布尔函数的Nega-Hadamard变换研究
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摘要
利用Nega-Hadamard变换的性质,研究了bent函数与其对偶函数Nega-Hadamard变换的关系,得出bent函数为negabent函数,其对偶函数也是negabent函数的结果;同时,得到了布尔函数在某个仿射子空间上NegaHadamard变换的性质;研究了布尔函数的导数与它的二进制移位关于Nega-Hadamard变换的关系。
By using the properties of Nega-Hadamard transform,this paper studied the relationship between bent function and its dual in the Nega-Hadamard transform. It showed that if bent function was a negabent,then its dual was again negabent.Also,it obtained the property of the Nega-Hadamard transform of Boolean function f in some affine subspace. At last,it provided the relationship between the Nega-Hadamard transforms of the derivatives of f and the dyadic shifts.
By using the properties of Nega-Hadamard transform,this paper studied the relationship between bent function and its dual in the Nega-Hadamard transform. It showed that if bent function was a negabent,then its dual was again negabent.Also,it obtained the property of the Nega-Hadamard transform of Boolean function f in some affine subspace. At last,it provided the relationship between the Nega-Hadamard transforms of the derivatives of f and the dyadic shifts.
引文
[1]Zhang Weiguo,Pasalic E.Constructions of resilient S-boxes with strictly almost optimal nonlinearity through disjiont linear codes[J].IEEE Trans on Information Theory,2014,60(3):1638-1651.
[2]Tang Deng,Zhang Weiguo,Tang Xiaohu.Construction of balanced Boolean functions with high nonlinearity and good autocorrelation properties[J].Designs Codes and Cryptography,2013,67(1):77-91.
[3]Rothaus O S.On“bent”functions[J].Journal of Combinatorial Theory Series A,1976,20(3):300-305.
[4]Khoo K,Gong Guang,Stinson D R.A new characterization of semibent and bent functions on finite fields[J].Designs,Codes,Cryptography,2006,38(2):279-295.
[5]Charpin P,Pasalic E,Tavernier C.On bent and semi-bent quadratic Boolean functions[J].IEEE Trans on Information Theory,2005,51(12):4286-4298.
[6]Carlet C.Two new classes of bent functions[C]//Advance in Cryptology-EUROCRYPT.Berlin:Spinger-Verlag,1994:77-101.
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[8]Parker M G,Pott A.On Boolean functions which are bent and negabent[C]//Proc of International Workshop on Sequences,Subsequences,and Consequences.Berlin:Spinger-Verlag,2007:9-23.
[9]Schmidt K U,Parker M G,Pott A.Negabent functions in the MaioranaMc Farland class[C]//Proc of the 5th International Conference on Sequences and Their Applications.Berlin:Spinger-Verlag,2008:390-402.
[10]Sarkar S.On the symmetric negabent Boolean functions[C]//Progress in Cryptology.Berlin:Spinger-Verlag,2009:136-143.
[11]Gangopadhyay S,Chaturvedi A.A new class of bent-negabent Boolean functions[EB/OL].http://eprint.iacr.org/2010/597.pdf.
[12]Stanica P,Gangopadhyay S,Chaturvedi A,et al.Nega-Hadamard transform,bent and negabent functions[C]//Proc of the 6th International Conference on Sequences and Their Applications.Berlin:Spinger-Verlag,2010:359-372.
[13]任传伦,刘凤梅,杨义先,等.关于negabent函数的若干讨论[J].通信学报,2011,32(8):179-182.
[14]Sarkar S.Characterizing negabent Boolean functions over finite fields[C]//Proc of the 7th International Conference on Sequences and Their Applications.Berlin:Spinger-Verlag,2012:77-88.
[15]卓泽朋,崇金凤,魏仕民.Nega-Hadamard变换和negabent函数[J].山东大学学报:理学版,2013,48(7):29-32.
[2]Tang Deng,Zhang Weiguo,Tang Xiaohu.Construction of balanced Boolean functions with high nonlinearity and good autocorrelation properties[J].Designs Codes and Cryptography,2013,67(1):77-91.
[3]Rothaus O S.On“bent”functions[J].Journal of Combinatorial Theory Series A,1976,20(3):300-305.
[4]Khoo K,Gong Guang,Stinson D R.A new characterization of semibent and bent functions on finite fields[J].Designs,Codes,Cryptography,2006,38(2):279-295.
[5]Charpin P,Pasalic E,Tavernier C.On bent and semi-bent quadratic Boolean functions[J].IEEE Trans on Information Theory,2005,51(12):4286-4298.
[6]Carlet C.Two new classes of bent functions[C]//Advance in Cryptology-EUROCRYPT.Berlin:Spinger-Verlag,1994:77-101.
[7]卓泽朋,崇金凤,肖国镇,等.Bent函数的对偶性[J].武汉大学学报:理学版,2012,58(1):86-88.
[8]Parker M G,Pott A.On Boolean functions which are bent and negabent[C]//Proc of International Workshop on Sequences,Subsequences,and Consequences.Berlin:Spinger-Verlag,2007:9-23.
[9]Schmidt K U,Parker M G,Pott A.Negabent functions in the MaioranaMc Farland class[C]//Proc of the 5th International Conference on Sequences and Their Applications.Berlin:Spinger-Verlag,2008:390-402.
[10]Sarkar S.On the symmetric negabent Boolean functions[C]//Progress in Cryptology.Berlin:Spinger-Verlag,2009:136-143.
[11]Gangopadhyay S,Chaturvedi A.A new class of bent-negabent Boolean functions[EB/OL].http://eprint.iacr.org/2010/597.pdf.
[12]Stanica P,Gangopadhyay S,Chaturvedi A,et al.Nega-Hadamard transform,bent and negabent functions[C]//Proc of the 6th International Conference on Sequences and Their Applications.Berlin:Spinger-Verlag,2010:359-372.
[13]任传伦,刘凤梅,杨义先,等.关于negabent函数的若干讨论[J].通信学报,2011,32(8):179-182.
[14]Sarkar S.Characterizing negabent Boolean functions over finite fields[C]//Proc of the 7th International Conference on Sequences and Their Applications.Berlin:Spinger-Verlag,2012:77-88.
[15]卓泽朋,崇金凤,魏仕民.Nega-Hadamard变换和negabent函数[J].山东大学学报:理学版,2013,48(7):29-32.