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Sidon sets, thin sets, and the nonlinearity of vectorial boolean functions

JOURNAL OF COMBINATORIAL THEORY SERIES A 2025年 212卷

作者： Nagy, Gabor P. Tech Univ Budapest Inst Math & Econ Muegyet Rkp 3 H-1111 Budapest Hungary Univ Szeged Bolya Inst Arad Vertanuk Tere 1 H-6720 Szeged Hungary

The vectorial nonlinearity of a vector-valued function is its distance from the set of affine functions. In 2017, Liu, Mesnager, and Chen conjectured a general upper bound for the vectorial linearity. Recently, Carlet established a lower bound in terms of differential uniformity. In this paper, we improve Carlet's lower bound. Our approach is based on the fact that the level sets of a vectorial boolean function are thin sets. In particular, level sets of APN functions are Sidon sets, hence the Liu-Mesnager-Chen conjecture predicts that in F 2 n , there should be Sidon sets of size at least 2 n / 2 + 1 for all n. This paper provides an overview of the known large Sidon sets in F 2 n , and examines the completeness of the large Sidon sets derived from hyperbolas and ellipses of the finite affine plane. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： Sidon sets vectorial boolean functions APN functions Hamming distance Error correcting codes

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Autocorrelations of vectorial boolean functions 7th

Autocorrelations of Vectorial Boolean Functions

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7th International Conference on Cryptology and Information Security in Latin America (LATINCRYPT)

作者： Canteaut, Anne Koelsch, Lukas Li, Chao Li, Chunlei Li, Kangquan Qu, Longjiang Wiemer, Friedrich INRIA Paris France Univ Rostock Rostock Germany Natl Univ Def Technol Coll Liberal Arts & Sci Changsha Peoples R China Hunan Engn Res Ctr Commercial Cryptog Theory & Te Changsha Peoples R China Univ Bergen Dept Informat N-5020 Bergen Norway Ruhr Univ Bochum Horst Gortz Inst IT Secur Bochum Germany Cryptosolut Essen Germany

ISBN: (纸本)9783030882389;9783030882372

Recently, Bar-On et al. introduced at Eurocrypt'19 a new tool, called the differential-linear connectivity table (DLCT), which allows for taking into account the dependency between the two subciphers E-0 and E-1 involved in differential-linear attacks. This paper presents a theoretical characterization of the DLCT, which corresponds to an autocorrelation table (ACT) of a vectorial boolean function. We further provide some new theoretical results on ACTs of vectorial boolean functions.

关键词： vectorial boolean functions Differential-linear connectivity table Autocorrelation table Absolute indicator

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Construction of balanced vectorial boolean functions with almost optimal nonlinearity and very low differential-linear uniformity

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FINITE FIELDS AND THEIR APPLICATIONS 2021年 76卷 101903-101903页

作者： Tang, Deng Mandal, Bimal Maitra, Subhamoy Shanghai Jiao Tong Univ Sch Elect Informat & Elect Engn Shanghai 200240 Peoples R China State Key Lab Cryptol POB 5159 Beijing 100878 Peoples R China Indian Inst Technol Madras Dept Math Chennai 600036 Tamil Nadu India Indian Stat Inst Appl Stat Unit 203 BT Rd Kolkata 700108 India

The differential-linear connectivity table (DLCT) of a vectorial boolean function was recently introduced by Bar-On et al. at EUROCRYPT'19, whose value at a point is related to the autocorrelation value of its component functions. Further, in INDOCRYPT'19, we proposed a new construction method for vectorial boolean functions with very low differentiallinear uniformity using Maiorana-McFarland bent functions. The difficulty of that construction method was to identify the permutations and the sub-functions that satisfy the conditions to attain good cryptographic properties. In this paper we discover novel techniques to construct such sub-functions to generate vectorial boolean functions with substantially improved cryptographic properties. Our proposed methods are based on ideas from combinatorics as well as finite fields. In particular, we construct the sub-functions to generate (4t, t - 1)-function, t >= 5, in a different manner than our Indocrypt'19 paper. Further our new methods help in obtaining sub-functions to generate balanced (4t + 2, t - 1)function and (2k, k)-function with very good nonlinearity and very low differential-linear uniformity, that were never demonstrated earlier. (c) 2021 Elsevier Inc. All rights reserved.

