Fermat-Euler quotients arose from the study of the first case of Fermat's Last Theorem, and have numerous applications in number theory. Recently they were studied from the cryptographic aspects by constructing many pseudorandom binary sequences, whose linear complexities and trace representations were calculated. In this work, we further study their correlation measures by introducing a new approach based on Dirichlet characters, Ramanujan sums and Gauss sums. Our results show that the $4$-order correlation measures of these sequences are very large. Therefore they may not be suggested for cryptography.

In this paper, we study the $k$-error linear complexity of binary sequences with periods $p^n$, which are derived from new generalized cyclotomic classes modulo a power of an odd prime $p$. We establish a recursive relation and then estimate the $k$-error linear complexity of the binary sequences with periods $p^n$, the results extend the case $p^2$ that has been studied in an earlier work of Wu et al. at 2019. Our results show that the $k$-error linear complexity of these sequences does not decrease dramatically for $k<(p^n-p^{n-1})/2$.