一个组合猜想及其相关问题的研究

项目名称： 一个组合猜想及其相关问题的研究

项目编号： No.11426072

项目类型： 专项基金项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 邓贵新

作者单位： 广西师范学院

项目金额： 3万元

中文摘要： 我们记正整数t的权w(t)为它的2进展开式中1的个数。设正整数t,k满足t小于2的k次幂。涂自然和邓映蒲在2009年提出了下面这个组合猜想：令X={(a,b): a,b是非负整数且小于2的k次幂，a+b与t模2^k-1同余，且w(a)+w(b)

中文关键词： 2进展开；权；布尔函数；；

英文摘要： Let w(t) denote the number of 1 in the 2-ary expansion of a positive integer t. In 2009, Ziran Tu and Yingpu Deng proposed the following combinatorial conjecture: Let X={(a,b): a and b are nonnegative integers smaller than 2^k, a+b is congruence to t modulo 2^k-1, and w(a)+w(b)<k}. Then |X| is smaller or equal to k-1 powers of 2. Under the assumption that the conjecture is true, they obtained two classes of Boolean functions which are both algebraic immunity optimal and have high nonlinearity. Since Boolean functions play an important role in cryptography, this conjecture was generalized to more general form soon and more classes of Boolean functions were constructed under the assumption that the general form is true. There are only a few cases of the conjecture which have been proved so far. This project will investigate the original conjecture by using probability and combinatorial methods.

英文关键词： 2-ary expansion；weight；Boolean functions；；