November  2019, 13(4): 613-627. doi: 10.3934/amc.2019038

Landscape Boolean functions

1. 

Department of Computing, Mathematics, and Physics, Western Norway University of Applied Sciences, 5020 Bergen, Norway

2. 

Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA 93943, USA

* Corresponding author: Pantelimon Stănică

Received  October 2018 Published  June 2019

In this paper we define a class of generalized Boolean functions defined on $ {\mathbb F}_2^n $ with values in $ {\mathbb Z}_q $ (we consider $ q = 2^k $, $ k\geq 1 $, here), which we call landscape functions (whose class contains generalized bent, semibent, and plateaued) and find their complete characterization in terms of their Boolean components. In particular, we show that the previously published characterizations of generalized plateaued Boolean functions (which includes generalized bent and semibent) are in fact particular cases of this more general setting. Furthermore, we provide an inductive construction of landscape functions, having any number of nonzero Walsh-Hadamard coefficients. We also completely characterize generalized plateaued functions in terms of the second derivatives and fourth moments.

Citation: Constanza Riera, Pantelimon Stănică. Landscape Boolean functions. Advances in Mathematics of Communications, 2019, 13 (4) : 613-627. doi: 10.3934/amc.2019038
References:
[1]

C. Carlet, On the secondary constructions of resilient and bent functions, Coding, Cryptography and Combinatorics, Birkhäuser, Basel, 23 (2004), 3-28.

[2]

C. Carlet, Boolean functions for cryptography and error correcting codes, In: Y. Crama, P. Hammer (eds.), Boolean Methods and Models, Cambridge Univ. Press, Cambridge, (2010), 257-397.

[3]

C. Carlet and S. Mesnager, On the supports of the Walsh transforms of Boolean functions, Boolean Functions: Cryptography and Applications, BFCA'04, (2005), 65-82.

[4] T. W. Cusick and P. Stănică, Cryptographic Boolean Functions and Applications, (Ed. 2), Academic Press, San Diego, 2017. 
[5]

S. HodžićW. Meidl and E. Pasalic, Full characterization of generalized bent functions as (semi)-bent spaces, their dual and the Gray image, IEEE Trans. Inform. Theory, 64 (2018), 5432-5440.  doi: 10.1109/TIT.2018.2837883.

[6]

S. Hodžić and E. Pasalic, Generalized bent functions - Some general construction methods and related necessary and sufficient conditions, Cryptogr. Commun., 7 (2015), 469-483.  doi: 10.1007/s12095-015-0126-9.

[7]

P. V. KumarR. A. Scholtz and L. R. Welch, Generalized bent functions and their properties, J. Combin. Theory - Ser. A, 40 (1985), 90-107.  doi: 10.1016/0097-3165(85)90049-4.

[8]

S. Maitra and P. Sarkar, Cryptographically significant Boolean functions with five valued Walsh spectra, Theoretical Comp. Sci., 276 (2002), 133-146.  doi: 10.1016/S0304-3975(01)00196-7.

[9]

T. MartinsenW. MeidlS. Mesnager and P. Stănică, Decomposing generalized bent and hyperbent functions, IEEE Trans. Inform. Theory, 63 (2017), 7804-7812.  doi: 10.1109/TIT.2017.2754498.

[10]

T. Martinsen, W. Meidl and P. Stănică, Generalized bent functions and their Gray images, Proc. of WAIFI 2016: Arithmetic of Finite Fields, LNCS, 10064 (2017), 160-173.

[11]

T. MartinsenW. Meidl and P. Stănică, Partial Spread and Vectorial Generalized Bent Functions, Des. Codes Cryptogr., 85 (2017), 1-13.  doi: 10.1007/s10623-016-0283-7.

[12]

S. Mesnager, On semi-bent functions and related plateaued functions over the Galois field $F_{2^n}$, Open Problems in Mathematics and Computational Science, Springer, (2014), 243-273.

[13]

S. MesnagerC. Tang and Y. Qi, Generalized plateaued functions and admissible (plateaued) functions, IEEE Trans. Inform. Theory, 63 (2017), 6139-6148.  doi: 10.1109/TIT.2017.2715804.

[14]

P. Mihăilescu, Primary Cyclotomic Units and a Proof of Catalan's Conjecture, J. Reine Angew. Math., 572 (2004), 167-195.  doi: 10.1515/crll.2004.048.

[15]

T. Ono, An Introduction to Algebraic Number Theory, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4613-0573-6.

[16]

K. U. Schmidt, Quaternary constant-amplitude codes for multicode CDMA, IEEE Trans. Inform. Theory, 55 (2009), 1824-1832.  doi: 10.1109/TIT.2009.2013041.

[17]

B. K. Singh, Generalized semibent and partially bent Boolean functions, Math. Sci. Lett., 3 (2014), 21-29. 

[18]

P. Solé and N. Tokareva, Connections between quaternary and binary bent functions, Prikl. Diskr. Mat., 1 (2009), 16-18, (see also, http://eprint.iacr.org/2009/544.pdf).

[19]

P. StănicăT. MartinsenS. Gangopadhyay and B. K. Singh, Bent and generalized bent Boolean functions, Des. Codes Cryptogr., 69 (2013), 77-94.  doi: 10.1007/s10623-012-9622-5.

[20]

C. TangC. XiangY. Qi and K. Feng, Complete characterization of generalized bent and $2^k$-bent Boolean functions, IEEE Trans. Inform. Theory, 63 (2017), 4668-4674.  doi: 10.1109/TIT.2017.2686987.

[21]

N. Tokareva, Generalizations of bent functions: A survey of publications, J. Appl. Ind. Math., 5 (2011), 110-129.  doi: 10.1134/S1990478911010133.

