2-分解H布尔函数和高非线性度布尔函数

doi:10.11896/j.issn.1002-137X.2016.07.030

摘要/Abstract

摘要： 以布尔函数的导数和自定义的e-导数为主要研究工具,研究满足一次扩散准则、可2-分解为两个子函数乘积的一类H布尔函数的非线性度、相关免疫性和代数免疫性等密码学性质。得到了这类H布尔函数的相关免疫阶与两个子函数的关系,以及这类H布尔函数的相关免疫阶可达到 n2 －1的结论。还得到了利用两个子函数使布尔函数的非线性度易于求解的方法,以及这类H布尔函数的最低代数次数零化子与两个子函数的关系。进一步地,在这类H布尔函数上述特点的基础上,利用导数和e-导数构造出了非线性度提高到2n－2+2n－3、具有相关免疫性和2阶代数免疫性的一族H布尔函数。由此,解决了提高布尔函数的非线性度问题,以及同时具有较高非线性度、扩散性、相关免疫性和较高阶代数免疫性的布尔函数的存在性问题。

关键词: H布尔函数,2-分解,e-导数,非线性度,代数免疫性,相关免疫性

Abstract: Using the derivative of the Boolean functions and the e-derivative defined by ourselves as research tools,we studied the cryptographic properties of a class of H Boolean function which satisfy one degree propagation and are divi-ded into the product of two subfunctions,including nonlinearity,correlation immunity and algebraic immunity and so on.We achieved the relationship between the correlation immunity of this kind of H Boolean function and the two subfunctions,and also arrived at a conclusion on the correlation immunity of this kind of H Boolean function which can reach n2 －1.Moreover,we obtained the relationship between the lowest algebraic degree annihilator of this kind of an H Boolean function and the two subfunctions.Further,using e-derivative and derivative of a Boolean function,we constructed a cluster of H Boolean function which has the nonlinearity 2n－2+2n－3,the correlation immunity and 2-order algebraic immunity from obtained H Boolean functions.In this way,we resolved the problem of improving the nonlinearity of a Boolean function,and the existence problem of a Boolean function having higher nonlinearity,propagation,correlation immunity and higher algebraic immunity.

Key words: H Boolean functions,2-divide,e-derivative,Nonlinearity,Algebraic immunity,Correlation immunity

黄景廉,王卓,李娟. 2-分解H布尔函数和高非线性度布尔函数[J]. 计算机科学, 2016, 43(7): 166-170. https://doi.org/10.11896/j.issn.1002-137X.2016.07.030

HUANG Jing-lian, WANG Zhuo and LI Juan. H Boolean Functions with Divided into Two Parts and High Nonlinearity Boolean Functions[J]. Computer Science, 2016, 43(7): 166-170. https://doi.org/10.11896/j.issn.1002-137X.2016.07.030

