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Nonexistence of generalized bent functions and the quadratic norm form equations

DESIGNS CODES AND CRYPTOGRAPHY 2025年 1-17页

作者： Lv, Chang Zhu, Yuqing Chinese Acad Sci Inst Informat Engn State Key Lab Cyberspace Secur Def Beijing 100093 Peoples R China Beijing Jiaotong Univ Sch Cyberspace Sci & Technol Beijing Key Lab Secur & Privacy Intelligent Transp Beijing 100044 Peoples R China

We present a new result on the nonexistence of generalized bent functions (GBFs) from (Z/tZ)n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathbb {Z}/t\mathbb {Z})<^>n$$\end{document} to Z/tZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}/t\mathbb {Z}$$\end{document} (called type [n, t]) for a large class. Assume p is an odd prime number. By showing certain quadratic norm form equations having no integral points, we obtain a universal result on the nonexistence of GBFs with type [n,2pe]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[n, 2p<^>e]$$\end{document} when p and n satisfy a certain inequality, and by computational methods with a widely accepted hypothesis, generalized Riemann Hypothesis, we also achieve some results on the nonexistence of GBFs for relatively small p.

关键词： generalized bent functions Quadratic norm form equations Integral points Cyclotomic fields

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On the Duals of generalized bent functions

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IEEE TRANSACTIONS ON INFORMATION THEORY 2022年第7期68卷 4770-4781页

作者： Wang, Jiaxin Fu, Fang-Wei Nankai Univ Chern Inst Math Tianjin 300071 Peoples R China Nankai Univ LPMC Tianjin 300071 Peoples R China

In this paper, we study the duals of generalized bent functions f : V-n -> Z(pk), where V-n is an n-dimensional vector space over F-p and p is an odd prime, k is a positive integer. It is known that weakly regular generalized bent functions always appear in pairs since the dual of a weakly regular generalized bent function is also a weakly regular generalized bent function. The duals of non-weakly regular generalized bent functions can be generalized bent or not generalized bent. By generalizing the construction of *** et al., 2016, we provide an explicit construction of generalized bent functions whose duals can be generalized bent or not generalized bent. We show that the well-known direct sum construction and the generalized indirect sum construction given in Wang and Fu, 2021. can provide secondary constructions of generalized bent functions whose duals can be generalized bent or not generalized bent. By using the knowledge on ideal decomposition in cyclotomic fields, we prove that f** (x) = f(- x) if f is a generalized bent function and its dual f* is also a generalized bent function. For any non-weakly regular generalized bent function f which satisfies that f(x) = f(- x) and its dual f* is generalized bent, we give a property and as a consequence, we prove that there is no self-dual generalized bent function f : V-n -> Z(pk) if p = 3 (mod 4) and n is odd. When p = 1 (mod 4) or p = 3 (mod 4) and n is even, we give a secondary construction of self-dual generalized bent functions. In the end, by the decomposition of generalized bent functions, we characterize the relations between the generalized bentness of the dual of a generalized bent function f and the bentness of the duals of bent functions associated with the generalized bent function f, as well as the relations of self-duality between a generalized bent function f and bent functions associated with the generalized bent function f.

关键词： generalized bent functions dual self-dual (non)-weakly regular Walsh transform

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New Secondary Constructions of generalized bent functions

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Chinese Journal of Electronics 2021年第6期30卷 1022-1029页

作者： YANG Zhiyao KE Pinhui CHEN Zhixiong Fujian Provincial Key Laboratory of Network Security and Cryptology Fujian Normal University College of Mathematics and Informatics Fujian Normal University Fujian Provincial Key Laboratory of Applied Mathematics Putian University

Three new secondary constructions of generalized bent functions are presented. We provide a secondary construction of generalized bent functions from indirect sum methods proposed by Carlet et al. A new secondary construction of generalized bent functions from four initial functions is also investigated. We demonstrate that many known constructions can be derived from our proposed construction as special cases by choosing proper initial functions and parameters. By modifying the new construction, a novel secondary construction of generalized bent functions from two initial generalized bent functions is obtained. For the binary case, the dual functions of the bent functions by our method are presented, which share the same formula as the indirect sum.

