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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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generalized bent functions and class group of imaginary quadratic fields

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Science China Mathematics 2001年第5期44卷 562-570页

作者：冯克勤 1. Department of Mathematical Sciences Tsinghua University 100084 Beijing China

Several new results on non-existence of generalized bent functions are presented. The results are related to the class number of imaginary quadratic fields.

关键词： generalized bent functions imaginary quadratic fields.

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generalized bent functions - Some General Construction Methods and Related Necessary and Sufficient Conditions

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN functions AND SEQUENCES 2015年第4期7卷 469-483页

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

In this article we present a broader theoretical framework useful in studying the properties of so-called generalized bent functions. We give the sufficient conditions (and in many cases also necessary) for generalized bent functions when these functions are represented as a linear combination of: generalized bent;Boolean bent;and a mixture of generalized bent and Boolean bent functions. These conditions are relatively easy to satisfy and by varying the variables that specify these linear combinations many different classes of generalized bent functions can be derived. In particular, based on these results, we provide some generic construction methods of these functions and demonstrate that some previous methods are just special cases of the results given in this article.

关键词： generalized boolean functions generalized bent functions Walsh-Hadamard 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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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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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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On generalized bent functions with Dillon's exponents

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INFORMATION PROCESSING LETTERS 2014年第4期114卷 222-227页

作者： Bajric, Samed Pasalic, Enes Ribic-Muratovic, Amela Sugata, Gangopadhyay Univ Primorska FAMNIT Koper Slovenia Univ Sarajevo Sarajevo 71000 Bosnia & Herceg Indian Inst Technol Dept Comp Sci & Engn Roorkee Uttar Pradesh India

In this paper we investigate the possibility of constructing bent functions over fields with odd characteristic. While in the binary case, and for n = 2k, the bent property of monomials of the form Tr-1(n)(ax(r(2k-1))) and binomials Tr-1(n)(x(2k-1) + ax(r(2k-1))) were investigated in several papers, generalized bent functions f :GF(p(n)) -> GF(p) of the form Tr-1(n)(Sigma(t)(i=1) a(i)x(rt(pk-1))), p being an odd prime and n = 2k, were not analyzed previously. In particular, the construction of vectorial (generalized) bent functions has not been addressed. It is shown that the necessary and sufficient bent conditions for both the single output function of the form f (x) = Tr-1(n)(Sigma(t)(i=1) a(i)x(ri(pk-1))) and the associated mapping F(x) = Tr-k(2k)(Sigma(t)(i=1) aix(ri(pk-1))), where F : GF(p(2k)) -> GF(p(k)), are very similar and can be expressed in terms of the image of a set V used in the direct sum decomposition of GF(p(2k)). Furthermore, it is observed that vectorial bent functions are easily constructed using the Maiorana-McFarland method. (C) 2013 Elsevier B.V. All rights reserved.

关键词： Cryptography generalized bent functions Trace binomials Dillon exponent Vectorial bent functions Planar mappings

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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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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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