参考文献

[1] Carlet C.Boolean Functions for Cryptography and Error Correcting Codes [M].Cambridge University Press,2010
[2] Carlet C.Vectorial Boolean functions for cryptography[M]∥Crama E Y,Hammer P,eds.Boolean Models and Methods Cambridge University Press,2006
[3] Xiong F,Qiao D,Wang H X,et al.A Novel Network Reliability Evaluating Algorithm with Ordered Binary Decision Diagram Based on Boolean Function[J].Journal of Electronics & Information Technology,2014,36(11):2786-2790(in Chinese) 熊飞,乔迪,王宏祥,等.一种基于有序二元决策图和布尔函数性质计算网络可靠性的算法[J].电子与信息学报,2014,36(11):2786-2790
[4] Cusick T W,Stanica P.Cryptographic Boolean Functions and Ap-plications[M].Academic Press,2009
[5] Gao G P,Liu W F.The Notes on the Linear Structures of Rotation Symmetric Boolean Functions[J].Journal of Electronics & Information Technology,2012,34(9):2273-2276(in Chinese) 高光普,刘文芬.关于旋转对称布尔函数线性结构的几点注记[J].电子与信息学报,2012,34(9):2273-2276
[6] Qu L J,Fu S J,Li C.Recent Progress in Properties of Cryptographic Functions[J].Journal of Cryptologic Research,2014,1(6):578-588(in Chinese) 屈龙江,付绍静,李超.密码函数安全性指标的研究进展[J].密码学报,2014,1(6):578-588
[7] Courtois N,Meier W.Algebraic attacks on stream ciphers with linear feedback[M]∥Advances in Cryptology－EUROCRYPT 2003.Warsaw,Poland,2003:345-359
[8] Carlet C,Zeng X Y.Further properties of several classes ofBoolean functions with optimum algebraic immunity[J].Designs,Codes and Cryptography,2009,52(3):303-338
[9] Xiong X W,Wei A G,Zhang Z J.Construction of Rotation Symmetric Boolean Functions with Good Cryptographic Properties[J].Journal of Electronics & Information Technology,2012,34(10):2358-2362(in Chinese) 熊晓雯,魏爱国,张智军.构造具有良好密码学性质的旋转对称布尔函数[J].电子与信息学报,2012,34(10):2358-2362
[10] Chen Y D,Zhang Y N,Tian W.Construction of Even-variable Rotation Symmetric Boolean Functions with Optimal Algebraic Immunity[J].Journal of Cryptologic Research,2014,1(5):437-448(in Chinese) 陈银冬,张亚楠,田威.具有最优代数免疫度的偶数元旋转对称布尔函数的构造[J].密码学报,2014,1(5):437-448
[11] Li Y,Yang M,Kan H B.Constructing and counting Boolean functions on even variables with maximum algebraic immunity[J].IEICE Transactions on Fundamentals,2010,93-A(3):640-643
[12] Rizomiliotis P.On the resistance of Boolean functions against algebraic attacks using univariate polynomial representation[J].IEEE Transactions on Information Theory,2010,56(8):4014-4024
[13] Wu B F,Lin D D.Constructing Boolean Functions with Good Cryptographic Properties by Concatenation[J].Journal of Cryptologic Research,2014,1(1):64-71(in Chinese) 吴保峰,林东岱.具有良好密码学性质的布尔函数的级联构造[J].密码学报,2014,1(1):64-71
[14] Wang Q,Peng J,Kan H,et al.Constructions of cryptographically significant Boolean functions using primitive polynomials[J].IEEE Transactions on Information Theory,2010,56(6):3048-3053
[15] Zhou Q F,Li X X,Qian H F.Construction of almost perfect algebraic immune resilient functions on even variables[J].Computer Engineering,2014,40(12):74-77(in Chinese) 周祁丰,李祥学,钱海峰.具有几乎完美代数免疫的偶数元弹性函数构造[J].计算机工程,2014,40(12):74-77
[16] Carlet C.On the higher order nonlinearities of algebraic immune functions[M]∥Advances in Cryptology－CRYPTO 2006.Springer Berlin Heidelberg,2006:584-601
[17] Carlet C,Feng K.An infinite class of balanced vectorial Boolean functions with optimum algebraic immunity and good nonlinearity[M]∥Coding and Cryptology.Springer Berlin Heidelberg,2009:1-11
[18] Feng K,Yang J.Vectorial Boolean functions with good cryptographic properties[J].International Journal of Foundations of Computer Science,2011,22(6):1271-1282
[19] Dong D,Qu L,Fu S,et al.New constructions of vectorial Boo-lean functions with good cryptographic properties[J].International Journal of Foundations of Computer Science,2012,23(3):749-760
[20] Lou Y,Han H,Tang C,et al.Constructing vectorial Booleanfunctions with high algebraic immunity based on group decomposition[J].International Journal of Computer Mathematics,2014,92(3):451-462
[21] 温巧燕,钮心忻,杨义先.现代密码学中的布尔函数[M].北京:科学出版社,2000
[22] Li C L,Zhang H G,Zeng X Y,et al.The lower bound on the second-order nonlinearity for a class of Bent functions[J].Chinese Journal of Computers,2012,35(8):1588-1593(in Chinese) 李春雷,张焕国,曾祥勇,等.一类Bent函数的二阶非线性度下界[J].计算机学报,2012,35(8):1588-1593
[23] Sun G H,Wu C K.On the nonlinearity,algebraic degree and algebraic immunity of some symmetric Boolean functions[J].Chinese Journal of Computers,2014,37(11):2247-2255(in Chinese) 孙光洪,武传坤.几类对称布尔函数的非线性度、代数次数和代数免疫阶[J].计算机学报,2014,37(11):2247-2255
[24] Zhou Y.Characterization of a Balanced Boolean Function with the Minimum of the Sum-of-squares Indicator[J].Journal of Cryptologic Research,2015,2(1):17-26(in Chinese) 周宇.具有最小平方和指标的平衡布尔函数性质刻画[J].密码学报,2015,2(1):17-26
[25] Li W,Wang Z,Huang J.The e-derivative of boolean functions and its application in the fault detection and cryptographic system[J].Kybernetes,2011,40(5/6):905-911
[26] Huang J L,Wang Z.The relationship between correlation im-mune and weight of H Boolean function[J].Journal on Communications,2012,33(2):110-118(in Chinese) 黄景廉,王卓.H布尔函数的相关免疫性与重量的关系[J].通信学报,2012,33(2):110-118
[27] Zhao M L.Method of detecting special logic function based on Boolean e-derivative[J].Journal of Zhejiang University (Science Edition),2014,41(4):424-426(in Chinese) 赵美玲.基于布尔e导数的特殊逻辑函数检测方法[J].浙江大学学报(理学版),2014,41(4):424-426

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