高非线性四谱值和五谱值布尔函数的构造  

Construction of highly nonlinear Boolean functions with four-valued and five-valued spectra

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作  者:郭飞 王子龙[2] 段明[1] GUO Fei;WANG Zilong;DUAN Ming(State Key Laboratory of Mathematical Engineering and Advanced Computing,Information Engineering University,Zhengzhou 450001,China;State Key Laboratory of Integrated Services Networks,Xidian University,Xi’an 710071,China)

机构地区:[1]信息工程大学数学工程与先进计算国家重点实验室,河南郑州450001 [2]西安电子科技大学空天地一体化综合业务网全国重点实验室,陕西西安710071

出  处:《通信学报》2025年第3期144-150,共7页Journal on Communications

基  金:国家自然科学基金资助项目(No.62472438,No.62172319)。

摘  要:四谱值和五谱值布尔函数对于密码学应用具有特殊的意义,通过修改Maiorana-McFarland类bent函数,给出了一种偶数元四谱值和五谱值布尔函数的构造,确定了所构造函数的Walsh谱分布,证明其非线性度和半bent函数一样高,为2^(n-1)-2^(n/2)(n为变元数),代数次数能取到3和理论上界n/2+1之间的任意值。并深入研究了该构造的一个子类,包含的函数具有五谱值和最高的代数次数n/2+1,且不存在非零线性结构。Boolean functions with four-valued and five-valued spectra are of special interest for cryptography applications.By modifying bent functions in the Maiorana-McFarland class,a construction of Boolean functions on even numbers of variables with four-valued and five-valued spectra was presented,and their spectral distributions were determined.The nonlinearity of these functions was proved to be as good as that of semi-bent functions,i.e.,2^(n-1)-2^(n/2)(n was the number of variables),and the algebraic degree could reach any value ranging from 3 to the theoretical upper bound n/2+1.Furthermore,a subclass of the construction was studied,which consisted of Boolean functions with fivevalued spectra,the highest algebraic degree n/2+1,and without non-zero linear structures.

关 键 词:布尔函数 四谱值函数 五谱值函数 非线性度 代数次数 

分 类 号:TN918.8[电子电信—通信与信息系统]

 

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