[22]

E. UyanÇ. Çalik and A. Doganaksoy, Counting Boolean functions with specified values in their Walsh spectrum, J. Comp. Appl. Math., 259 (2014), 522-528.  doi: 10.1016/j.cam.2013.06.035.

[23]

L. C. Washington, Introduction to Cyclotomic Fields (2nd ed.), Graduate Texts in Mathematics 83, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-1934-7.

[24]

Y. L. Zheng and X. M. Zhang, On plateaued functions, IEEE Trans. Inform. Theory, 47 (2001), 1215-1223.  doi: 10.1109/18.915690.

show all references

References:
[1]

C. Carlet, On the secondary constructions of resilient and bent functions, Coding, Cryptography and Combinatorics, Birkhäuser, Basel, 23 (2004), 3-28.

[2]

C. Carlet, Boolean functions for cryptography and error correcting codes, In: Y. Crama, P. Hammer (eds.), Boolean Methods and Models, Cambridge Univ. Press, Cambridge, (2010), 257-397.

[3]

C. Carlet and S. Mesnager, On the supports of the Walsh transforms of Boolean functions, Boolean Functions: Cryptography and Applications, BFCA'04, (2005), 65-82.

[4] T. W. Cusick and P. Stănică, Cryptographic Boolean Functions and Applications, (Ed. 2), Academic Press, San Diego, 2017. 
[5]

S. HodžićW. Meidl and E. Pasalic, Full characterization of generalized bent functions as (semi)-bent spaces, their dual and the Gray image, IEEE Trans. Inform. Theory, 64 (2018), 5432-5440.  doi: 10.1109/TIT.2018.2837883.

[6]

S. Hodžić and E. Pasalic, Generalized bent functions - Some general construction methods and related necessary and sufficient conditions, Cryptogr. Commun., 7 (2015), 469-483.  doi: 10.1007/s12095-015-0126-9.

[7]

P. V. KumarR. A. Scholtz and L. R. Welch, Generalized bent functions and their properties, J. Combin. Theory - Ser. A, 40 (1985), 90-107.  doi: 10.1016/0097-3165(85)90049-4.

[8]

S. Maitra and P. Sarkar, Cryptographically significant Boolean functions with five valued Walsh spectra, Theoretical Comp. Sci., 276 (2002), 133-146.  doi: 10.1016/S0304-3975(01)00196-7.

[9]

T. MartinsenW. MeidlS. Mesnager and P. Stănică, Decomposing generalized bent and hyperbent functions, IEEE Trans. Inform. Theory, 63 (2017), 7804-7812.  doi: 10.1109/TIT.2017.2754498.

[10]

T. Martinsen, W. Meidl and P. Stănică, Generalized bent functions and their Gray images, Proc. of WAIFI 2016: Arithmetic of Finite Fields, LNCS, 10064 (2017), 160-173.

[11]

T. MartinsenW. Meidl and P. Stănică, Partial Spread and Vectorial Generalized Bent Functions, Des. Codes Cryptogr., 85 (2017), 1-13.  doi: 10.1007/s10623-016-0283-7.

[12]

S. Mesnager, On semi-bent functions and related plateaued functions over the Galois field $F_{2^n}$, Open Problems in Mathematics and Computational Science, Springer, (2014), 243-273.

[13]

S. MesnagerC. Tang and Y. Qi, Generalized plateaued functions and admissible (plateaued) functions, IEEE Trans. Inform. Theory, 63 (2017), 6139-6148.  doi: 10.1109/TIT.2017.2715804.

[14]

P. Mihăilescu, Primary Cyclotomic Units and a Proof of Catalan's Conjecture, J. Reine Angew. Math., 572 (2004), 167-195.  doi: 10.1515/crll.2004.048.

[15]

T. Ono, An Introduction to Algebraic Number Theory, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4613-0573-6.

[16]

K. U. Schmidt, Quaternary constant-amplitude codes for multicode CDMA, IEEE Trans. Inform. Theory, 55 (2009), 1824-1832.  doi: 10.1109/TIT.2009.2013041.

[17]

B. K. Singh, Generalized semibent and partially bent Boolean functions, Math. Sci. Lett., 3 (2014), 21-29. 

[18]

P. Solé and N. Tokareva, Connections between quaternary and binary bent functions, Prikl. Diskr. Mat., 1 (2009), 16-18, (see also, http://eprint.iacr.org/2009/544.pdf).

[19]

P. StănicăT. MartinsenS. Gangopadhyay and B. K. Singh, Bent and generalized bent Boolean functions, Des. Codes Cryptogr., 69 (2013), 77-94.  doi: 10.1007/s10623-012-9622-5.

[20]

C. TangC. XiangY. Qi and K. Feng, Complete characterization of generalized bent and $2^k$-bent Boolean functions, IEEE Trans. Inform. Theory, 63 (2017), 4668-4674.  doi: 10.1109/TIT.2017.2686987.

[21]

N. Tokareva, Generalizations of bent functions: A survey of publications, J. Appl. Ind. Math., 5 (2011), 110-129.  doi: 10.1134/S1990478911010133.

[22]

E. UyanÇ. Çalik and A. Doganaksoy, Counting Boolean functions with specified values in their Walsh spectrum, J. Comp. Appl. Math., 259 (2014), 522-528.  doi: 10.1016/j.cam.2013.06.035.

[23]

L. C. Washington, Introduction to Cyclotomic Fields (2nd ed.), Graduate Texts in Mathematics 83, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-1934-7.

[24]

Y. L. Zheng and X. M. Zhang, On plateaued functions, IEEE Trans. Inform. Theory, 47 (2001), 1215-1223.  doi: 10.1109/18.915690.

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