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### 大线性复杂度三值自相关的二元三阶分圆序列的构造

1. 1. 湖北大学 数学与统计学学院, 武汉 430062;
2. 陕西国际商贸学院 商学院, 陕西 咸阳 712046
• 收稿日期:2015-04-24 修回日期:2015-06-10 出版日期:2015-09-10 发布日期:2015-09-17
• 通讯作者: 李胜华(1972-),女,湖北麻城人,副教授,博士,主要研究方向:序列设计、密码学,lishmag2014@163.com
• 作者简介:赵晗诺(1988-),女,陕西西安人,硕士,主要研究方向:序列设计;罗炼飞(1991-),男,湖北武汉人,硕士研究生,主要研究方向:密码编码理论、序列设计。
• 基金资助:
湖北省教育厅中青年项目(Q20101004);应用数学湖北省重点实验室开放基金资助项目(O24017)。

### Construction of binary three-order cyclotomic sequences with 3-valued autocorrelation and large linear complexity

1. 1. Faculty of Mathematics and Statistics, Hubei University, Wuhan Hubei 430062, China;
2. School of Business, Shaanxi Institute of International Trade and Commerce, Xianyang Shaanxi 712046, China
• Received:2015-04-24 Revised:2015-06-10 Online:2015-09-10 Published:2015-09-17

Abstract: In order to obtain the sequences with a few autocorrelation values and large linear complexity, a new class of binary cyclotomic sequences of order 3 with period p were constructed, where p is a prime and p≡1(mod 3). The autocorrelation was computed based on cyclotomy, and the condition for p that assures the 3-valued autocorrelation was discussed. The condition is that p should be the form p=a2+12 for an integer a. The linear complexity is p-1 if p is the form, or 2(p-1)/3 otherwise. By computer experiments, all ps' satisfying the form were found, the corresponding sequences were given, and the autocorrelation and linear complexity were confirmed. The linear complexity was the same as that of the known ternary cyclotomic sequence of order 3. Compared with the related known binary cyclotomic sequences of even order, the linear complexity was the same or better in most cases. The method in this paper can be extended to construct other cyclotomic sequences of odd order with a few autocorrelation values and large linear complexity. Since the cyclotomic sequences of larger odd order also have better balance, they can be applied to stream ciphers and communication systems.