Monomial Boolean functions with large high-order nonlinearities

Exhibiting an explicit Boolean function with a large high-order nonlinearity is an important problem in cryptography, coding theory, and computational complexity. We prove lower bounds on the second-order, third-order, and higher order nonlinearities of some monomial Boolean functions. We prove lowe...

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Bibliographic Details
Published in:	Information and computation Vol. 297; p. 105152
Main Authors:	Gao, Jinjie, Kan, Haibin, Li, Yuan, Xu, Jiahua, Wang, Qichun
Format:	Journal Article
Language:	English
Published:	Elsevier Inc 01.03.2024
Subjects:	High-order nonlinearity Linear kernel Lower bound Monomial Boolean function Trace function Lower bound High-order nonlinearity Trace function Monomial Boolean function Linear kernel
ISSN:	0890-5401, 1090-2651
Online Access:	Get full text
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Summary:	Exhibiting an explicit Boolean function with a large high-order nonlinearity is an important problem in cryptography, coding theory, and computational complexity. We prove lower bounds on the second-order, third-order, and higher order nonlinearities of some monomial Boolean functions. We prove lower bounds on the second-order nonlinearities of functions trn(x7) and trn(x2r+3) where n=2r. Among all monomial Boolean functions, our bounds match the best second-order nonlinearity lower bounds by Carlet [IEEE Transactions on Information Theory 54(3), 2008] and Yan and Tang [Discrete Mathematics 343(5), 2020] for odd and even n, respectively. We prove a lower bound on the third-order nonlinearity for functions trn(x15), which is the best third-order nonlinearity lower bound. For any r, we prove that the r-th order nonlinearity of trn(x2r+1−1) is at least 2n−1−2(1−2−r)n+r2r−1−1−O(2n2). For r≪log2⁡n, this is the best lower bound among all explicit functions.
ISSN:	0890-5401 1090-2651
DOI:	10.1016/j.ic.2024.105152