TY - JOUR

T1 - Decomposing self-dual bent functions

AU - Kutsenko, Aleksandr

N1 - The work was carried out within the framework of the state contract of the Sobolev Institute of Mathematics (project no. FWNF-2022-0018).

PY - 2024/1

Y1 - 2024/1

N2 - Bent functions are Boolean functions in even number of variables that have maximal nonlinearity. They have flat Walsh–Hadamard spectrum and are of interest for their applications in algebra, coding theory and cryptography. A bent function is called self-dual if it coincides with its dual bent function. In current work we study the decomposition of the form (f0,f1,…,f2k-1) of the vector of values of a self-dual bent function, formed by the concatenation of 2 k Boolean functions fj in n- k variables. We treat the cases k= 1 , 2 . Based on a spectral characterization, we introduce a notion of self-dual near-bent function in odd number of variables and prove that there exists a one-to-one correspondence between the notions of self-duality for even and odd number of variables. As a result the characterization for the decomposition (f, f1) is obtained. For the decomposition f= (f, f1, f2, f3) we prove that if sign vectors of subfunctions fj are linearly dependent, then all these subfunctions are bent. We prove that for n⩾ 6 the converse does not hold, that is the obtained condition is sufficient only. These results are also generalized for the case of an arbitrary bent function. Three new iterative constructions of self-dual bent functions are proposed. One of them allows to build a class of self-dual bent functions which cannot be decomposed into the concatenation of four bent functions. Based on the constructions a new iterative lower bound on the cardinality of the set of self-dual bent functions is obtained.

AB - Bent functions are Boolean functions in even number of variables that have maximal nonlinearity. They have flat Walsh–Hadamard spectrum and are of interest for their applications in algebra, coding theory and cryptography. A bent function is called self-dual if it coincides with its dual bent function. In current work we study the decomposition of the form (f0,f1,…,f2k-1) of the vector of values of a self-dual bent function, formed by the concatenation of 2 k Boolean functions fj in n- k variables. We treat the cases k= 1 , 2 . Based on a spectral characterization, we introduce a notion of self-dual near-bent function in odd number of variables and prove that there exists a one-to-one correspondence between the notions of self-duality for even and odd number of variables. As a result the characterization for the decomposition (f, f1) is obtained. For the decomposition f= (f, f1, f2, f3) we prove that if sign vectors of subfunctions fj are linearly dependent, then all these subfunctions are bent. We prove that for n⩾ 6 the converse does not hold, that is the obtained condition is sufficient only. These results are also generalized for the case of an arbitrary bent function. Three new iterative constructions of self-dual bent functions are proposed. One of them allows to build a class of self-dual bent functions which cannot be decomposed into the concatenation of four bent functions. Based on the constructions a new iterative lower bound on the cardinality of the set of self-dual bent functions is obtained.

KW - Bent-decomposition

KW - Gram matrix

KW - Near-bent

KW - Rayleigh quotient

KW - Self-dual bent

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85172987829&origin=inward&txGid=e9a97763c13f3a853f07cfe5ca446e18

UR - https://www.mendeley.com/catalogue/2ccd75b0-940e-30d8-8e58-39edaa7a3c82/

U2 - 10.1007/s10623-023-01298-2

DO - 10.1007/s10623-023-01298-2

M3 - Article

VL - 92

SP - 113

EP - 144

JO - Designs, Codes, and Cryptography

JF - Designs, Codes, and Cryptography

SN - 0925-1022

IS - 1

ER -