Advances in Applied Mathematics
Vol.3 No.02(2014), Article ID:13492,7 pages DOI:10.12677/AAM.2014.32010

T-D猜想上多输出布尔函数构造

Yiran Chen, Meng Zhou

Key Laboratory of Mathematics, Information and Behaviour of the Ministry of Education, School of Mathematics and System Science, Beihang University, Beijing

Email: 13352225@qq.com, zm1613@sina.com

Copyright © 2014 by authors and Hans Publishers Inc.

Received: Feb. 18th, 2014; revised: Mar. 20th, 2014; accepted: Mar. 28th, 2014

ABSTRACT

An improvement has been made on the construction method of Boolean Functions and the relevant conclusions of combinatorial conjecture proposed by Ziran Tu. We generalized their results and extended to the vectorial case. A class of bent Boolean functions F with the maximum algebraic immunity is presented by a more general construction method. Then by modifying F, we get new vectorial balanced functions with optimum algebraic degree, good nonlinearity and good algebraic immunity even maximum algebraic immunity for some cases.

Keywords:Vectorial Boolean Functions, Algebraic Immunity, Bent Function, Balancedness, Nonlinearity

T-D猜想上多输出布尔函数构造

1. 引言

2. 布尔函数基本知识

2.1. 单输出布尔函数

2.2. 多输出(向量)布尔函数

3. 一些组合猜想

[7] 中涂自然和邓映蒲说明了虽然至今无法精确证明此猜想，但已经可以证明当时猜想成立。D.Tang等人在文献[8] 又给出了一个类似的新组合猜想：

，则

，则

4. 最优代数免疫度的多输出布尔函数

4.1. 代数免疫度

4.2. 一类具有最优代数免疫度的Bent函数

，有

1)，即对任意

(因为)

2)，即对一些

5. 具有最优代数免疫度的平衡布尔函数

(是证明定理1中构造的平衡函数)。

6. 总结与展望

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