TY - JOUR

T1 - Derivatives of bent functions in connection with the bent sum decomposition problem

AU - Shaporenko, Alexander

N1 - Acknowledgements: The work is supported by Mathematical Center in Akademgorodok under Agreement No. 075-15-2022-281 with the Ministry of Science and Higher Education of the Russian Federation. The author would like to thank Natalia Tokareva for her support and attention to this work. The author is also very grateful to the reviewers for their valuable remarks and comments.

PY - 2023/5

Y1 - 2023/5

N2 - In this paper, we investigate when a balanced function can be a derivative of a bent function. We prove that every nonconstant affine function in an even number of variables n is a derivative of (2n-1-1)∣ Bn-2∣ 2 bent functions, where Bn is the set of all bent functions in n variables. Based on this result, we propose a new iterative lower bound for the number of bent functions. We study the property of balanced functions that depend linearly on at least one of their variables to be derivatives of bent functions. We show the connection between this property and the “bent sum decomposition problem”. We use this connection to prove that if a balanced quadratic Boolean function is a derivative of a Boolean function, then this function is a derivative of a bent function.

AB - In this paper, we investigate when a balanced function can be a derivative of a bent function. We prove that every nonconstant affine function in an even number of variables n is a derivative of (2n-1-1)∣ Bn-2∣ 2 bent functions, where Bn is the set of all bent functions in n variables. Based on this result, we propose a new iterative lower bound for the number of bent functions. We study the property of balanced functions that depend linearly on at least one of their variables to be derivatives of bent functions. We show the connection between this property and the “bent sum decomposition problem”. We use this connection to prove that if a balanced quadratic Boolean function is a derivative of a Boolean function, then this function is a derivative of a bent function.

KW - Bent functions

KW - Bent sum decomposition problem

KW - Boolean functions

KW - Derivatives of a bent function

KW - Lower bound for the number of bent functions

UR - https://www.scopus.com/inward/record.url?eid=2-s2.0-85145178721&partnerID=40&md5=e0bc7bc2fc6db7002cf92fcb5ed55f39

UR - https://www.mendeley.com/catalogue/437613a8-db3d-3997-bb9f-b23f18eebe8c/

U2 - 10.1007/s10623-022-01167-4

DO - 10.1007/s10623-022-01167-4

M3 - Article

VL - 91

SP - 1607

EP - 1625

JO - Designs, Codes, and Cryptography

JF - Designs, Codes, and Cryptography

SN - 0925-1022

IS - 5

ER -