p $-Walsh–Hadamard transform, $ \mu_p $-bent function.
Mathematics Subject Classification: Primary: 06E30, 94C10; Secondary: 05A05.
Citation: Bimal Mandal, Aditi Kar Gangopadhyay. A note on generalization of bent boolean functions. Advances in Mathematics of Communications, 2021, 15 (2) : 329-346. doi: 10.3934/amc.2020069
References:
[1]

A. Canteaut, S. Carpov, C. Fontaine, T. Lepoint, M. Naya-Plasencia, P. Paillier and R. Sirdey, Stream ciphers: A practical solution for efficient homomorphic-ciphertext compression, Journal of Cryptology, 31 (2018), 885-916.  doi: 10.1007/s00145-017-9273-9.  Google Scholar

[2]

C. Carlet, P. Méaux and Y. Rotella, Boolean functions with restricted input and their robustness, application to the FLIP cipher, IACR Transactions on Symmetric Cryptology, 3 (2017), 192-227.   Google Scholar

[3] T. W. Cusick and P. Stǎnicǎ, Cryptographic Boolean Functions and Applications, 2nd Edition, Elsevier-Academic Press, London, 2017.   Google Scholar
[4]

J. F. Dillon, Elementary Hadamard difference sets, Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing, Congressus Numerantium, Utilitas Math., Winnipeg, Man., (1975), 237-249.   Google Scholar

[5]

H. Dobbertin, Construction of bent functions and balanced Boolean functions with high nonlinearity, Fast Software Encryption (FSE 1994), LNCS, Springer-Verlag, (1995), 61-74.  doi: 10.1007/3-540-60590-8_5.  Google Scholar

[6]

P. Erdös and A. Rényi, On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl, 5 (1960), 17-61.   Google Scholar

[7]

S. Gangopadhyay, A. K. Gangopadhyay, S. Pollatos and P. Stǎnicǎ, Cryptographic Boolean functions with biased inputs, Cryptography and Communications, 9 (2017), 301-314.  doi: 10.1007/s12095-015-0174-1.  Google Scholar

[8]

S. Gangopadhyay, G. Paul, N. Sinha and P. Stǎnicǎ, Generalized nonlinearity of $S$-boxes, Advances in Mathematics of Communications, 12 (2018), 115-122.  doi: 10.3934/amc.2018007.  Google Scholar

[9]

H. Hatami, A remark on Bourgain's distributional inequality on the Fourier spectrum of Boolean functions, Online Journal of Analytic Combinatorics, (2006), Art. 3, 6 pp.  Google Scholar

[10]

M. Heidari, S. S. Pradhan and R. Venkataramanan, Boolean functions with biased inputs: Approximation and noise sensitivity, IEEE International Symposium on Information Theory (ISIT), (2019), 1192-1196.  doi: 10.1109/ISIT.2019.8849233.  Google Scholar

[11]

S. Kavut, S. Maitra and D. Tang, Construction and search of balanced Boolean functions on even number of variables towards excellent autocorrelation profile, Designs, Codes and Cryptography, 87 (2019), 261-276.  doi: 10.1007/s10623-018-0522-1.  Google Scholar

[12]

M. Khairallah, A. Chattopadhyay, B. Mandal and S. Maitra, On hardware implementation of Tang-Maitra Boolean functions, Arithmetic of Finite Fields, Lecture Notes in Comput. Sci., Springer, Cham, 11321 (2018), 111-127.  doi: 10.1007/978-3-030-05153-2_6.  Google Scholar

[13]

Y. Lu and Y. Desmedt, Bias analysis of a certain problem with applications to $E_0$ and Shannon ciper, Information Security and Cryptology—ICISC 2010, Lecture Notes in Comput. Sci., Springer, Heidelberg, 6829 (2011), 16-28.  doi: 10.1007/978-3-642-24209-0_2.  Google Scholar

[14]

B. Mandal, S. Maitra and P. Stǎnicǎ, On the existence and non-existence of some classes of bent-negabent functions, submitted. Google Scholar

[15]

S. Maitra, B. Mandal, T. Martinsen, D. Roy and P. Stǎnicǎ, Tools in analyzing linear approximation for Boolean functions related to FLIP, Progress in Cryptology—INDOCRYPT 2018, Lecture Notes in Comput. Sci., Springer, Cham, 11356 (2018), 282-303.  doi: 10.1007/978-3-030-05378-9_16.  Google Scholar

[16]

S. Maitra, B. Mandal, T. Martinsen, D. Roy and P. Stǎnicǎ, Analysis on Boolean function in a restricted (biased) domain, IEEE Transactions on Information Theory, 66 (2020), 1219-1231.  doi: 10.1109/TIT.2019.2932739.  Google Scholar

[17]

R. L. McFarland, A family of noncyclic difference sets, Journal of Combinatorial Theory, Series A, 15 (1973), 1-10.   Google Scholar

[18]

S. Mesnager, Z. C. Zhou and C. S. Ding, On the nonlinearity of Boolean functions with restricted input, Cryptography and Communications, 11 (2019), 63-76.  doi: 10.1007/s12095-018-0293-6.  Google Scholar

