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Cryptographic properties of cyclic binary matrices
A note on generalization of bent boolean functions
1. | CARAMBA, INRIA Nancy-Grand Est., 54600, France |
2. | Department of Mathematics, Indian Institute of Technology Roorkee, 247667, India |
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.
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. |
[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.
![]() |
[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.
|
[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. |
[6] |
P. Erdös and A. Rényi,
On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl, 5 (1960), 17-61.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
S. Mesnager, Bent Functions, Fundamentals and Results, Springer, [Cham], 2016.
doi: 10.1007/978-3-319-32595-8. |
[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. |
[21] |
R. O'Donnell, Analysis of Boolean Functions, Cambridge University Press, New York, 2014.
doi: 10.1017/CBO9781139814782.![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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.
![]() |
[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.
|
[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. |
[6] |
P. Erdös and A. Rényi,
On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl, 5 (1960), 17-61.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
S. Mesnager, Bent Functions, Fundamentals and Results, Springer, [Cham], 2016.
doi: 10.1007/978-3-319-32595-8. |
[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. |
[21] |
R. O'Donnell, Analysis of Boolean Functions, Cambridge University Press, New York, 2014.
doi: 10.1017/CBO9781139814782.![]() ![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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