几类高非线性度密码函数的构造  

Several classes of cryptographic Boolean functions with high nonlinearity

作　　者：刘欢 伍高飞 LIU Huan;WU Gaofei(School of Cyber Engineering,Xidian University,Xi’an 710071,China;Hubei Key Laboratory of Applied Mathematics,Faculty of Mathematics and Statistics,Hubei University,Wuhan 430062,China)

机构地区：[1]西安电子科技大学网络与信息安全学院,陕西西安710071 [2]湖北大学数学与统计学学院,应用数学湖北省重点实验室,湖北武汉430062

出　　处：《西安电子科技大学学报》2023年第6期237-250,共14页Journal of Xidian University

基　　金：陕西省自然科学基础研究计划(2021JQ-192);应用数学湖北省重点实验室(湖北大学)开放基金(HBAM202102)。

摘　　要：布尔函数在密码学中有着重要应用。Bent函数作为非线性度最大的布尔函数一直是对称密码学的热点研究对象。从频谱的角度来看,bent函数在Walsh-Hadamard变换下具有均匀频谱。Negabent函数是bent函数的推广,它在nega-Hadamard变换下具有均匀频谱。广义negabent函数是指在广义nega-Hadamard变换下具有均匀频谱的函数。Bent函数自1976年被提出以来,人们对其进行了广泛和深入的研究。然而,对于negabent函数和广义negabent函数的相关研究则较少。文中分析了广义negabent函数和广义bent-negabent函数的性质,并构造出一系列广义negabent函数、广义bent-negabent函数和广义semibent-negabent函数。首先,通过分析广义布尔函数的nega-互相关函数与广义nega-Hadamard变换之间的关系,提出一个广义negabent函数的判据。基于该判据,构造了一类广义negabent函数。其次,利用直和构造给出了两类形如f(x)=c_(1)f_(1)(x^((1)))+c_(2)f_(2)(x^((2)))+…+c_(r)f_(r)(x^((r)))的广义negabent函数。最后,利用直和构造得到了几类Z_(8)上的广义bent-negabent函数和广义semibent-negabent函数。文中提出了一些广义negabent函数构造的新方法,丰富了广义negabent函数的结果。Boolean functions have important applications in cryptography.Bent functions have been a hot research topic in symmetric cryptography as Boolean functions have maximum nonlinearity.From the perspective of spectrum,bent functions have a flat spectrum under the Walsh-Hadamard transform.Negabent functions are a class of generalized bent functions,which have a uniform spectrum under the nega-Hadamard transform.A generalized negabent function is a function with a uniform spectrum under the generalized nega-Hadamard transform.Bent functions has been extensively studied since its introduction in 1976.However,there are few research on negabent functions and generalized negabent functions.In this paper,the properties of generalized negabent functions and generalized bent-negabent functions are analyzed.Several classes of generalized negabent functions,generalized bent-negabent functions,and generalized semibent-negabent functions are constructed.First,by analyzing a link between the nega-crosscorrelation of generalized Boolean function and the generalized nega-Hadamard transformation,a criterion for generalized negabent functions is presented.Based on this criterion,a class of generalized negabent functions is constructed.Secondly,two classes of generalized negabent functions of the form f(x)=c_(1)f_(1)(x^((1)))+c_(2)f_(2)(x^((2)))+…+c_(r)f_(r)(x^((r)))are constructed by using the direct sum construction.Finally,generalized bent-negabent functions and generalized semibent-negabent functions over Z_(8) are obtained by using the direct sum construction.Some new methods for constructing generalized negabent functions are given in this paper,which will enrich the results of negabent functions.

关 键 词：布尔函数 广义negabent函数 广义BENT函数 nega-Hadamard bent-negabent函数 

分 类 号：TN918.1[电子电信—通信与信息系统]