[19]

S. Mesnager, Bent Functions, Fundamentals and Results, Springer, [Cham], 2016. doi: 10.1007/978-3-319-32595-8.  Google Scholar

[20]

A. Montanaro and T. J. Osborne, Quantum Boolean functions, Chicago J. Theor. Comput. Sci., (2010), Art 1, 45 pp. doi: 10.4086/cjtcs.2010.001.  Google Scholar

[21] R. O'Donnell, Analysis of Boolean Functions, Cambridge University Press, New York, 2014.  doi: 10.1017/CBO9781139814782.  Google Scholar
[22]

M. G. Parker, Generalised S-Box Nonlinearity, NESSIE Public Document, 11.02.03: NES/DOC/UIB/WP5/020/A. Google Scholar

[23]

M. G. Parker, The constabent properties of Goley-Devis-Jedwab sequences, Int. Symp. Information Theory, Sorrento, Italy, (2000). Google Scholar

[24]

C. Riera and M. G. Parker, Generalized bent criteria for Boolean functions. Ⅰ, IEEE Transactions on Information Theory, 52 (2006), 4142-4159.  doi: 10.1109/TIT.2006.880069.  Google Scholar

[25]

O. S. Rothaus, On "bent" functions, Journal of Combinatorial Theory, Series A, 20 (1976), 300-305.  doi: 10.1016/0097-3165(76)90024-8.  Google Scholar

[26]

K.-U. Schmidt, M. G. Parker and A. Pott, Negabent functions in Maiorana-McFarland class, equences and Their Applications—SETA 2008, Lecture Notes in Comput. Sci., Springer, Berlin, 5203 (2008), 390-402.  doi: 10.1007/978-3-540-85912-3_34.  Google Scholar

[27]

P. Stǎnicǎ, S. Gangopadhyay, A. Chaturvedi, A. K. Gangopadhyay and S. Maitra, Investigations on bent and negabent function via nega-Hadamard transform, IEEE Transactions on Information Theory, 58 (2012), 4064-4072.  doi: 10.1109/TIT.2012.2186785.  Google Scholar

[28]

D. Tang and S. Maitra, Constructions of $n$-variable ($n\equiv 2 \bmod 4$) balanced Boolean functions with maximum absolute value in autocorrelation spectra $<2^{\frac{n}{2}}$, IEEE Transactions on Information Theory, 64 (2018), 393-402.  doi: 10.1109/TIT.2017.2769092.  Google Scholar

[29]

D. Tang, S. Kavut, B. Mandal and S. Maitra, Modifying Maiorana-McFarland type bent functions for good cryptographic properties and efficient implementation, SIAM Journal on Discrete Mathematics, 33 (2019), 238-256.  doi: 10.1137/18M1202864.  Google Scholar

show all references

References:
[1]

A. Canteaut, S. Carpov, C. Fontaine, T. Lepoint, M. Naya-Plasencia, P. Paillier and R. Sirdey, Stream ciphers: A practical solution for efficient homomorphic-ciphertext compression, Journal of Cryptology, 31 (2018), 885-916.  doi: 10.1007/s00145-017-9273-9.  Google Scholar

[2]

C. Carlet, P. Méaux and Y. Rotella, Boolean functions with restricted input and their robustness, application to the FLIP cipher, IACR Transactions on Symmetric Cryptology, 3 (2017), 192-227.   Google Scholar

[3] T. W. Cusick and P. Stǎnicǎ, Cryptographic Boolean Functions and Applications, 2nd Edition, Elsevier-Academic Press, London, 2017.   Google Scholar
[4]

J. F. Dillon, Elementary Hadamard difference sets, Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing, Congressus Numerantium, Utilitas Math., Winnipeg, Man., (1975), 237-249.   Google Scholar

[5]

H. Dobbertin, Construction of bent functions and balanced Boolean functions with high nonlinearity, Fast Software Encryption (FSE 1994), LNCS, Springer-Verlag, (1995), 61-74.  doi: 10.1007/3-540-60590-8_5.  Google Scholar

[6]

P. Erdös and A. Rényi, On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl, 5 (1960), 17-61.   Google Scholar

[7]

S. Gangopadhyay, A. K. Gangopadhyay, S. Pollatos and P. Stǎnicǎ, Cryptographic Boolean functions with biased inputs, Cryptography and Communications, 9 (2017), 301-314.  doi: 10.1007/s12095-015-0174-1.  Google Scholar

[8]

S. Gangopadhyay, G. Paul, N. Sinha and P. Stǎnicǎ, Generalized nonlinearity of $S$-boxes, Advances in Mathematics of Communications, 12 (2018), 115-122.  doi: 10.3934/amc.2018007.  Google Scholar

[9]

H. Hatami, A remark on Bourgain's distributional inequality on the Fourier spectrum of Boolean functions, Online Journal of Analytic Combinatorics, (2006), Art. 3, 6 pp.  Google Scholar

[10]

M. Heidari, S. S. Pradhan and R. Venkataramanan, Boolean functions with biased inputs: Approximation and noise sensitivity, IEEE International Symposium on Information Theory (ISIT), (2019), 1192-1196.  doi: 10.1109/ISIT.2019.8849233.  Google Scholar

