Nonlinearity of Boolean Functions: An Algorithmic Approach Based on Multivariate Polynomials

We review and compare three algebraic methods to compute the nonlinearity of Boolean functions. Two of them are based on Gröbner basis techniques: the first one is defined over the binary field, while the second one over the rationals. The third method improves the second one by avoiding the Gröbner...

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Bibliographic Details
Published inSymmetry (Basel) Vol. 14; no. 2; p. 213
Main Authors Bellini, Emanuele, Sala, Massimiliano, Simonetti, Ilaria
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.02.2022
Subjects
Algorithms
Asymptotic methods
Asymptotic properties
Boolean
Boolean algebra
Boolean functions
Codes
Coding theory
Complexity
Cryptography
Integer programming
Mathematical analysis
Methods
Nonlinearity
Polynomials
Walsh transforms
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ISSN2073-8994
2073-8994
DOI10.3390/sym14020213

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Summary:We review and compare three algebraic methods to compute the nonlinearity of Boolean functions. Two of them are based on Gröbner basis techniques: the first one is defined over the binary field, while the second one over the rationals. The third method improves the second one by avoiding the Gröbner basis computation. We also estimate the complexity of the algorithms, and, in particular, we show that the third method reaches an asymptotic worst-case complexity of O(n2n) operations over the integers, that is, sums and doublings. This way, with a different approach, the same asymptotic complexity of established algorithms, such as those based on the fast Walsh transform, is reached.
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content type line 14
ISSN:2073-8994
2073-8994
DOI:10.3390/sym14020213