| bent函数和半bent函数的二阶非线性度下界 |
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| 引用本文： | 李雪莲, 胡予濮, 高军涛. bent函数和半bent函数的二阶非线性度下界[J]. 电子与信息学报, 2010, 32(10): 2521-2525. doi: 10.3724/SP.J.1146.2010.00191 |
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| 作者姓名： | 李雪莲 胡予濮 高军涛 |
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| 作者单位： | 西安电子科技大学应用数学系,西安,710071;西安电子科技大学计算机网络与信息安全教育部重点实验室,西安,710071 |
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| 基金项目： | 国家973计划项目，国家自然科学基金项目，广西信息与通讯技术重点实验室基金(20902)资助课题 |
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| 摘 要： | 该文研究了形如f(x,y)的n+1变元bent函数和半bent函数的二阶非线性度，其中xGF(2n), yGF(2)。首先给出了f(x,y)的2n-1个导数非线性度的精确值；然后推导出了函数f(x,y)的其余2n个导数的非线性度紧下界。进而给出了f(x,y)的二阶非线性度的紧下界。通过比较可知所得下界要优于现有的一般结论。结果表明f(x,y)具有较高的二阶非线性度，可以抵抗二次函数逼近和仿射逼近攻击。
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| 关 键 词： | 密码学 布尔函数 Walsh变换 非线性度 |
| 收稿时间： | 2010-03-04 |
| 修稿时间： | 2010-04-30 |
The Lower Bounds on the Second Order Nonlinearity of Bent Functions and Semi-bent Functions |
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| Li Xue-Lian, Hu Yu-Pu, Gao Jun-Tao. The Lower Bounds on the Second Order Nonlinearity of Bent Functions and Semi-bent Functions[J]. Journal of Electronics & Information Technology, 2010, 32(10): 2521-2525. doi: 10.3724/SP.J.1146.2010.00191 |
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| Authors: | Li Xue-lian Hu Yu-pu Gao Jun-tao |
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| Affiliation: | (Department of Applied Mathematics, Xidian University, Xi’ an 710071, China)
(Key Laboratory of Computer networks &; Information Security, Xidian University, Xi’ an 710071, China) |
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| Abstract: | This paper studies the lower bounds on the second order nonlinearity of bent functions and semi-bent functions f(x,y) with n+1 variables, where x∈GF(2n), y∈GF(2). Firstly, the values of the nonlinearity of the 2n-1 derivatives of the Boolean function f(x,y) are given. Then, the tight lower bounds on the other 2n derivatives of f(x,y) are deduced. Furthermore, the tight lower bounds on the second order nonlinearity of f(x,y) are presented. The derived bounds are better than the existing general ones. The results show that these functions f(x,y) have higher second order nonlinearity, and can resist the quardratic and affine approximation attacks. |
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| Keywords: | Cryptography Boolean functions Walsh transforms Nonlinearity |
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