Based on the properties of trace functions and quadratic forms, this paper presents value distributions of Walsh spectrum of the Plateaued functions of the form Tr（R（x）） with n=3r or 4r variables, where r 〉 1 is ...Based on the properties of trace functions and quadratic forms, this paper presents value distributions of Walsh spectrum of the Plateaued functions of the form Tr（R（x）） with n=3r or 4r variables, where r 〉 1 is an odd integer. Our results can be used to determine the numbers of non-zero Walsh spectrum values and the nonlinearities of these functions, and estimate their resiliency orders. Especially, the value distributions can be used to deduce the tight lower bounds of the second order nonlinearity of two classes of Boolean functions. It is demonstrated that our bounds are better than the previously obtained bounds.展开更多
This paper proposes a general method to construct 1-resilient Boolean functions by modifying the Tu-Deng and Tang-Carlet-Tang functions. Cryptographic properties such as algebraic degree, nonlinearity and algebraic im...This paper proposes a general method to construct 1-resilient Boolean functions by modifying the Tu-Deng and Tang-Carlet-Tang functions. Cryptographic properties such as algebraic degree, nonlinearity and algebraic immunity are also considered. A sufficient condition of the modified func- tions with optimal algebraic degree in terms of the Siegenthaler bound is proposed. The authors obtain a lower bound on the nonlinearity of the Tang-Carlet-Tang functions, which is slightly better than the known result. If the authors do not break the ＂continuity＂ of the support and zero sets, the functions constructed in this paper have suboptimal algebraic immunity. Finally, four specific classes of 1-resilient Boolean functions constructed from this construction and with the mentioned good cryptographic properties are proposed. Experimental results show that there are many 1-resilient Boolean functions have higher nonlinearities than known l-resilient functions modified by Tu-Deng and Tang- Carlet-Tang functions.展开更多
基金Acknowledgments This work was supported in part by 973 Project of China （No. 2007CB311201）, the Notional Natural Science Foundation（No. 60833008, 60803149）, and the Foundation of Guangxi Key Laboratory of Information and Communication（No. 20902）.
文摘Based on the properties of trace functions and quadratic forms, this paper presents value distributions of Walsh spectrum of the Plateaued functions of the form Tr（R（x）） with n=3r or 4r variables, where r 〉 1 is an odd integer. Our results can be used to determine the numbers of non-zero Walsh spectrum values and the nonlinearities of these functions, and estimate their resiliency orders. Especially, the value distributions can be used to deduce the tight lower bounds of the second order nonlinearity of two classes of Boolean functions. It is demonstrated that our bounds are better than the previously obtained bounds.
基金supported by the National Key Basic Research Program of China under Grant No.2013CB834203the National Natural Science Foundation of China under Grant Nos.61472417 and 61472120the Research Council of Norway
文摘This paper proposes a general method to construct 1-resilient Boolean functions by modifying the Tu-Deng and Tang-Carlet-Tang functions. Cryptographic properties such as algebraic degree, nonlinearity and algebraic immunity are also considered. A sufficient condition of the modified func- tions with optimal algebraic degree in terms of the Siegenthaler bound is proposed. The authors obtain a lower bound on the nonlinearity of the Tang-Carlet-Tang functions, which is slightly better than the known result. If the authors do not break the ＂continuity＂ of the support and zero sets, the functions constructed in this paper have suboptimal algebraic immunity. Finally, four specific classes of 1-resilient Boolean functions constructed from this construction and with the mentioned good cryptographic properties are proposed. Experimental results show that there are many 1-resilient Boolean functions have higher nonlinearities than known l-resilient functions modified by Tu-Deng and Tang- Carlet-Tang functions.
