A family of weightwise (almost) perfectly balanced boolean functions with optimal algebraic immunity

Tang, Deng; Liu, Jian

doi:10.1007/s12095-019-00374-6

Cited by 20 publications

(18 citation statements)

References 16 publications

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“…as in Definition 11. Property 4 (Properties of WPB families, [1,3,4]). Let m ∈ N * and n = 2 m , the n-variable CMR function f n , an n-variable LM function g n and an n-variable TL function h n have the following properties:…”

Section: Families Of Wpb Functionsmentioning

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: Families Of Wpb Functionsmentioning

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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“…Given an n-variable Boolean function f , if for every integer k ∈ {1, 2, • • • , n − 1}, the restriction of the function f to the subset E n,k = {x ∈ F n 2 | wt(x) = k} is balanced and f (0, 0, • • • , 0) = f (1, 1, • • • , 1), then f is called weightwise perfectly balanced Boolean function, where wt(x) denotes the Hamming weight of the vector x ∈ F n 2 . In 2019, a large family of weightwise (almost) perfectly balanced Boolean functions with optimal algebraic immunity on F n 2 and good algebraic immunity on some subsets of vectors in F n 2 , especially on the subsets of vectors with constant Hamming weight was proposed [12]. That was the first time that weightwise (almost) perfectly balanced Boolean functions with good local algebraic immunities are presented.…”

Section: Introductionmentioning

confidence: 99%

A new construction of weightwise perfectly balanced Boolean functions

Zhang¹,

Su²

2023

AMC

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In this paper, we first introduce a class of quartic Boolean functions. And then, the construction of weightwise perfectly balanced Boolean functions on 2 m variables are given by modifying the support of the quartic functions, where m is a positive integer. The algebraic degree, the weightwise nonlinearity, and the algebraic immunity of the newly constructed weightwise perfectly balanced functions are discussed at the end of this paper.

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“…Being WAPB function relevant in a cryptographic context, all these works aim to produce W(A)PB functions having good parameters relatively to the other cryptographic criteria such as restricted and global nonlinearity, algebraic immunity and degree. For instance, the functions proposed in [TL19] have optimal algebraic immunity, while the family described in [LM19] has good nonlinearity on all the slices, also called weightwise nonlinearities. In fact, the weightwise nonlinearity is the criterion that got the most attention in these constructions, often used to compare the different families.…”

Section: Introductionmentioning

confidence: 99%

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 family of weightwise (almost) perfectly balanced boolean functions with optimal algebraic immunity

Cited by 20 publications

References 16 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

A new construction of weightwise perfectly balanced Boolean functions

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

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