A new construction of weightwise perfectly balanced Boolean functions

Zhang, Rui; Su, Sihong

doi:10.3934/amc.2021020

Cited by 12 publications

(2 citation statements)

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“…for each 1 ≤ k ≤ n − 1 where the slice E k,n denotes the set of F n 2 with all vectors of Hamming weight k, f globally balanced, and f (0 n ) = 0. Since then, several articles studied the properties on restricted sets, and multiple articles focused on WPB functions such as [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

S0-equivalent classes, a new direction to find better weightwise perfectly balanced functions, and more

Gini,

Méaux

2023

Preprint

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We investigate the concept of S0 equivalent class, n-variable Boolean functions up to the addition of a symmetric function null in 0n and 1n, as a tool to study weightwise perfectly balanced functions. On the one hand we show that weightwise properties, such as being weightwise perfectly balanced, the weightwise nonlinearity and weightwise algebraic immunity, are invariant of these classes. On the other hand, we analyse the variation of global parameters inside the same class, showing for example that there is always a function with high degree, algebraic immunity, or nonlinearity in the S0 equivalent class of a function. Finally, we discuss how these results extend to other equivalence relations and their applications in cryptography MSC Classification: 94A60 , 94D10

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Section: Introductionmentioning

confidence: 99%

S0-equivalent classes, a new direction to find better weightwise perfectly balanced functions, and more

Gini,

Méaux

2023

Preprint

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“…Recently, the modifications of this construction presented in [MSL21] allow one to obtain WPB functions still with optimal algebraic immunity but with higher weightwise nonlinearity. A line of work started in 2020 with [MS21] provide recursive constructions based on the modification of the support of a low degree function over F n 2 ; more precisely on linear [MS21], quadratic [MS21,LS20] or quartic functions [ZS21]. Finally, in the recent preprint [MPJ + 22] an experimental approach with evolutionary algorithms is considered to find 8-variable WPB with high weightwise nonlinearity.…”

mentioning

confidence: 99%

On the weightwise nonlinearity of weightwise perfectly balanced functions

Gini

Méaux

2022

Discrete Applied Mathematics

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Weightwise Almost Perfectly Balanced Functions: Secondary Constructions for All n and Better Weightwise Nonlinearities

Gini

Méaux²

2022

Lecture Notes in Computer Science

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The design of FLIP stream cipher presented at Eurocrypt 2016 motivates the study of Boolean functions with good cryptographic criteria when restricted to subsets of F n 2 . Since the security of FLIP relies on properties of functions restricted to subsets of constant Hamming weight, called slices, several studies investigate functions with good properties on the slices, i.e. weightwise properties. A major challenge is to build functions balanced on each slice, from which we get the notion of Weightwise Almost Perfectly Balanced (WAPB) functions. Although various constructions of WAPB functions have been exhibited since 2017, building WAPB functions with high weightwise nonlinearities remains a difficult task. Lower bounds on the weightwise nonlinearities of WAPB functions are known for very few families, and exact values were computed only for functions in at most 16 variables. In this article, we introduce and study two new secondary constructions of WAPB functions. This new strategy allows us to bound the weightwise nonlinearities from those of the parent functions, enabling us to produce WAPB functions with high weightwise nonlinearities. As a practical application, we build several novel WAPB functions in up to 16 variables by taking parent functions from two different known families. Moreover, combining these outputs, we also produce the 16-variable WAPB function with the highest weightwise nonlinearities known so far.

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A new construction of weightwise perfectly balanced Boolean functions

Cited by 12 publications

References 12 publications

S0-equivalent classes, a new direction to find better weightwise perfectly balanced functions, and more

S0-equivalent classes, a new direction to find better weightwise perfectly balanced functions, and more

On the weightwise nonlinearity of weightwise perfectly balanced functions

Weightwise Almost Perfectly Balanced Functions: Secondary Constructions for All n and Better Weightwise Nonlinearities

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