关键词： Maiorana-McFarland bent functions vectorial boolean functions Autocorrelation Differential-linear uniformity Finite fields

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Construction of Resilient boolean and vectorial boolean functions with High Nonlinearity

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IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES 2019年第10期E102A卷 1397-1401页

作者： Li, Luyang Zheng, Dong Zhao, Qinglan Xian Univ Posts & Telecommunicat Xian 710121 Shaanxi Peoples R China

boolean functions and vectorial boolean functions are the most important components of stream ciphers. Their cryptographic properties are crucial to the security of the underlying ciphers. And how to construct such functions with good cryptographic properties is a nice problem that worth to be investigated. In this paper, using two small nonlinear functions with t - 1 resiliency, we provide a method on constructing t-resilient n variables boolean functions with strictly almost optimal non-linearity > 2(n-1) - 2(n/2) and optimal algebraic degree n - t - 1. Based on the method, we give another construction so that a large class of resilient vectorial boolean functions can be obtained. It is shown that the vectorial boolean functions also have strictly almost optimal nonlinearity and optimal algebraic degree.

关键词： boolean function vectorial boolean functions stream ciphers nonlinearity resiliency algebraic degree

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On weak differential uniformity of vectorial boolean functions as a cryptographic criterion

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APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING 2016年第5期27卷 359-372页

作者： Aragona, Riccardo Calderini, Marco Maccauro, Daniele Sala, Massimiliano 1. Trento Italy 2. Perugia Italy

We study the relation among some security parameters for vectorial boolean functions which prevent attacks on the related block cipher. We focus our study on a recently-introduced security criterion, called weak differential uniformity, which prevents the existence of an undetectable trapdoor based on imprimitive group action. We present some properties of functions with low weak differential uniformity, especially for the case of power functions and 4-bit S-Boxes.

关键词： Permutation vectorial boolean functions Power functions Weak differential uniformity

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Constructions of vectorial boolean functions With Good Cryptographic Properties

Constructions of Vectorial Boolean Functions With Good Crypt...

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中国密码学会2017年密码算法学术会议

作者： Lu-Yang Li Wei-Guo Zhang Science and Technology on Communication Security Laboratory Chengdu 610041 State Key Laboratory of Integrated Services Networks Xidian University Xi'an 710071

vectorial boolean functions are used as the nonlinear components in stream *** security of the cipher system depends on the cryptographic properties of the *** widely accepted criteria are balancedness,high nonlinearity,correlation immune and so *** this paper,we present a new construction method of balanced vectorial boolean functions with strictly almost optimal *** the first time,the first-order correlation immune functions with currently best known nonlinearity 2n-1-2n/2-1-2[n/4」 are *** is also shown that our functions can have good algebraic degree.

关键词： vectorial boolean functions stream ciphers nonlinearity balancedness correlation immunity

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On Verification of Restricted Extended Affine Equivalence of vectorial boolean functions 5

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5th International Workshop on the Arithmetic of Finite Fields (WAIFI)

作者： Ozbudak, Ferruh Sinak, Ahmet Yayla, Oguz Middle E Tech Univ Dept Math TR-06800 Ankara Turkey Middle E Tech Univ Inst Appl Math TR-06800 Ankara Turkey Austrian Acad Sci Johann Radon Inst Computat & Appl Math A-4040 Linz Austria Necmettin Erbakan Univ Dept Math & Comp Sci TR-42090 Konya Turkey

ISBN: (纸本)9783319162768;9783319162775

vectorial boolean functions are used as substitution boxes in cryptosystems. Designing inequivalent functions resistant to known attacks is one of the challenges in cryptography. In doing this, finding a fast technique for determining whether two given functions are equivalent is a significant problem. A special class of the equivalence called restricted extended affine (REA) equivalence is studied in this paper. We update the verification procedures of the REA-equivalence types given in the recent work of Budaghyan and Kazymyrov (2012). In particular, we solve the system of linear equations simultaneously in the verification procedures to get better complexity. We also present the explicit number of operations of the verification procedures of these REA-equivalence types. Moreover, we construct two new REA-equivalence types and present the verification procedures of these types with their complexities.

