几类高非线性度密码函数的构造

doi:10.19665/j.issn1001-2400.20230416

Abstract

Abstract:

Boolean functions have important applications in cryptography.Bent functions have been a hot research topic in symmetric cryptography as Boolean functions have maximum nonlinearity.From the perspective of spectrum,bent functions have a flat spectrum under the Walsh-Hadamard transform.Negabent functions are a class of generalized bent functions,which have a uniform spectrum under the nega-Hadamard transform.A generalized negabent function is a function with a uniform spectrum under the generalized nega-Hadamard transform.Bent functions has been extensively studied since its introduction in 1976.However,there are few research on negabent functions and generalized negabent functions.In this paper,the properties of generalized negabent functions and generalized bent-negabent functions are analyzed.Several classes of generalized negabent functions,generalized bent-negabent functions,and generalized semibent-negabent functions are constructed.First,by analyzing a link between the nega-crosscorrelation of generalized Boolean function and the generalized nega-Hadamard transformation,a criterion for generalized negabent functions is presented.Based on this criterion,a class of generalized negabent functions is constructed.Secondly,two classes of generalized negabent functions of the form f(x)=c₁f₁(x⁽¹⁾)+c₂f₂(x⁽²⁾)+…+c_rf_r(x⁽^r⁾) are constructed by using the direct sum construction.Finally,generalized bent-negabent functions and generalized semibent-negabent functions over Z₈ are obtained by using the direct sum construction.Some new methods for constructing generalized negabent functions are given in this paper,which will enrich the results of negabent functions.

Key words: Boolean function, generalized negabent function, generalized bent function, nega-Hadamard, bent-negabent function

CLC Number:

TN918.1

LIU Huan, WU Gaofei. Several classes of cryptographic Boolean functions with high nonlinearity[J].Journal of Xidian University, 2023, 50(6): 237-250.

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References 34

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	构造方法^[23]	条件
f为Z₈上的广义	f(x,y)=4g(x)+(4h(x)+4g(x))y	当g,h为negabent函数
negabent函数	f(x,y)=4g(x)+(4h(x)+2g(x))y	当g,h$\oplus$k为negabent函数且$\mathcal{N}$_h(u)= $\mathcal{N}$_g_$\oplus$_h(u)
f为Z₁₆上的广义	f(x,y)=8g(x)+(8h(x)+8k(x)+8g(x))y	当g,h$\oplus$k为negabent函数
negabent函数	f(x,y)=8g(x)+(8h(x)+8k(x)+4g(x))y	当g,g$\oplus$k,g$\oplus$h$\oplus$k 为negabent函数, 且$\mathcal{N}$_h_$\oplus$_k(u)= $\mathcal{N}$_g_$\oplus$_h_$\oplus$_k(u)
	f(x,y,z)=8g(x)+(8h(x)+4)y+(8k(x)+4)z+ (8l(x)+4)(1+y)(1+z)-4yz	当g$\oplus$l,g$\oplus$k,g$\oplus$h,g$\oplus$h$\oplus$k 为 negabent函数,且$\mathcal{N}$_g_$\oplus$_k(u)= $\mathcal{N}$_g_$\oplus$_h_$\oplus$_k(u)

Several classes of cryptographic Boolean functions with high nonlinearity

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