Abstract

In this paper, we present septenary Gordon–Mills–Welch sequences (GMWSs) with a period of N = 2400 that are formed in finite GF[(7^m)]ⁿ = GF(7^S) fields. Checking polynomials h_GMWS(x) are obtained in the form of a product of both primitive and irreducible polynomials h_сi(x) with a degree of S = 4. The formation of GMWSs by summing sequences with polynomials h_сi(x) is shown to require knowledge of the symbols of the M-sequence (MS) with polynomial h_MS(x) and decimation indices determined by the exponents of the roots of polynomials h_сi(x). It is determined that, compared to the binary case, septenary summable sequences can have an initial shift that is a multiple of N/(p – 1) = 400. It is shown that for each of the 160 primitive polynomials of degree S = 4 in the GF(7⁴) field, it is possible to form seven GMWSs with equivalent linear complexity l_s from 12 to 84. Compared to septenary MSs, the maximal gain in structural secrecy is 21 times.

中文翻译：

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数字信息传输系统的 Gordon-Mills-Welch 序列的形成

摘要

在本文中，我们提出了在有限GF [(7 ^m )] ⁿ = GF (7 ^S ) 场中形成的周期为N = 2400的七阶 Gordon-Mills-Welch 序列 (GMWS) 。检查多项式h _GMWS ( x ) 以S = 4 的原始多项式和不可约多项式h _{с i} ( x )的乘积的形式获得。通过对多项式h _{с i} ( x的序列求和来形成 GMWSs) 表明需要了解具有多项式h _MS ( x ) 的 M 序列 (MS) 的符号，以及由多项式h _{с i} ( x )的根的指数确定的抽取指数。已确定，与二进制情况相比，七进制可求和序列的初始移位是N /( p – 1) = 400 的倍数。结果表明，对于 160 个S = 4的原始多项式中的每一个在GF (7 ⁴ ) 域中，可以形成七个具有等效线性复杂度l _{s 的GMWS}从 12 到 84。与 septenary MS 相比，结构保密性的最大增益是 21 倍。