关键词： vectorial boolean functions EA-equivalence REA-equivalence

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Asymptotic nonlinearity of vectorial boolean functions

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES boolean functions AND SEQUENCES 2014年第2期6卷 103-115页

作者： Dib, Stephanie Inst Math Luminy F-13288 Marseille 9 France

The nonlinearity of a boolean function F : F-2(m) -> F-2 is the minimum Hamming distance between f and all affine functions. The nonlinearity of a S-box f : F-2(m) -> F-2(n) is the minimum nonlinearity of its component (boolean) functions upsilon center dot f, upsilon epsilon F-2(n)\ functions. This notion quantifies the level of resistance of the S-box to the linear attack. In this paper, the distribution of the nonlinearity of (m, n)-functions is investigated. When n = 1, it is known that asymptotically, almost all m-variable boolean functions have high nonlinearities. We extend this result to (m, n)-functions.

关键词： vectorial boolean functions Nonlinearity S-boxes Bent functions

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Determination of cryptographic tables and properties related to the revised boomerang and its application to a fundamental S-box

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES boolean functions AND SEQUENCES 2025年第2期17卷 465-492页

作者： Eddahmani, Said Mesnager, Sihem Univ Paris VIII Dept Math F-93526 St Denis France Univ Sorbonne Paris Nord Lab Geometry Anal & Applicat LAGA CNRSUMR 7539 F-93430 Villetaneuse France Telecom Paris F-91120 Palaiseau France

In symmetric cryptography, vectorial boolean functions over the finite field F2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_{2<^>n}$$\end{document} are used to construct strong S-boxes. A strong S-box must meet various criteria to resist known attacks, including differential, linear, boomerang, and their variants. To evaluate an S-box's resistance, several tables are utilized, such as the Difference Distribution Table (DDT) and the Boomerang Connectivity Table (BCT). Recent developments in boomerang attacks have revisited the concept of the boomerang switch effect, illustrating the effectiveness of this technique. As a result, a new tool called the Boomerang Difference Table (BDT) was introduced as an alternative to the traditional BCT. Additionally, two novel tables have been proposed: the Upper Boomerang Connectivity Table (UBCT) and the Lower Boomerang Connectivity Table (LBCT). These tables are enhancements over the BCT and facilitate a systematic evaluation of boomerangs that can return over multiple rounds. This paper focuses on the new tools for measuring the revisited version of boomerang attacks and the related tables UBCT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\texttt {UBCT}$$\end{document}, LBCT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\texttt {LBCT}$$\end{document}, as well as the so-called Extended Boomerang Connectivity Table (EBCT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepa

关键词： vectorial boolean functions Upper Boomerang Connectivity Table (UBCT) Lower Boomerang Connectivity Table (LBCT) Extended Boomerang Connectivity Table (EBCT) Boomerang connectivity uniformity

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On Dillon's property of (n, m)-functions

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES boolean functions AND SEQUENCES 2024年第6期16卷 1449-1473页

作者： Abbondati, Matteo Calderini, Marco Villa, Irene Univ Trento Dept Math Via Sommar 14 I-38123 Povo TN Italy Univ Montpellier CNRS LIRMM Montpellier France Univ Genoa Dept Math Via Dodecaneso 35 I-16146 Genoa Italy

Dillon observed that an APN function F over F-2(n) with n greater than 2 must satisfy the condition {F(x) + F(y) + F(z) + F(x + y + z) :x, y, z is an element of F-2(n)} = F-2(n). Recently, Taniguchi (Cryptogr. Commun. 15, 627-647 2023) generalized this condition to functions defined from F-2(n) to F-2(m), with m > n, calling it the D-property. Taniguchi gave some characterizations of APN functions satisfying the D-property and provided some families of APN functions from F-2(n) to F-2(n+1) satisfying this property. In this work, we further study the D-property for (n, m)-functions with m >= n. We give some combinatorial bounds on the dimension m for the existence of such functions. Then, we characterize the D-property in terms of the Walsh transform and for quadratic functions we give a characterization of this property in terms of the ANF. We also give a simplification on checking the D-property for quadratic functions, which permits to extend some of the APN families provided by Taniguchi. We further focus on the class of the plateaued functions, providing conditions for the D-property. To conclude, we show a connection of some results obtained with the higher-order differentiability and the inverse Fourier transform.

关键词： vectorial boolean functions D-property APN functions Walsh transform Plateaued functions

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