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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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vectorial boolean functions and induced algebraic equations

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IEEE TRANSACTIONS ON INFORMATION THEORY 2006年第2期52卷 528-537页

作者： Golic, JD Telecom Italia Secur Innovat I-10148 Turin Italy

A general mathematical framework behind algebraic cryptanalytic attacks is developed. The framework relates to finding algebraic equations induced by vectorial boolean functions and, in particular, equations of low algebraic degree. The equations may involve only a subset of input variables and may or may not be conditioned on the values of output variables. In addition, the equations may have a constrained form interesting for the so-called fast algebraic attacks. A possible divide-and-conquer effect is pointed out and the notion of algebraic immunity order, naturally extending the notion of correlation immunity order, is defined. An application of general results to stream ciphers known as combiners with or without memory, with possibly multiple outputs, is studied in particular detail and the concept of divide-and-conquer algebraic attacks is introduced. Special properties of combiners with finite input memory, such as nonlinear filter generators, are also established. It is also pointed out that Grijbner basis algorithms may be used for finding low-degree induced algebraic equations.

关键词： algebraic attacks algebraic equations combiners with memory Grobner basis algorithms vectorial boolean functions

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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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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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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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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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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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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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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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NEW CONSTRUCTIONS OF vectorial boolean functions WITH GOOD CRYPTOGRAPHIC PROPERTIES

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INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE 2012年第3期23卷 749-760页

作者： Dong, Deshuai Qu, Longjiang Fu, Shaojing Li, Chao Natl Univ Def Technol Coll Sci Changsha 410073 Hunan Peoples R China Natl Univ Def Technol Sch Comp Sci Changsha 410073 Hunan Peoples R China

vectorial boolean functions play an important role in cryptography. How to construct vectorial boolean functions with good cryptographic properties is a nice problem that worth to be investigated. In this paper we present several constructions of balanced vectorial boolean functions with high algebraic immunity, high (or optimum) algebraic degree, and very high nonlinearity. In some cases, the constructed functions also achieve optimum algebraic immunity.

关键词： vectorial boolean functions algebraic immunity nonlinearity

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