关键词： cryptography Boolean functions indirect sum methods proper initial functions new secondary constructions generalized bent functions

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generalized partially bent functions, generalized perfect arrays, and cocyclic Butson matrices

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN functions AND SEQUENCES 2024年第2期16卷 323-337页

作者： Armario, J. A. Egan, R. Flannery, D. L. Univ Seville Dept Matemat Aplicada 1 Avda Reina Mercedes S-N Seville 41012 Spain Dublin City Univ Sch Math Sci Dublin Ireland Univ Galway Sch Math & Stat Sci Galway Ireland

In a recent survey, Schmidt compiled equivalences between generalized bent functions, group invariant Butson Hadamard matrices, and abelian splitting relative difference sets. We establish a broader network of equivalences by considering Butson matrices that are cocyclic rather than strictly group invariant. This result has several applications;for example, to the construction of Boolean functions whose expansions are generalized partially bent functions, including cases where no bent function can exist.

关键词： generalized bent functions Butson Hadamard matrices generalized perfect arrays Cocycles

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Nonexistence results of generalized bent functions from Zn2 to Zm

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JOURNAL OF COMBINATORIAL THEORY SERIES A 2023年第1期198卷

作者： Leung, Ka Hin Li, Shuxing Mao, Songtao Natl Univ Singapore Dept Math Singapore 119260 Singapore Simon Fraser Univ Dept Math 8888 Univ Dr Burnaby BC V5A 1S6 Canada Johns Hopkins Univ Dept Comp Sci 3400 North Charles St Baltimore MD 21218 USA

New nonexistence results of generalized bent functions from Zn2 to Zm are presented. In particular, we prove that there exists no generalized bent function from Z42 to Zm for every odd M. (c) 2023 Elsevier Inc. All ri... 详细信息

关键词： generalized bent functions Vanishing sums Field descent Nonexistence

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Construction methods for generalized bent functions

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DISCRETE APPLIED MATHEMATICS 2018年 238卷 14-23页

作者： Hodzic, S. Pasalic, E. Univ Primorska FAMNIT Glagoljaska 6 Koper 6000 Slovenia Univ Primorska IAM Glagoljaska 6 Koper 6000 Slovenia

generalized bent (gbent) functions is a class of functions f : Z(2)(n) -> Z(q), where q >= 2 is a positive integer, that generalizes a concept of classical bent functions through their co-domain extension. A lot of research has recently been devoted towards derivation of the necessary and sufficient conditions when f is represented as a collection of Boolean functions. Nevertheless, apart from the necessary conditions that these component functions are bent when n is even (respectively semi-bent when n is odd), no general construction method has been proposed yet for n odd case. In this article, based on the use of the well-known Maiorana-McFarland (MM) class of functions, we give an explicit construction method of gbent functions, for any even q > 2 when n is even and for any q of the form q = 2(r) (for r > 1) when n is odd. Thus, a long-term open problem of providing a general construction method of gbent functions, for odd n, has been solved. The method for odd n employs a large class of disjoint spectra semi-bent functions with certain additional properties which may be useful in other cryptographic applications. (C) 2017 Elsevier B.V. All rights reserved.

关键词： generalized bent functions Walsh-Hadamard transform (generalized) Maiorana-McFarland class Gray maps

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Full Characterization of generalized bent functions as (Semi)-bent Spaces, Their Dual, and the Gray Image

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IEEE TRANSACTIONS ON INFORMATION THEORY 2018年第7期64卷 5432-5440页

作者： Hodzic, Samir Meidl, Wilfried Pasalic, Enes Univ Primorska FAMNIT Koper 6000 Slovenia OEAW Johann Radon Inst Computat & Appl Math A-4040 Linz Austria Univ Primorska IAM FAMNIT Koper 6000 Slovenia

A natural generalization of bent functions is a class of functions from F-2(n) to Z(2k) which is known as generalized bent (gbent) functions. The construction and characterization of gbent functions are commonly described in terms of the Walsh transforms of the associated Boolean functions. Using similar approach, we first determine the dual of a gbent function when n is even. Then, depending on the parity of n, it is shown that the Gray image of a gbent function is (k - 1) or (k - 2) plateaued, which generalizes previous results for k = 2,3, and 4. We then completely characterize gbent functions as algebraic objects. More precisely, again depending on the parity of n, a gbent function is a (k - 1)- dimensional affine space of bent functions or semi-bent functions with certain interesting additional properties, which we completely describe. Finally, we also consider a subclass of functions from F-2(n) to Z(2k), called Zq - bent functions (which are necessarily gbent), which essentially gives rise to relative difference sets similarly to standard bent functions. Two examples of this class of functions are provided and it is demonstrated that many gbent functions are not Zq-bent.

