逐重量完美平衡布尔函数的构造  

Construction of weightwise perfectly balanced Boolean functions

作　　者：赵庆兰[1] 王富佳 秦宝东 ZHAO Qinglan;WANG Fujia;QIN Baodong(School of Cyberspace Security,Xi’an University of Posts and Telecommunications,Xi’an 710121,China)

机构地区：[1]西安邮电大学网络空间安全学院,陕西西安710121

出　　处：《通信学报》2025年第2期97-107,共11页Journal on Communications

基　　金：国家自然科学基金资助项目(No.62372370,No.61902314,No.62072371)。

摘　　要：在FLIP等同态友好流密码的背景下,逐重量完美平衡布尔函数成为密码学中研究的热点问题,但已有的研究结果构造的逐重量完美平衡布尔函数的k-重量非线性度离其上界仍有距离,基于此,提出一种新的逐重量完美平衡布尔函数的构造方法。首先,对于m≥4的正整数,给出了一类2^(m)元的八次基础布尔函数,并利用代数正规型确定其k-重量分布。随后,通过修改此类基础函数的支撑集,构造出了一类2^(m)元逐重量完美平衡布尔函数,从理论上证明了其在每个非平凡等重量子集上都是平衡的。此外,分析了所提构造方法与同类构造方法之间的区别,证明了逐重量完美平衡布尔函数的代数次数。最后,与目前已有的同类函数进行比较,结果表明,新构造的8元函数在k=3和k=4时分别超过现有的k-重量非线性度,达到18和26,新构造的16元函数在k=13时的k-重量非线性度从目前最高值152提高到了160。In the context of homomorphic-friendly stream ciphers such as FLIP,weightwise perfectly balanced Boolean functions have become a hot topic in cryptography in recent years.The k-weight nonlinearity of weightwise perfectly balanced Boolean functions constructed by existing research is still far from its upper bound.Based on this,a new construction of weightwise perfectly balanced Boolean functions was introduced.Initially,for positive integers m≥4,a class of 2^(m)variable Boolean functions with algebraic degree 8 was given,and their k-weight distribution was determined using algebraic normal forms.Subsequently,by modifying the support set of these basic functions,a class of 2^(m)-variable weightwise perfectly balanced Boolean functions was constructed.It was theoretically proven that they were balanced on every nontrivial subset with the same weight vector.Additionally,the difference between the construction methods presented and those of similar constructions was analyzed.The algebraic degree of the weightwise perfectly balanced Boolean functions is proven.Compared with existing constructions,the new 8-variable WPB function outperforms the existing kweight nonlinearity at values of 3 and 4,reaching 18 and 26,respectively,and the new 16-variable WPB function shows enhanced k-weight nonlinearity at k=13,achieving 160,surpassing the highest value of 152 for the existing constructions.

关 键 词：FLIP 逐重量完美平衡布尔函数 代数次数 重量非线性度 

分 类 号：TN92[电子电信—通信与信息系统]