一类旋转对称bent函数的构造
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摘要
考虑到已知的旋转对称bent函数不多且其代数次数较低,在变元数n=2 m为偶数的情况下,对已知置换和旋转对称序列进行线性仿射变换和级联求和,得到一个特殊的Maiorana-McFarland类函数。可以证明,新函数是旋转对称bent函数,其代数次数任意且可达m。新函数的密码学特性优良,可用于流密码非线性部件的算法设计。
Considering that there are only a few rotation symmetric bent functions,and that they all have lower algebraic degrees,a corresponding linear affine transformation and concatenation summation are carried out for the known permutation and the rotation symmetric sequence in the case that the number of variables nis even(n=2 m),and a special Maiorana-McFarland class function is obtained.It can be proved that the new is rotation symmetric bent function,whose algebraic degree is arbitrary and can reach m.This new function has excellent cryptographic properties and can be used in the algorithm design of nonlinear components of stream ciphers.
Considering that there are only a few rotation symmetric bent functions,and that they all have lower algebraic degrees,a corresponding linear affine transformation and concatenation summation are carried out for the known permutation and the rotation symmetric sequence in the case that the number of variables nis even(n=2 m),and a special Maiorana-McFarland class function is obtained.It can be proved that the new is rotation symmetric bent function,whose algebraic degree is arbitrary and can reach m.This new function has excellent cryptographic properties and can be used in the algorithm design of nonlinear components of stream ciphers.
引文
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[11]SU S H,TANG X H.Systematic constructions of rotation symmetric bent functions,2-rotation symmetric bent functions,and bent idempotent functions[J/OL].IEEE Transactions on Information Theory,2017,63(7):4658-4667[2017-10-02].https://doi.org/10.1109/tit.2016.2621751.
[12]TANG C M,QI Y F,ZHOU Z C,et al.Two infinite classes of rotation symmetric bent functions with simple representation[C/OL]//Applicable Algebra in Engineering Communication&Computing,AAECC(2017).Berlin:Springer,2017:1-12[2017-10-02].https://doi.org/10.1007/s00200-017-0337-8.
[13]CESMELIOGLU A,MEIDL W,POTT A.Generalized Maiorana-McFarland class and normality of p-ary bent functions[J/OL].Finite Fields and Their Applications,2013,24(C):105-117[2017-10-02].https://doi.org/10.1016/j.ffa.2013.06.001.
[2]CARLET C.Boolean Functions for Cryptography and Error Correcting Codes[M/OL]//CRAMA Y,HAMMER P.Boolean Models&Methods in Mathematics.Cambridge:Cambridge University Press,2007:1-118[2017-10-02].https://doi.org/10.1017/CBO9780511780448.
[3]GAO G,CUSICK T W,LIU W.Families of rotation symmetric functions with useful cryptographic properties[J/OL].IET Information Security,2014,8(6):297-302[2017-10-02].http://dx.doi.org/10.1049/iet-ifs.2013.0241.
[4]GAO G,ZHANG X,LIU W F,et al.Constructions of Quadratic and Cubic Rotation Symmetric Bent Functions[J/OL].IEEE Transactions on Information Theory,2012,58(7):4908-4913[2017-10-02].https://doi.org/10.1109/tit.2012.2193377.
[5]PIEPRZYK J,QU C X.Rotation-symmetric functions and fast hashing[C/OL]//BOYD C,DAWSON E.Information Security and Privacy.ACISP 1998.Lecture Notes in Computer Science,vol 1438.Berlin:Springer,1998:20-31[2017-10-02].https://doi.org/10.1007/bfb0053731.
[6]KAVUT S.Results on rotation symmetric Sboxes[J/OL].Information Sciences,2012,201(19):93-113[2017-10-02].https://doi.org/10.1016/j.ins.2012.02.030.
[7]ROTHAUS O S.On“bent”functions[J/OL].Journal of Combinatorial Theory,1976,20(3):300-305[2017-10-02].https://doi.org/10.1016/0097-3165(76)90024-8.
[8]FILIOL E,FONTAINE C.Highly nonlinear balanced Boolean functions with a good correlation-immunity[C/OL]//NYBERG K.Advances in Cryptology—EUROCRYPT’98.EUROCRYPT 1998.Lecture Notes in Computer Science,vol 1403.Berlin:Springer,1998:475-488[2017-10-02].https://doi.org/10.1007/bfb0054147.
[9]STNIC P,MAITRA S.Rotation symmetric Boolean functions-Count and cryptographic properties[J/OL].Discrete Applied Mathematics,2003,156(10):1567-1580[2017-10-02].https://doi.org/10.1016/j.dam.2007.04.029.
[10]高光普,程庆丰,王磊.三次旋转对称Bent函数的构造[J/OL].密码学报,2015,2(4):372-380[2017-10-02].http://www.cnki.net/kcms/doi/10.13868/j.cnki.jcr.000086.html.
[11]SU S H,TANG X H.Systematic constructions of rotation symmetric bent functions,2-rotation symmetric bent functions,and bent idempotent functions[J/OL].IEEE Transactions on Information Theory,2017,63(7):4658-4667[2017-10-02].https://doi.org/10.1109/tit.2016.2621751.
[12]TANG C M,QI Y F,ZHOU Z C,et al.Two infinite classes of rotation symmetric bent functions with simple representation[C/OL]//Applicable Algebra in Engineering Communication&Computing,AAECC(2017).Berlin:Springer,2017:1-12[2017-10-02].https://doi.org/10.1007/s00200-017-0337-8.
[13]CESMELIOGLU A,MEIDL W,POTT A.Generalized Maiorana-McFarland class and normality of p-ary bent functions[J/OL].Finite Fields and Their Applications,2013,24(C):105-117[2017-10-02].https://doi.org/10.1016/j.ffa.2013.06.001.