[11]

S. Kavut, S. Maitra and D. Tang, Construction and search of balanced Boolean functions on even number of variables towards excellent autocorrelation profile, Designs, Codes and Cryptography, 87 (2019), 261-276.  doi: 10.1007/s10623-018-0522-1.  Google Scholar

[12]

M. Khairallah, A. Chattopadhyay, B. Mandal and S. Maitra, On hardware implementation of Tang-Maitra Boolean functions, Arithmetic of Finite Fields, Lecture Notes in Comput. Sci., Springer, Cham, 11321 (2018), 111-127.  doi: 10.1007/978-3-030-05153-2_6.  Google Scholar

[13]

Y. Lu and Y. Desmedt, Bias analysis of a certain problem with applications to $E_0$ and Shannon ciper, Information Security and Cryptology—ICISC 2010, Lecture Notes in Comput. Sci., Springer, Heidelberg, 6829 (2011), 16-28.  doi: 10.1007/978-3-642-24209-0_2.  Google Scholar

[14]

B. Mandal, S. Maitra and P. Stǎnicǎ, On the existence and non-existence of some classes of bent-negabent functions, submitted. Google Scholar

[15]

S. Maitra, B. Mandal, T. Martinsen, D. Roy and P. Stǎnicǎ, Tools in analyzing linear approximation for Boolean functions related to FLIP, Progress in Cryptology—INDOCRYPT 2018, Lecture Notes in Comput. Sci., Springer, Cham, 11356 (2018), 282-303.  doi: 10.1007/978-3-030-05378-9_16.  Google Scholar

[16]

S. Maitra, B. Mandal, T. Martinsen, D. Roy and P. Stǎnicǎ, Analysis on Boolean function in a restricted (biased) domain, IEEE Transactions on Information Theory, 66 (2020), 1219-1231.  doi: 10.1109/TIT.2019.2932739.  Google Scholar

[17]

R. L. McFarland, A family of noncyclic difference sets, Journal of Combinatorial Theory, Series A, 15 (1973), 1-10.   Google Scholar

[18]

S. Mesnager, Z. C. Zhou and C. S. Ding, On the nonlinearity of Boolean functions with restricted input, Cryptography and Communications, 11 (2019), 63-76.  doi: 10.1007/s12095-018-0293-6.  Google Scholar

[19]

S. Mesnager, Bent Functions, Fundamentals and Results, Springer, [Cham], 2016. doi: 10.1007/978-3-319-32595-8.  Google Scholar

[20]

A. Montanaro and T. J. Osborne, Quantum Boolean functions, Chicago J. Theor. Comput. Sci., (2010), Art 1, 45 pp. doi: 10.4086/cjtcs.2010.001.  Google Scholar

[21] R. O'Donnell, Analysis of Boolean Functions, Cambridge University Press, New York, 2014.  doi: 10.1017/CBO9781139814782.  Google Scholar
[22]

M. G. Parker, Generalised S-Box Nonlinearity, NESSIE Public Document, 11.02.03: NES/DOC/UIB/WP5/020/A. Google Scholar

[23]

M. G. Parker, The constabent properties of Goley-Devis-Jedwab sequences, Int. Symp. Information Theory, Sorrento, Italy, (2000). Google Scholar

[24]

C. Riera and M. G. Parker, Generalized bent criteria for Boolean functions. Ⅰ, IEEE Transactions on Information Theory, 52 (2006), 4142-4159.  doi: 10.1109/TIT.2006.880069.  Google Scholar

[25]

O. S. Rothaus, On "bent" functions, Journal of Combinatorial Theory, Series A, 20 (1976), 300-305.  doi: 10.1016/0097-3165(76)90024-8.  Google Scholar

[26]

K.-U. Schmidt, M. G. Parker and A. Pott, Negabent functions in Maiorana-McFarland class, equences and Their Applications—SETA 2008, Lecture Notes in Comput. Sci., Springer, Berlin, 5203 (2008), 390-402.  doi: 10.1007/978-3-540-85912-3_34.  Google Scholar

[27]

P. Stǎnicǎ, S. Gangopadhyay, A. Chaturvedi, A. K. Gangopadhyay and S. Maitra, Investigations on bent and negabent function via nega-Hadamard transform, IEEE Transactions on Information Theory, 58 (2012), 4064-4072.  doi: 10.1109/TIT.2012.2186785.  Google Scholar

[28]

D. Tang and S. Maitra, Constructions of $n$-variable ($n\equiv 2 \bmod 4$) balanced Boolean functions with maximum absolute value in autocorrelation spectra $<2^{\frac{n}{2}}$, IEEE Transactions on Information Theory, 64 (2018), 393-402.  doi: 10.1109/TIT.2017.2769092.  Google Scholar

[29]

D. Tang, S. Kavut, B. Mandal and S. Maitra, Modifying Maiorana-McFarland type bent functions for good cryptographic properties and efficient implementation, SIAM Journal on Discrete Mathematics, 33 (2019), 238-256.  doi: 10.1137/18M1202864.  Google Scholar

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