计算机应用 ›› 2015, Vol. 35 ›› Issue (9): 2542-2545.DOI: 10.11772/j.issn.1001-9081.2015.09.2542

• 信息安全 • 上一篇    下一篇

大线性复杂度三值自相关的二元三阶分圆序列的构造

李胜华1, 赵晗诺1,2, 罗炼飞1   

  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

LI Shenghua1, ZHAO Hannuo1,2, LUO Lianfei1   

  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

摘要: 对于一类周期为素数p,p≡1(mod 3)的二元三阶分圆序列提出了一种构造方法,确保其少自相关值及大线性复杂度。利用分圆的知识计算其自相关值,并进一步考虑序列的自相关值为三值时,素数p应满足的条件。此时p应满足p=a2+12,a为整数。当p满足此形式时,序列的线性复杂度为p-1,否则为2(p-1)/3。通过计算机实验,找出了满足所给形式的p,并能生成对应的序列集,验证了序列的自相关性及线性复杂度。新序列的线性复杂度和已有的三元三阶分圆序列的相同;和二元偶数阶分圆序列的相比,大部分相同或较优(已有的有些情况为(p-1)/2、(p+1)/2或1+(p-1)/6)。所提出的构造方法可推广至其他少自相关值、大线性复杂度的奇数阶分圆序列集的构造上。大奇数阶分圆序列的平衡性也会提高,能被较好地应用于密码与通信系统中。

关键词: 伪随机序列, 分圆序列, 分圆数, 自相关值, 极小多项式, 线性复杂度

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.

Key words: pseudorandom sequence, cyclotomic sequence, cyclotomic number, autocorrelation, minimal polynomial, linear complexity

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