Xiao, Zeng, Li and Helleseth proposed new generalized cyclotomic binary sequences \(s^{\infty }\) of period \(p^m\) and showed that these sequences are almost balanced and have very large linear complexity if p is a non-Wieferich prime and \(m=2\). Wu, Xu, Chen and Ke determined the values of the k-error linear complexity for \(m=2\) in terms of the theory of Fermat quotients and the results indicated that sequences \(s^{\infty }\) have good stability. Edemskiy, Li, Zeng and Helleseth studied the linear complexity of \(s^{\infty }\) for general integers \(m\ge 2\). Furthermore, Ouyang and Xie constructed new \(2p^{m}\)-periodic binary sequences \({\widehat{s}}^{\infty }\) and \({\widetilde{s}}^{\infty }\) and proved that the sequences \({\widehat{s}}^{\infty }\) and \({\widetilde{s}}^{\infty }\) are of high linear complexity when \(m\ge 2\). In this paper we shall show that despite a high linear complexity the sequences \(s^{\infty }\), \({\widehat{s}}^{\infty }\) and \({\widetilde{s}}^{\infty }\) have some undesirable features which may not suggest them for cryptography. The properties of multiplicative character sums modulo \(p^m\) play an important role in the proof of this paper.

Xiao，Zeng，Li和Helleseth提出了周期为（p ^ m \）的新的广义环原子二元序列\（s ^ {\ infty} \），并证明如果p为a ，这些序列几乎是平衡的并且具有非常大的线性复杂度非Wieferich素数和\（m = 2 \）。Wu，Xu，Chen和Ke根据Fermat商的理论确定了\（m = 2 \）的k误差线性复杂度的值，结果表明序列\（s ^ {\ infty} \具有稳定性好。Edemskiy，Li，Zeng和Helleseth研究了一般整数\（m \ ge 2 \）的\（s ^ {\ infty} \）的线性复杂度。此外，欧阳和谢构造了新的\（2p ^ {m} \）周期二进制序列\（{\ widehat {s}} ^ {\ infty} \）和\（{\ widetilde {s}} ^ {\ infty } \）并证明\（{\ widehat {s}} ^ {\ infty} \）和\（{\ widetilde {s}} ^ {\ infty} \）的序列在\（m \ ge 2 \）。在本文中，我们将显示，尽管线性复杂度很高，但序列\（s ^ {\ infty} \），\（{\ widehat {s}} ^ {\ infty} \）和\（{\ widetilde {s} } ^ {\ infty} \）具有某些不良功能，可能无法建议使用它们进行加密。乘法字符和的性质取模\（p ^ m \） 在本文的证明中起着重要作用。