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. Google Scholar

[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.Google Scholar

[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.Google Scholar

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

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

[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. Google Scholar

[17]

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

[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).Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

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

[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. Google Scholar

[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. Google Scholar

[24]

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

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. Google Scholar

[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.Google Scholar

[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.Google Scholar

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

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

[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. Google Scholar

[17]

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

[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).Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

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

[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. Google Scholar

[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. Google Scholar

[24]

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

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