关键词： generalized bent functions Zq-bent functions Gray maps dual of generalized bent function relative difference sets

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Further Results on generalized bent functions and Their Complete Characterization

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IEEE TRANSACTIONS ON INFORMATION THEORY 2018年第7期64卷 5441-5452页

作者： Mesnager, Sihem Tang, Chunming Qi, Yanfeng Wang, Libo Wu, Baofeng Feng, Keqin Univ Paris VIII Dept Math F-93526 St Denis France Univ Paris XIII CNRS LAGA UMR 7539 Sorbonne Paris Cite F-93430 Villetaneuse France Telecom ParisTech F-75013 Paris France China West Normal Univ Sch Math & Informat Nanchong 637002 Peoples R China Hangzhou Dianzi Univ Sch Sci Hangzhou 310018 Zhejiang Peoples R China Jinan Univ Coll Informat Sci & Technol Guangzhou 510632 Guangdong Peoples R China Chinese Acad Sci Inst Informat Engn Beijing 100093 Peoples R China Tsinghua Univ Dept Math Sci Beijing 100084 Peoples R China

This paper contributes to increase our knowledge on generalized bent functions (including generalized bent Boolean functions and generalized p-ary bent functions with odd prime p) by bringing new results on their characterization and construction in arbitrary characteristic. More specifically, we first investigate relations between generalized bent functions and bent functions by the decomposition of generalized bent functions. This enables us to completely characterize generalized bent functions and Z(pk)-bent functions by some affine space associated with the generalized bent functions. We also present the relationship between generalized bent Boolean functions with an odd number of variables and generalized bent Boolean functions with an even number of variables. Based on the well-known Maiorana-McFarland class of Boolean functions, we present some infinite classes of generalized bent Boolean functions. In addition, we introduce a class of generalized hyperbent functions that can be seen as generalized Dillon's PS functions. Finally, we solve an open problem related to the description of the dual function of a weakly regular generalized bent Boolean function with an odd number of variables via the Walsh-Hadamard transform of their component functions, and we generalize these results to the case of odd prime.

关键词： bent functions hyperbent functions generalized Walsh-Hadamard transform generalized bent functions generalized hyperbent functions

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On the Non-Existence of Certain Classes of generalized bent functions

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IEEE TRANSACTIONS ON INFORMATION THEORY 2017年第1期63卷 738-746页

作者： Lv, Chang Li, Jianing Chinese Acad Sci Inst Informat Engn State Key Lab Informat Secur Beijing 100093 Peoples R China Chinese Acad Sci Acad Math & Syst Sci NCMIS Key Lab Math Mechanizat Beijing 100093 Peoples R China

We obtain new non-existence results of generalized bent functions from Z(q)(n) to Z(q) (called type [n, q]). The first case is a class with types, where q = 2(p1)(r1) p(2)(r2). The second case contains two types [1 &l... 详细信息

关键词： generalized bent functions cyclotomic fields class groups Stickelberger relations

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Nonexistence of two classes of generalized bent functions

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DESIGNS CODES AND CRYPTOGRAPHY 2017年第3期85卷 471-482页

作者： Li, Jianing Deng, Yingpu Chinese Acad Sci Acad Math & Syst Sci NCMIS Key Lab Math Mechanizat Beijing 100190 Peoples R China Univ Chinese Acad Sci Beijing 100049 Peoples R China

We obtain some new nonexistence results of generalized bent functions from Z(q)(n) to Z(q) (called type [n, q]) in the case that there exist cyclotomic integers in Z[zeta(q)] with absolute value q (n/2). This result generalizes two previous nonexistence results [n, q] = [1, 2x7] of Pei (Lect Notes Pure Appl Math 141:165-172, 1993) and [3, 2 x 23(e)] of Jiang and Deng (Des Codes Cryptogr 75: 375-385, 2015). We also remark that by using a same method one can get similar nonexistence results of GBFs from Z(2)(n) to Z(m).

关键词： generalized bent functions Cyclotomic fields Prime ideal factorizations Class groups

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