Generalized bent (gbent) functions is a class of functions $f: \mathbb{Z}_2^n\rightarrow \mathbb{Z}_q$, where $q \geq 2$ is a positive integer, thatgeneralizes a concept of classical bent functions through their co-domainextension. A lot of research has recently been devoted towards derivation ofthe necessary and sufficient conditions when $f$ is represented as a collectionof Boolean functions. Nevertheless, apart from the necessary conditions thatthese component functions are bent when $n$ is even (respectively semi-bentwhen $n$ is odd), no general construction method has been proposed yet for $n$odd case. In this article, based on the use of the well-knownMaiorana-McFarland (MM) class of functions, we give an explicit constructionmethod 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 problemof providing a general construction method of gbent functions, for odd $n$, hasbeen solved. The method for odd $n$ employs a large class of disjoint spectrasemi-bent functions with certain additional properties which may be useful inother cryptographic applications.
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机译：广义弯曲（gbent）函数是一类函数$ f：\ mathbb {Z} _2 ^ n \ rightarrow \ mathbb {Z} _q $，其中$ q \ geq 2 $是一个正整数，泛化了经典弯曲的概念通过其共同域扩展来发挥功能。最近，许多研究致力于将$ f $表示为布尔函数集合时的必要条件和充分条件的推导。然而，除了在$ n $偶数时弯曲这些分量函数的必要条件（当$ n $是奇数时，分别是半benbent）之外，尚未提出针对$ n $奇数情况的通用构造方法。在本文中，基于对著名的Maiorana-McFarland（MM）函数类的使用，我们给出了gbent函数的显式构造方法，当$ n $是偶数时，对于$ q> 2 $的任何偶数，对于$ q当$ n $为奇数时，形式为$ q = 2 ^ r $（对于$ r> 1 $）的$。因此，解决了一个长期的开放问题，即为奇数$ n $提供gbent函数的一般构造方法。奇数$ n $的方法采用了一大类不相交的频谱半弯曲函数，具有某些附加属性，这在其他加密应用程序中可能很有用。
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